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2024 Senior Theses - Graduated with Distinction

Angikar ghosal.

Representation Theoretic Formulation of Quantum Error Correcting Codes Advisor: Robert Calderbank

Benjamin Goldstein

Soap-Film-Like Surfaces of Revolution Advisor: Demetre Kazaras

Noah Harris

Black Hole Thermodynamics, Large N Gauge Theories, and Deriving the AdS/CFT Correspondence Advisor: Paul Aspinwall

Long-Time Behavior of Some ODEs with Partial Damping Advisor: Kyle Liss

Aram Lindroth

Towards a Functional Equation for the $\mathbb{A}^1$-Logarithmic Zeta Function Advisor: Kirsten Wickelgren

Emmanuel Mokel

Monitoring Nonstationary Variance to Assess Convergence of Markov Chain Monte Carlo Advisor: Jonathan Mattingly

Nathan Nguyen

Towards Solving Variational Graphon Problem for Random Hypergraphs Advisor: Nicholas Cook

Nathanael Ong

On the Betti Numbers of Rank 2 Compact Locally Symmetric Spaces Advisor: Mark Stern

Jean-Luc Rabideau

Random Restrictions in the p-Biased Measure Advisor: Henry Pfister

Riki Shimizu

Unveil Sleep Spindles with Concentration of Frequency and Time (ConceFT) Advisor: Hau-Tieng Wu

December 2023

Quantum State Tomography via Tensor Ring Representation Advisor: Jianfeng Lu

Jesse Zhang

Answer Filtration with Filtration: Toward a Theory of Lifetime Filtration for Multiparameter Persistence Modules Advisor: Ezra Miller

Alex Burgin

The Schrodinger Maximal Function and Generalizations Advisor: Lillian Pierce

Nick Chakraborty

Improve Accuracy and Speed of Manifold Reconstruction and De-Noising from Scattered Data in R 2 Advisor: Hongkai Zhao

Jeffrey Cheng

Mixing in Measure Preserving Dynamical Systems Advisor: Tarek Elgindi

Carson Dudley

A Mathematical Model of a Peritoneal Staphylococcus Aureus infection Advisor: Anna Nelson

Riley Fisher

Pattern Formation in Evolving Domains Advisor: Tom Witelski

Multitaper Wave-Shape F-Test For Detecting Non-Sinusoidal Oscillations Advisor: Hau-Tieng Wu

Diffusing on multiple fibers Advisor: Ingrid Daubechies and Shira Faigenbaum

December 2022

Symmetric Formulas for Products of Permutations Advisor: Benjamin Rossman

A homotopic variant of policy gradients for the linear quadratic regulator problem Advisor: Andrea Agazzi

Nathan Geist

Homological algebra of modules over real polyhedral groups Advisor: Ezra Miller

Braden Hoagland

Percolation Processes on Dynamically Grown Graphs Advisor: Rick Durrett

Daniel Hwang

Analyzing the bistability of the minimally bistable ERK network using the discriminant locus Advisor: Maggie Regan

Wallace Peaslee

Dolbeault Cohomology of Non-Compact Metric Graphs Advisor: Joseph Rabinoff

Mathematical Modeling of TIE1 and Endothelial Metabolism Advisor: Michael Reed

December 2021

Some Mathematical Problems in Quantum Computing and Quantum Information Advisor: Robert Calderbank

Anuk Dayaprema

Solitons for the closed G2 Laplacian flow in the cohomogeneity-one setting Advisor: Mark Haskins

Ziyang Ding

At the Intersection of Deep Sequential and State-space Model Framework Advisor: Sayan Mukherjee

Lucas Fagan

Schur Polynomials and Crystal Graphs Advisor: Spencer Leslie

Resolving Simpson’s Paradox in NC Public School Grading System Advisor: Greg Herschlag

Phoebe Klett

Implementing non-canonical Sylvan Resolutions Advisor: Ezra Miller

Jianyou Wang

Deep Reinforcement Adaptive Computational Processor Advisor: Vahid Tarokh

Alex Damian

Theoretical Guarantees for Signal Recovery Advisor: Hau-tieng Wu

Blythe Davis

The Spherical Manifold Realization Problem Advisor: Faramarz Vafaee

Onkar Gujral

Khovanov Homology and Knot Concordance dvisor: Adam Levine

Xiayimei Han

Hodge Representations of Calabi-Yau 3 Folds Advisor:  Colleen Robles

Remy Kassem

Symmetry Detection of Unknown Volumes from Projected Variations Advisor: Xiuyuan Cheng

Joey Li

Algebraic Data Structures for Decomposing Multipersistence  Modules Advisor: Ezra Miller 

Evaluating Bayesian Convolutional Neural Networks in the Clinic Advisor: Paul Bendich

Jonathan Michala

Uniqueness of Ranked Pairs Advisor: Hubert Bray 

Benjamin Nativi

An Analogue of Gauss Composition for Binary Cubic Forms Advisor: Aaron Pollack

Computing Values of Symmetric Square L-Functions using Ichino's Pullback Formula Advisor: Aaron Pollack

Junmo Ryang

Embedding Lagrangian Surfaces Advisor: Robert Bryant

Irina Cristali

Poisson Percolation on the Square Lattice Advisors: Rick Durrett, Matthew Junge

Creating Musical Rubato Using Deep Learning Advisor: Ezra Miller

Zhenhua Liu

Stationary One-Sided Area Minimizing Hypersurfaces with Isolated Singularities Advisors: William Allard, Hubert Bray, Robert Bryant

Xueying Wang

Unfolding High-Dimensional Convex Polyhedra Advisor: Ezra Miller

Claire Wiebe

Analyzing the Effects of Partisan Correlation on Election Outcomes using Order Statistics Advisor: Jonathan Mattingly

Gaitling Zhou

Elliptic Curves over Dedekind Domains Advisor: William Pardon

(you can search for archived versions of these theses here )

• Surabhi Beriwal  Statistical analysis of fruit fly wing vein topology  (2018) [with E. Miller]
• Trung Can  The Heisenberg-Weyl Group, Finite Symplectic Geometry, and their applications   (2018) [with R. Calderbank]
• Feng Gui  On Calibrations for Area Minimizing Cones  (2018) with [H. Bray]
• Neel Kurupassery   Cryptographic Primitives in Artin Groups of Type I k (m)    (2018)  [with M. Abel]
• Eric Peshkin  T he quantification of markers of economic development from time-series satellite imagery using deep learning   (2018) with [with P. Bendich and D. Thomas]
• Weiyao Wang   Understanding Operator Reed-Muller Codes Through the Weyl Transform   (2018) [with R. Calderbank]
• Alexander Pieloch  The Topology of Moduli Spaces of Real Algebraic Curves  (2017) [with R. Hain]
• Samadwara Reddy  The Vietoris–Rips Complexes of Finite Subsets of an Ellipse of Small Eccentricity  (2017) [with H. Adams]
• Lindsey Brown  An Application of Abstract Algebra to the Neural Code for Sound Localization in Barn Owls  (2016) [with M. Reed]
• David Builes  The Large Cardinal Hierarchy  (2016) [with R. Hodel]
• Kyle Casey  Siegel Modular Forms  (2016) [with L. Saper]
• Bryan Runjing Liu  Modeling the Effects of Positive and Negative Feedback in Kidney Blood Flow Control  (2016) [with A. Layton]
• Francois Thelot A Maximum Entropy Based Approach for the Description of the Conformational Ensemble of Calmodulin from Paramagnetic NMR (2016) [with M. Maggioni and B. Donald]
• Will Victor  Efficient algorithms for Traffic Data Analysis  (2016)[computer science with P. Agarwal]
• Paul Ziquan Yang  Morphisms with Only Mild Singular Fibers and Bertini Theorems over Finite Fields  (2016) [with C. Schoen]
• Rex Zhitao Ying  Approximation Algorithms of Dynamic Time Warping and Edit Distance  (2016) [computer science with P. Agarwal]
• Roger Zou  Deformable Graph Model for Trackng Epithelial Cell Sheets in Florescence Microscopy  (2016)[computer science with C. Tomasi]
• Anne Talkington  Modeling the Dynamics of Cancerous Cells in vivo  (2015) [with R. Durrett]
• Rowena Gan  Geometry of Impressionist Music  (2015) [with E. Miller]
• David Hemminger  Augmentation Rank of Satellites with Braid Pattern  (2015) [with L. Ng and C. Cornwell]
• Mandy Jiang  Dynamic random network model for human papilloma virus transmission  (2015) [with M. Ryser]
• Hunter Nisonoff  Efficient Partition Function Estimatation in Computational Protein Design  (2015) [with M. Maggioni]
• Eugene Rabinovich  The Conformal Manifold in N=(2,2) SCFTs    (2015)  [physics  with R. Plesser]
• Marshall Ratliff  Introducing the Cover tree to Music Information Retrieval  (2015) [with P. Bendich]
• Brett Schnobrich  Heisenberg-Weyl Group, Subspace Packings, and Image Processing  (2015) [with R. Calderbank]
• Christy Vaughn  Stochastic Study of Gerrymandering  (2015) [with J. Mattingly]
• Aashiq Dheeraj  A Stochastic Spatial Model for Tumor Growth  (2014) [with R. Durrett]
• Joshua Izzard  Rank p 2  Representations of Semisimple Lie Algebras  (2014) [with J. Getz]
• Kathleen Lan  Coalescing random walks on n-block Markov chains  (2014) [with K. McGoff]
• Leslie Lei Lei  Infinite Swapping Simulated Tempering  (2014) [with J. Lu]
• Julia Ni  A convex approach to tree-based wavelet compression  (2014) [with A. Thompson]
• Jiarou Ivy Shen  Merge times and hitting times of time-inhomogeneous Markov chains  (2014) [with D. Sivakoff]
• Daniel Stern  Low-Order Lagrangians Depending on a Metric and a Matter Field of Arbitrary Rank  (2014) [with H. Bray]
• Daniel Vitek  Knot Contact Homology and the Augmentation Polynomial  (2014) [with C. Cornwell]
• Alexander Wertheim  Complex Multiplication on Elliptic Curves  (2014) [with L. Saper]
• Luxi Wei  Modeling Credit Risk using Rating and Environmental Factors  (2014) [with R. Durrett]
• Timothy Chang  On the existence of a simple winning strategy in the T(4.3) knot game  (2013) [with D. Herzog]
• Conrad de Peuter  Modeling basketball games as alternating renewal-reward processes and predicting match outcomes  (2013) [with R. Durrett]
• Bryan Jacobson  A practical approximation of persistent local homology  (2013) [with P. Bendich]
• Kara Karpman  Simulating mucociliary transport using the method of regularized Stokelets  (2013) [with A. Layton]
• Carmen Lopez  Modeling the folate pathway in Escherichia coli  (2013) [with A. Layton]
• James Mallernee  Strategy and honesty based comparison of preferential ballot voting methods  (2013) [with H. Bray]
• William Zhang  Evolutionary dynamics in host pathogen model  (2013) [with R. Durrett]
• Ben Bellis  Investigation of a Local Computation of the Signature from the Triangulation of a Manifold  (2012) [with M. Stern]
• Adrian Chan  Pricing financial derivatives with multi-task machine learning and mixed effects method  (2012) [with J. Bouvrie]
• Kyu Won Choi  Relative contributions of common jumps in realized correlations  (2012) [with A. Petters]
• Veronica Ciocanel  Analysis of the nonlinear dynamics of the forced planar string pendulum  (2012) [with T. Witelski]
• Kaveh Danesh  A branching process model of ovarian cancer  (2012) [with R. Durrett]
• Theo Frehlinghuysen  Carbon sequestration via forest management techniques  (2012) [with D. Kraines]
• Yingyi Shen  A study of edge toric ideals using associated graphs  (2012) [with S. Mapes]
• Daniel Thielman  Complex-balanced steady state of chemical reaction networks that contain an Eulerian cycle  (2012) [with C. Berkesch]
• Kaitlin Daniels  Noise driven Transitions between stable equilibria in stochastic dynamical systems  (2011) [with A. Athreya]
• Alan Guo  Lattice point methods for combinatorial games  (2011) [with E. Miller]
• Nils Hultgren  Centrality and network analysis: A perturbative approach to dynamical importance  (2011) [with I. Matic]
• Hans Kist  Estimating carbon sequestration potential in the boreal forests  (2011) [with D. Kraines]
• Misha Lavrov  Invariants in Legendrian links in the solid torus  (2011) [with D. Rutherford]
• Philip Pham  Tubuloglomerular feedback signal transduction in the loops of Henle  (2011) [with A. Layton]
• Thames Sae Sue  A simple cardiac model exhibiting stationary discordant alternans  (2011) [with D. Schaeffer]
• Max Tabachnik  An analysis of preferential ballot voting methods  (2011) [with H. Bray]
• Bo Waggoner  A model of the foot and ankle in running  (2011) [with E. Bouzarth]
• Wutichai Chongchitmate  Classification of Legendrian knots and links  (2010) [with L. Ng]
• Jason D. Lee  Multiscale analysis of dynamic graphs  (2010) [with M. Maggioni]
• Jeremy Semko  Statistical analysis simulations of coarsening droplets coating a hydrophobic surface  (2010) [with T. Witelski]
• Amy Wen  Model of feedback-mediated dynamics of coupled nephrons with compliant thick ascending limbs  (2010) [with A. Layton]
• Jason Ferguson  Factorization of Primes in Biquadratic Extensions of Q  (2009) [with C. Schoen]
• Jared Haftel  A Closer Look at ADC multivariate GARCH  (2009) [with M. Huber]
• Mark Hallen  Improving accuracy and scope of quantitative FRAP analysis  (2009) [with A. Layton]
• Andy Ng  Retinoid Transport in the Vision cycle  (2009) [with J. Mercer]
• Aaron Pollack  Relations between special derivations arising from modular forms  (2009) [with R. Hain]
• Jesse Thorner  Simplicial homology and DeRham’s theorem  (2009) [with W. Allard]
• Barry Wright III  Objective measures of preferential ballot voting systems  (2009) [with H. Bray]
• Michael Bauer  Existence and stability of patterns arising from square wave forcing of the damped Mathieu equation  (2008) [with A. Catlla]
• Tirasan Khandhawit  On Legandrean and transverse knots  (2008) [with L. Ng]
• Aalok Shah  An overview of fast marching and optimal control methods for trajectory optimization  (2008) [with W. Allard]
• Charles Staats III  Application of discrete geometry to the construction of Laurent-rational zeros  (2008) [with S. Sharif]
• Elliott Wolf  Computational pathways to Godel’s first incompletness theorem  (2008) [with R. Hodel]
• Lingen Zhang  The motion of sets of vortices  (2008) [with T. Witelski]
• Morgan Brown  An algorithm for tracking persistence pairing of a discrete homotopy of Morse functions on S 2   (2007) [with J.Harer]
• Brandon Levin  Class field theory and the problem of representing primes by binary quadratic forms  (2007) [with L. Saper]
• Stepan Paul  Lines and conics relative to degenerating divisors in CP 2   (2007) [with J. Davis]
• James Zou  3-D reconstruction and topological analysis of root architecture  (2007) [with J. Harer]
• Pradeep Baliga  Dynamic cellular automata model of toll plaza traffic flows  (2006) [with W. G. Mitchener]
• Adam Chandler  Dynamic cellular automata model of toll plaza traffic flows  (2006) [with W. G. Mitchener]
• Matthew Fischer  Mapping model of cardiac-membrane dynamics  (2006) [with D. Schaeffer]
• Qinzheng Tian  Simulation of Newtonian fluid flow between rotating cylinders  (2006) [with T. Witelski]
• Yee Lok Wong  Models of instant runoff voting  (2006) [with J. Mattingly]
• Oaz Nir  Mechanical arms and algebraic topology  (2005) [with J.Harer]
• Mayank Varia  Explicit calculation of the L invariant for Kummer surfaces  (2005) [with J. Hanke]
• David Arthur  On the higher Hasse-Witt matrices and related in variants  (2004) [with W. Pardon]
• Suzy Borgschulte  A mathematical approach to the panting of dogs  (2004) [with M. Reed]
• Lauren M. Childs  Scaling population dynamics from the macroscopic to the microscopic  (2004) [with T. Kepler]
• Ryan Letchworth  Wavelet methods for numerical solutions of differential equations  (2004) [with S. Roudenko]
• David Marks  Coadjoint orbits and geometric quantization  (2004) [with M.R. Plesser]
• Lori Peacock  Distributions of the small eigenvalues of Wishart matrices  (2004) [with B. Rider]
• Lindsay C. Piechnik  Smooth reflexive 4-polytopes have quadratic triangulations  (2004) [with C. Haase]
• Matthew Toups  A solution to the D0-D4 system of equations  (2004) [with M. Stern]
• Jenna VanLiere  Mathematically modelling the growth and diversification of T-cell populations  (2004) [with T. Kepler]
• Matthew J. Atwood  Evaluating singular and nearly singular integrals numerically  (2003) [with J.T. Beale]
• Marie Guerraty  Controlling alternans in a cardiac map model  (2003) [with M. Romeo]
• Meredith C. Houlton  Classification of critical curves and preliminary analysis of caustics  (2003) [with A. Petters]
• Steven R. Nicklas  Envy and satisfaction in the public goods game  (2003) [with D. Kraines]
• Dane R. Voris  A numerical approach to the M t /M t /N t  queue with abandonment  (2003) [with B. Rider]
• Melanie Wood  Invariants and relations of the action of the absolute Galois group on dessins d’enfants and the algebraic fundamental group of the punctured sphere  (2003) [with R. Hain]
• Thomas W. Finley  Efficient Myrinet routing  (2002) [with W. Allard]
• Samuel W. Malone  Alternative Price Processes for Black-Scholes: Empirical Evidence and Theory  (2002) [with A. Petters]
• Carl Miller  Exponential Iterated Integrals and the Solvable Completion of Fundamental Groups  (2001) [with R. Hain]
• Daniel Neill  Optimality under Noise: Higher Memory Strategies for the Alternating Prisoner’s Dilemma  (2001) Computer Science [with D. Kraines]
• Luis Von Ahn  Models of the language of set theory and Zermelo Frankel axioms  (2000) [with R. Hodel]
• Christopher Beasley  Superconformal theories from Branes at Singularities  (1999) Physics [with R. Plesser]
• Alexander Brodie  Measurable Cardinals  (1999) [with R. Hodel]
• Jeffrey DiLisi  The Biology and Mathematics of the Hypothalamic-Pituitary-Testicular Axis  (1999) [with M. Reed]
• Garrett Mitchener  Lattices and Sphere Packing  (1999) [with R. Hain]
• Andrew O. Dittmer  Generalized Formulas for Circular Polygons  (1998) [with R. Hain]
• Richard R. Schneck  Set Theory and Cardinal Arithmetic  (1997) [with R. Hodel]
• Tung T. Tran  Counting Independent Subsets in Nearly Regular Graphs  (1997) [with G. Lawler]
• Paul A. Dreyer  Knot theory and the human pretzel game  (1995) [with J. Harer]
• Paul J. Koss  Effects of noise on the iterated prisoner’s dilemma  (1995) [with D. Kraines]
• Jeff Vanderkam  Eigenfunctions of an acoustic system  (1994) [with T. Beale]
• Linie Chang  Mathematics and immunology: Modeling antigen and antibody interactions  (1993) [with M. Reed]
• Sang H. Chin  Action of the Torelli group on the 3-fold cover of G-hole torus  (1993) [with R. Hain]
• Jennifer Slimowitz  Transitions of gaps between the integers N satisfying N q < j (1993) [with M. Reed]
• David Jones  Primality testing, factoring and continued fractions  (1992) [with C. Schoen]
• Will Schneeberger  The axiom diamond  (1992) [with J. Shoenfield]
• Jeanne Nielsen  Triply periodic minimal surfaces in  R 3  (1991) [with R. Bryant
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Undergraduate Honors & Senior Thesis

Students excelling in their major coursework and interested in pure math should consider Departmental Honors. Departmental Honors means you will graduate “With Distinction” as opposed to College Honors which is “With Honors”.  The most important component of graduating with Departmental Honors is researching and writing a Senior Thesis.

Requirements for Departmental Honors:

• Must complete a B.S. Mathematics Degree.
• Must satisfactorily complete at least one three-quarter sequence 402-3-4, 424-5-6, or 441-2-3; or two two-quarter sequences from this list. Exceptions must be approved by the chair of the Departmental Honors Committee.
• Must earn a GPA of 3.5 or better in Math coursework completed at the UW.
• Must write a senior thesis (earn a numerical grade for MATH 496).
• Must have a 3.3 minimum cumulative GPA at UW.

Please note: If you are not interested in the College Honors or Departmental Honors in Mathematics, you may still write a Senior Thesis. The process is the same as above, but it does not need to be approved by the Honors Committee.

Research credit (Math 498) may be available with faculty permission.

Beginning of your final year at the UW : think about a thesis topic and seek out a faculty supervisor.  Read below for more details about selecting a topic.

First week of classes the quarter before you expect to graduate: submit a thesis proposal form to the Dept. Honors Committee.  The form is online here: Math Dept. Honors Thesis Proposal Form

Last day of your final quarter: Once your advisor approves the thesis, email it to [email protected] and cc your faculty advisor.  You may also wish to upload it to the University Libraries archive .

Nature of the thesis

The senior thesis shall be an expository account of a topic in pure or applied mathematics related to the student’s area of interest. (Original results or proofs are welcome but are definitely not expected.) The thesis must contain some nontrivial mathematical arguments. (E.g., a non-technical essay on “fractals in nature” would not be acceptable.) The thesis should normally be about 20 to 30 pages in length (double spaced, Times New Roman 12pt font, 1” margins). These figures are guidelines, not rigid requirements. The topic should be something that cannot simply be read out of a standard textbook. Writing the thesis should involve:

• obtaining material from the periodical literature, or
• consulting several books and synthesizing material from them, or
• reading an account of a topic in a book that is substantially more advanced than the student’s regular coursework, digesting it, and putting it into readable form.

Choosing a topic

Finding a topic is the students’ responsibility, although consultation with faculty members is encouraged. The topic must be approved by a faculty member of the Mathematics Department who will supervise the work (the “supervisor”) and by the chair of the Departmental Honors Committee. A Senior Thesis Topic Proposal form can be found at the link above, and should be filled out by the student with the supervisor's support (the Dept. will check in with your supervisor). The topic proposal must be submitted to the chair of the Departmental Honors Committee no later than the end of the first week of classes the quarter preceding the quarter in which the student expects to graduate. Exceptions to this deadline may be granted only by the chair of the Departmental Honors Committee. Students contemplating writing a thesis are strongly encouraged to start thinking about a topic in the autumn quarter of their senior year.

Writing the thesis

The student must register for Math 496 (Honors Senior Thesis) during the last quarter of thesis work. The student may receive three credit hours of W-course credit for writing the thesis. Normally, the students will register for a reading course (Math 498) with the supervisor during the preceding quarter (s). The student will receive three hours of credit for each of these courses, but in exceptional cases, with the approval of the supervisor, the number or credit hours may be increased. The supervisor may allow the student to replace Math 498 with a suitable topics course; however, it is still expected that the student will meet periodically with the supervisor.

There is no specific required thesis template for an undergraduate thesis.  Some students may choose to use a modified version of the graduate thesis templates, but this is not required.

Approval of thesis

The student shall submit a draft of the thesis to the supervisor for comments and criticisms, and then shall submit a final version with appropriate revisions. The supervisor shall read the thesis and certify its acceptability with respect to both content and exposition. In order to ensure sufficient time for these things, the student must submit the first draft no later than three weeks before the last day classes of the quarter in which the student expects to graduate, and the final draft no later than the last day of classes. Exceptions to these deadlines may be granted only by the chair of the Departmental Honors Committee.

Once the thesis has been approved by your faculty supervisor, you will need to email the document to [email protected] (required) as well as submit it to the ResearchWorks archive , part of the University Libraries (optional but strongly recommended).  Submission to the archive will allow your thesis to be included in the dissemination and preservation of scholarly work.  Your thesis will be made publicly available.

Interdisciplinary theses

Theses which are concerned with the application of some part of mathematics to some others field are acceptable, as long as they contain some substantial mathematics. In exceptional cases the student may wish to work most closely with a faculty member in another department in preparing the thesis. However, in such cases the thesis topic and the thesis itself must still be approved by a member of the Mathematics Department.

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Guidelines for Senior Theses in the Mathematics Department

A senior thesis can form a valuable part of a student's experience in the Mathematics Major . It is intended to allow students to cover significant areas of mathematics not covered in course work, or not covered there in sufficient depth. The work should be independent and creative. It can involve the solution of a serious mathematics problem, or it can be an expository work, or variants of these. Both the process of doing independent research and mathematics exposition, as well as the finished written product and optional oral presentation, can have a lasting positive impact on a student's educational and professional future.

Recent Senior Theses

Supervision

Supervision by a qualified member of the field of mathematics at Cornell is the normal requirement for a senior thesis. Other arrangements are possible, however, provided they are made with the assistance of the student's major advisor, and with the approval of the Mathematics Major Committee .

Finding a supervisor/Encouraging students. It should be emphasized that both the writing and the supervising of a senior thesis are optional activities, both for students and faculty. Students interested in doing this will need to find a suitable supervisor — perhaps with the aid of their major advisor or another faculty member whom they know. Advisors and other faculty who encounter students whom they think would benefit from this activity are invited to mention this option to them and assist them in finding a supervisor.

Standard venues for senior theses. One obvious way in which a senior thesis can be produced is through an independent research course (MATH 4900); another way is through an REU experience, either at Cornell or elsewhere. (If the REU work was accomplished or initiated elsewhere, a "local expert" will still be needed to supervise or "vouch for" the work as a senior thesis.) In yet a third way, a student may present a faculty member with a solution or partial solution to an interesting problem. In such cases, this could form the core of a senior thesis. Faculty are invited to encourage such work from their students.

Public Lecture

A public lecture in which the results of the senior thesis are presented is welcome but optional. This should be arranged by the thesis supervisor in conjunction with the undergraduate coordinator and adequately advertised. Department faculty and graduate students are encouraged to attend these presentations.

Submitting the Completed Thesis

The supervisor must approve the student's thesis. No later than April 15th, the student must submit a completed thesis to the thesis supervisor. If the supervisor asks the student to make changes, the student has until April 30th to do so. By April 30th, the student must give the supervisor two paper copies and an electronic copy of the thesis in final form. The electronic copy will be posted on the department's web site. [Students who expect to graduate in January must submit a completed thesis by November 15th and the final form by November 30th.]

Form of the Thesis

Ideally, the final document should be TeXed or prepared in some equivalent technical document preparation system. The document must have large left margins (one and one-half inches or slightly larger). The title page should contain:

• The student's name and graduating class.
• The title of the senior thesis.
• The name of the faculty supervisor. (If there is more than one supervisor, list both. If one of the supervisors is not in the Mathematics Department, list the department and institution.)
• The date of completion of the thesis.

This information will be used to produce a standard frontispiece page, which will be added to the document in its library copies.

Merit of the Senior Thesis

Judgment as to the merit of a senior thesis will be based largely on the recommendation of the faculty member supervising the thesis. The Mathematics Major Committee will use this recommendation both in its determination of honors and in its decision on whether to place the thesis in our permanent library collection.

Honors Consideration

The senior thesis will automatically be considered by the Mathematics Major Committee as one of the ingredients for deciding on an honors designation for the student. Students may receive honors without a thesis and are not guaranteed honors with one. However, an excellent senior thesis combined with an otherwise excellent record can elevate the level of honors awarded.

Mathematics Library Collection of Senior Theses

Meritorious senior theses will be catalogued, bound, and stored in the Mathematics Library.

Department of Mathematics - UC Santa Barbara

Senior Thesis Options

To enter the honors program in mathematics, a student must have completed 120 units of coursework with an overall grade-point average of at least 3.5 and at least 24 upper-division mathematics and statistics units with a grade-point average of at least 3.5 (excluding Mathematics 100A-B, 193, 195A-B, and PSTAT 133A-B-C and 193).

To complete the honors program, the student must maintain a grade-point average of at least 3.5 in all upper-division and graduate mathematics and statistics courses (excluding Mathematics 100A-B, 193, 195A-B, and PSTAT 133A-B-C and 193) and as well as complete one of the following: (a) a senior thesis, Math 197A-B; (b) a two-quarter graduate sequence; or (c) together with an advisor, submit a Distinction in the Major proposal for an interdisciplinary program of three mathematically oriented courses outside the math department to the undergraduate committee for its approval. Option C does not apply to economics/mathematics or financial mathematics majors.

Distinction in the Major for each option will be awarded at graduation pending final approval by the Department of Mathematics Undergraduate Committee. Written projects will be submitted to the committee, and grades will be evaluated for coursework options. To apply for the Honors Program in Mathematics, please email the Mathematics Undergraduate Advisors at  [email protected] .

• Senior Thesis

A thesis is a more ambitious undertaking than a project. Most thesis writers within Applied Mathematics spend two semesters on their thesis work, beginning in the fall of senior year.  Students typically enroll in Applied Mathematics 91r or 99r (or Economics 985, if appropriate) during each semester of their senior year.  AM 99r is graded on a satisfactory/unsatisfactory basis.  Some concentrators will have completed their programs of study before beginning a thesis; in situations where this is necessary, students may take AM 91r for letter-graded credit, for inclusion in Breadth section (v) of the plan of study.  In the spring semester, the thesis itself may serve as the substantial paper on which the letter grade is based.  Econ 985 is also letter-graded, and may be included in the Breadth section of the plan of study in place of AM 91r.

Another, somewhat uncommon option, is that a project that meets the honors modeling requirement (either through Applied Mathematics 115 or 91r) can be extended to a thesis with about one semester of work.  Obviously the more time that is spent on the thesis, the more substantial the outcome, but students are encouraged to write a thesis in whatever time they have. It is an invaluable academic experience.

The thesis should make substantive use of mathematical, statistical or computational modeling,  though the level of sophistication will vary as appropriate to the particular problem context.  It is expected that conscientious attention will be paid to the explanatory power of mathematical modeling of the phenomena under study, going beyond data analysis to work to elucidate questions of mechanism and causation rather than mere correlation. Models should be designed to yield both understanding and testable predictions. A thesis with a suitable modeling component will automatically satisfy the English honors modeling requirement; however a thesis won't satisfy modeling Breadth section (v) unless the student also takes AM 91r or Econ 985.

Economics 985 thesis seminars are reserved for students who are writing on an economics topic. These seminars are full courses for letter-graded credit which involve additional activities beyond preparation of a thesis. They are open to Applied Mathematics concentrators with suitable background and interests.

Students wishing to enroll in AM 99r or 91r should follow the application instructions on my.harvard.

Thesis Timeline

The timeline below is for students graduating in May. The thesis deadline for May 2024 graduates is Monday, April 1 at 2:00PM. For off-cycle students, a similar timeline applies, offset by one semester. The thesis due date for March 2025 graduates is Friday, November 22, 2024. Late theses are not accepted.

Mid to late August:

Students often find a thesis supervisor by this time, and work with their supervisor to identify a thesis problem. Students may enroll in Econ 985 (strongly recommended when relevant), AM 91r, or AM 99r to block out space in their schedule for the thesis.

Early December:

All fourth year concentrators are contacted by the Office of Academic Programs. Those planning to submit a senior thesis are requested to supply certain information. This is the first formal interaction with the concentration about the thesis.

Mid-January:

A tentative thesis title approved by the thesis supervisor is required by the concentration.

Early February:

The student should provide the name and contact information for a recommended second reader, together with assurance that this individual has agreed to serve. Thesis readers are expected to be teaching faculty members of the Faculty of Arts and Sciences or SEAS. Exceptions to this requirement must be first approved by the Directors, Associate Director, or Assistant Director of Undergraduate Studies. For AM/Economics students writing a thesis on a mathematical economics topic for the March thesis deadline, the second reader will be chosen by the Economics Department. For AM/Economics students writing for the November deadline, the student should recommend the second reader.

On the thesis due date:

Thesis due at 2pm. Late theses are not accepted. Electronic copies in PDF format should be delivered by the student to the two readers and to [email protected] (which will forward to the Directors of Undergraduate Studies, Associate and Assistant Director of Undergraduate Studies) on or before that date and time. An electronic copy should also be submitted via the SEAS  online submission tool  on or before that date. SEAS will keep this electronic copy as a non-circulating backup and will use it to print a physical copy of the thesis to be deposited in the Harvard University Archives. During this online submission process, the student will also have the option to make the electronic copy publicly available via DASH, Harvard’s open-access repository for scholarly work.

Contemporaneously, the two readers will receive a rating sheet to be returned to the Office of Academic Programs before the beginning of the Reading Period, together with their copy of the thesis and any remarks to be transmitted to the student.

The Office of Academic Programs will send readers' comments to the student in late May, after the degree meeting to decide honors recommendations.

Thesis Readers

The thesis is evaluated by two readers, whose roles are further delineated below.  The first reader is the thesis adviser.  The second and reader is recommended by the student and adviser, who should secure the agreement of the individual concerned to serve in this capacity.  The reader must be approved by the Directors, Associate Director, or Assistant Director of Undergraduate Studies.  The second reader is normally are teaching members of the Faculty of Arts and Sciences, but other faculty members or comparable professionals will usually be approved, after being apprised of the responsibilities they are assuming.   For theses in mathematical economics, the choice of the second reader is made in cooperation with the Economics department.  The student and thesis adviser will be notified of the designated second reader by mid-March.

The roles of the thesis adviser and of the outside reader are somewhat different.  Ideally, the adviser is a collaborator and the outside reader is an informed critics.  It is customary for the adviser's report to comment not only on the document itself but also on the background and context of the entire effort, elucidating the overall accomplishments of the student.  The supervisor may choose to comment on a draft of the thesis before the final document is submitted, time permitting.  The outside reader is being asked to evaluate the thesis actually produced, as a prospective scientific contribution — both as to content and presentation.  The reader may choose to discuss their evaluation with the student, after the fact, should that prove to be mutually convenient.

The thesis should contain an informative abstract separate from the body of the thesis.  At the degree meeting, the Committee on Undergraduate Studies in Applied Mathematics will review the thesis, the reports from the two readers and the student’s academic record. The readers (and student) are told to assume that the Committee consists of technical professionals who are not necessarily conversant with the subject matter of the thesis so their reports should reflect this audience.

The length of the thesis should be as long as it needs to be to make the arguments made, but no longer!

Thesis Examples

The most recent thesis examples across all of SEAS can be found on the Harvard DASH (Digital Access to Scholarship at Harvard) repository . Search the FAS Theses and Dissertations collection for "applied mathematics" to find dozens of examples.

Note: Additional samples of old theses can be found in McKay Library. Theses awarded Hoopes' Prizes can be found in Lamont Library.

Recent thesis titles

Theses submitted in 2021, theses submitted in 2020, theses submitted in 2019, theses submitted in 2018 , senior thesis submission information for a.b. programs.

Senior A.B. theses are submitted to SEAS and made accessible via the Harvard University Archives and optionally via  DASH  (Digital Access to Scholarship at Harvard), Harvard's open-access repository for scholarly work.

In addition to submitting to the department and thesis advisors & readers, each SEAS senior thesis writer will use an online submission system to submit an electronic copy of their senior thesis to SEAS; this electronic copy will be kept at SEAS as a non-circulating backup. Please note that the thesis won't be published until close to or after the degree date. During this submission process, the student will also have the option to make the electronic copy publicly available via DASH.  Basic document information (e.g., author name, thesis title, degree date, abstract) will also be collected via the submission system; this document information will be available in  HOLLIS , the Harvard Library catalog, and DASH (though the thesis itself will be available in DASH only if the student opts to allow this). Students can also make code or data for senior thesis work available. They can do this by posting the data to the Harvard  Dataverse  or including the code as a supplementary file in the DASH repository when submitting their thesis in the SEAS online submission system.

Whether or not a student opts to make the thesis available through DASH, SEAS will provide an electronic record copy of the thesis to the Harvard University Archives. The Archives may make this record copy of the thesis accessible to researchers in the Archives reading room via a secure workstation or by providing a paper copy for use only in the reading room.  Per University policy , for a period of five years after the acceptance of a thesis, the Archives will require an author’s written permission before permitting researchers to create or request a copy of any thesis in whole or in part. Students who wish to place additional restrictions on the record copy in the Archives must contact the Archives  directly, independent of the online submission system. 

Students interested in commercializing ideas in their theses may wish to consult Dr. Fawwaz Habbal , Senior Lecturer on Applied Physics, about patent protection. See Harvard's policy for information about ownership of software written as part of academic work.

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Senior thesis information.

Both senior thesis and senior seminar satisfy the Bates W3 writing requirement and highlight mathematical research, writing, presentation, and group collaboration. Senior thesis is a good choice for students wanting to improve all these, with special emphasis on mathematical research on a topic chosen by the student. Senior theses also involve significant amounts of writing, presentations, and check-ins with other math thesis writers.

To ensure that each senior thesis writer has an enriching experience, the math department limits how many theses each faculty member advises, typically to no more than two theses per semester per advisor. To help the department determine senior thesis advisors, each junior math major who would like to write a senior thesis completes a request form by NOON on the last day of Winter Semester classes of the junior year, that is, by 12:00pm (noon) on Friday, April 12, 2024 . Some details:

• The request form seeks background information on the student, the student’s preferences regarding senior thesis, the student’s reasoning behind their preferences, and a description of the proposed senior thesis project. The project description should include enough information to show that the student has given their topic serious thought and that the project is feasible, given the student’s background and given the amount of time the student has to do the research.
• The math department strongly advises juniors to discuss senior thesis topics and ideas with faculty members before writing a request. The request form asks whether you have had such discussions.
• Students should plan to work at least 12 hours per week on thesis, and at least 15 hours per week if pursuing an Honors thesis.
• The math department meets to consider all senior thesis and senior seminar proposals. The department chair typically notifies students of the results of the meeting during Short Term.
• The mathematics department keeps copies of past senior theses in our lounge in Hathorn 209. We encourage prospective senior thesis writers to look through these past theses as part of deciding whether to write a thesis: past theses provide topic ideas, writing structures, and a sense of the scope of a senior thesis.

Types of thesis

• One-semester thesis: A one-semester thesis may be either in the fall (MATH 457) or winter (MATH 458). One-semester theses are due by the Friday of the final examination period of the semester in which the student is writing their thesis.
• Two-semester thesis: Two-semester theses (MATH 457 and MATH 458) not in the Honors Program are due by the last day of classes of the winter semester.
• Honors thesis: Honors theses (MATH 457 and MATH 458) are always two-semester theses and follow the procedures and deadlines of the Honors Program . While all capstone experiences expect students to demonstrate mathematical reading skills and ability to communicate mathematics, a thesis earning Honors in Mathematics is distinguished by an exceptional level of achievement in these areas. Students preferring to write an Honors thesis state this preference at the time of their senior thesis proposal. The Department then decides which students to nominate for the Honors Program, based on the thesis work presented at the end of the first semester.
• Double thesis with another major: A double thesis is a single year-long project that satisfies the thesis requirements of both mathematics and another department, and as such, requires a significant amount of mathematics. A student writing a double thesis signs up for their math thesis in one semester (either MATH 457 or MATH 458) and the other department thesis in the other semester. The math department requires the student to present a talk or poster in the “math semester.” A student who applies thesis course credit to another major may not apply that same credit to the Mathematics Major. The Department will not approve a proposal for a one-semester double thesis.

Completing the thesis

• Students turn in their thesis to their advisor, in a format determined by the advisor, and students give the department chair a final printed copy of the thesis to be placed on permanent display in the mathematics lounge.
• one-semester thesis students present a poster or a talk;
• two-semester non-honors thesis students present a talk in Fall Semester and a poster or a talk in Winter Semester;
• Honors thesis students present a talk in Fall Semester and give their Honors defense during a College-designated Honors defense time period;
• when there is a choice of a poster or a talk, this decision is to be made with the thesis advisor.

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Department of Mathematics

The honors program is a two-semester sequence (Math 99a, "Senior Research" in fall, followed by Math 99b, "Senior Research" in spring) during which senior mathematics majors carry out independent research and the  writing and oral presentation of a senior thesis. Only students who major in the BS in Mathematics or BS in Applied Math may choose the option of writing a thesis in order to be considered for Honors, High Honors or Highest Honors in mathematics. View university resources for undergraduate research support here .

Rules for Senior Honors Thesis

•  A committee of two faculty members, one of whom will be the official instructor, will supervise the work.
• A written thesis proposal must be prepared at the beginning of the first semester, and be approved and signed by both the committee and the Undergraduate Advising Head (UAH), prior to registration for the course. In order to register for the course, the student must requests to take the course in Work Day (see "Request Prerequisite or Permission to Enroll" section under Planning and Registration) .
• A mid-year evaluation must be done by the committee by the end of the first semester, and a written report submitted to the UAH. Students judged to have made insufficient progress will not be permitted to continue in the second semester.
• A thesis in a department-approved format must be submitted to the committee by the last day of classes in the second semester or by a deadline set by the committee members. At a minimum, the thesis must have a title page , a signature page , and an abstract, description of the work performed and the conclusions reached, and references. Students are encouraged to learn the Latex typesetting system ( this repository holds a LaTeX file that serves as a template for senior honors theses in the Brandeis University Department of Mathematics, developed by Chami Lamelas, '22).  We also encourage you to submit your thesis electronically to the library using the following link: Submitting your Thesis to the Library
• The student must defend their thesis in a public oral examination of at least 30 minutes duration by the end of the second semester exam period. The talk should take place in the Math Department and should be accessible to junior math majors. A list of thesis defense talks will be published on the website.
• Written evaluations of the thesis and of the defense must be submitted by the committee to the UAH by the Friday preceding the department degree meeting.
• The level of distinction will be determined by the UAH from evaluation of both the thesis and the student’s academic record.
• Students wishing to graduate in seven semesters must start their thesis research in the spring semester of their junior year, and follow the same rules moved forward one semester.
• The required forms for items (2), (3), and (6) are available on the department website.
• Supervisors outside of the Math Department are acceptable, either in other Brandeis departments or outside of the university. All the rules above apply, including the deadlines. The defense must take place in the Brandeis Math Department. The outside advisor may be an ex officio member of the supervising committee, and advise the committee on the evaluation of the work performed. The grade for Math 99a/b will be determined by the Brandeis instructor of the course. 

Student and Committee Forms for Senior Honors Research

• Student Proposal for Honors Thesis . This form is to be prepared by the student at the beginning of the first semester, and approved and signed by the committee and UAH, prior to registration for the course. You will also need your Instructor to sign an additional form, the  Course Change Form  (see #2) so that you can register.
• In order to register for the course--Course Change Form : This form is to be prepared by the student and signed by both the student and the Instructor. The student then sends the signed form to "[email protected]".

Mid-term Assessment Form . This form is to be prepared by the committee by the end of the first semester, and then submitted by the committee to the UAH.

• Committee Report on Senior Honors Thesis . This form is to be prepared by the committee after the thesis defense and is due to the UAH the Friday before Degree Meetings.

Resources and LATEX Template

View university resources for undergraduate research support.

View funding resources and deadlines for undergraduate research support.

View  how to submit your thesis electronically to Scholar Works. View a template   LaTeX  file that students can choose to use to fill in their information (name, thesis title, advisor etc.),  created by Chami Lamelas '22.

Completed Senior Honors Theses

• Ryan Xie '21: " Mathematically Modeling the Neuron Network Involved in Sleep Regulation " Thesis Advisor: Prof. Jonathan Touboul
• Chami Lamelas, '22  “ Vorticity-Stream Solver for Microfluidic Devices and Applications to Blood Cell Sorters ”   Thesis Advisor: Profs. Thomas Fai and An Huang
• Mathematics Placement

Applied Mathematics

• Undergraduate Program

Each of the Applied Mathematics concentrations allows exceptional students to pursue honors, which involves in-depth project work with faculty.

Outline of Honors Requirements

• Excellence in grades
• Completion of an in-depth, original research project carried out under the guidance of a Brown-affiliated faculty advisor
• Completion of an honors thesis describing this research 
• Completion of two semesters of independent study courses while working on the honors thesis

Deadlines and requirements for honors differ between joint concentrations due to the different needs and scales of each program. Check the details for each concentration below:

• Be in good academic standing by the end of the seventh (or penultimate) semester.
• Earn grades of A or S-with-distinction in at least 70% of the Brown University courses used for concentration credit, excluding calculus and linear algebra, or be in the upper 20% of the student's cohort (as measured by the fraction of grades of A or S-with-distinction among courses used for concentration credit, excluding calculus and linear algebra) by the end of the seventh (or penultimate) semester. (Since S-with-distinctions do not appear on the internal academic record or the official transcript, the department will consult directly with the Registrar’s Office to confirm a student’s grades in concentration courses.)
• Secure a faculty advisor and at least one second reader for the proposed honors thesis project. One of the advisors/readers must be an Applied Mathematics faculty member. 
• Meet regularly, as agreed upon, with their honors thesis advisor and provide regular written drafts on the thesis project.
• Complete an honors thesis that is approved by the faculty advisor and second reader(s) prior to the deadline in the students eighth (or final) semester. Deadlines and guidance about the honors thesis are described below. 
• Email a copy of their approved thesis to Student Affairs Manager Candida Hall ( [email protected] ) in Applied Math for archival purposes prior to the deadline.
• Give an oral presentation of the honors thesis at an approved venue, usually the senior thesis day in Applied Mathematics.
• Complete two semesters of independent study courses while working on the honors thesis, such as APMA 1970/1971 or BIOL 1950/1960 or ECON 1960/1970 or CS 1970. One of these courses can be used to fulfill the senior seminar requirement of the APMA ScB, but they cannot be used to fulfill other concentration requirements. 
• Obtain permission to pursue honors from the department by submitting a completed Honors Declaration Form to our Student Affairs Manager Candida Hall by the deadline (usually at the beginning of the seventh semester). The honors declaration form requires signatures from the thesis advisor, second reader, and concentration advisor, as well as, a brief description of the proposed thesis research. It also requires a preliminary check of eligibility requirements, including the fraction of quality grades.
• Secure a faculty advisor and at least one second reader for the proposed honors thesis project. One of the advisors/readers must be an Applied Mathematics faculty member and one must be a Biomed-affiliated faculty member. 
• Email a copy of their approved thesis to our Student Affairs Manager, Candida Hall ( [email protected] ) in Applied Math for archival purposes prior to the deadline.
• Give an oral presentation of the honors thesis at an approved venue, usually at the senior thesis day in Applied Mathematics or in Biology.
• Complete two semesters of independent study courses while working on the honors thesis, such as APMA 1970/1971 or BIOL 1950/1960. One of these courses can be used to fulfill the research course requirement of the concentration, but they cannot be used to fulfill other concentration requirements. These extra independent study courses are included in the calculation of quality grades described above.
• Obtain permission to pursue honors from the department by submitting a completed Honors Declaration Form to our Student Affairs Manager, Candida Hall ( [email protected] ) by the deadline (usually at the beginning of the seventh semester). The honors declaration form requires signatures from the thesis advisor, second reader, and concentration advisor, as well as, a brief description of the proposed thesis research. It also requires a preliminary check of eligibility requirements, including the fraction of quality grades.
• APMA-CS concentrators can choose to pursue honors within either APMA or CS, but their primary thesis advisor must be in the department that they choose. Students wishing to do honors research with a non-APMA or CS advisor should contact the Directors of Undergraduate Studies in APMA and CS to discuss options.
• Students pursuing honors within APMA should follow the APMA requirements described above.
• Students pursuing honors within CS should follow the CS requirements and deadlines described here: http://cs.brown.edu/degrees/undergrad/concentrating-in-cs/honors/
• Students pursuing honors in CS must also email a copy of their approved thesis to our Student Affairs Manager, Candida Hall ( [email protected] ) in Applied Math for archival purposes.
• APMA-Econ concentrators can choose to pursue honors within either APMA or Econ, but their primary thesis advisor must be in the department that they choose.
• Students pursuing honors within APMA should follow the APMA requirements described above. 
• Students pursuing honors within Econ should follow the Econ requirements and deadlines described here: https://economics.brown.edu/academics/undergraduate/honors-and-capstones/thesis
• Students pursuing honors in Econ must also email a copy of their approved thesis to our Student Affairs Manager, Candida Hall ( [email protected] ) in Applied Math for archival purposes.

APMA Deadlines for Honors Program

Applied Mathematics Honors Declaration Form

Honors Thesis Guidelines:

Mathematical Content :

• Research problem: The thesis should be written on a mathematical problem or on an application that is approached using mathematical techniques. The thesis should demonstrate that the research question is significant and important.
• Thoroughness: The thesis should put the research problem into a broader context, address it in a convincing and thorough manner, and use mathematical approaches that are sound, feasible, and appropriate to the research problem.
• Depth: The thesis should involve mathematics at the level of 1000-level APMA courses and should demonstrate a solid understanding of the mathematics used in the thesis.

Writing Quality :

• Organization: The thesis should have a clear and coherent organization that effectively develops the central idea. There is an introduction that includes a clear statement of the research problem and an outline of the research method. Throughout the paper, arguments are presented clearly and in logical order, and the conclusions are precise and concise. The thesis does not contain awkward or unexpected transitions.
• Clarity: The thesis must be clearly written; in particular, the mathematical content must be clear to the intended audience. It should be clear from the writing that the student has a correct and complete understanding of the mathematical content of the thesis. Assertions are clearly stated and well supported.
• Citations: All sources used in the thesis should be referenced and cited completely and correctly: it should become clear what information from other sources has been integrated into the thesis and where that information came from. The bibliography should also contain an accurate and reasonably complete list of related works and papers.
• Grammar and Orthography: The thesis should be properly formatted and free of errors of grammar, spelling, and punctuation. The tone should be professional.

In the College of Arts and Sciences   .

Course Offerings    

Mathematics is the language of modern science; basic training in the discipline is essential for those who want to understand, as well as for those who want to take part in, the important scientific developments of our time. Acquaintance with mathematics is also extremely useful for students in the social sciences and valuable for anyone interested in the full range of human culture and the ways of knowing the universe in which we live.

The Department of Mathematics faculty has strong and broad groups specializing in algebra, number theory, combinatorics, real and complex analysis, Lie groups, topology and geometry, logic, probability and statistics, mathematical physics, and applied mathematics. Additionally, several other departments at Cornell offer courses which involve a significant amount of advanced mathematical content.  These include computer science, economics, operations research, physics, and statistics. Certain courses in these and other disciplines can be readily integrated into the math major though the various concentrations which are offered.

The department offers a rich variety of undergraduate courses. Additionally, some of the introductory graduate courses are suitable for undergraduates who have completed a rigorous foundation of 4000-level coursework in mathematics. Under some conditions, a student may carry out an independent reading or research project for college credit under the supervision of a faculty member.

Members of the department are available to discuss with students the appropriate course for their levels of ability and interest, and students are urged to avail themselves of this help. Students who want to take any of the courses numbered 3000 or above are invited to confer with the instructor before enrolling.

Website: math.cornell.edu

T. Holm, chair; X. Cao, director of undergraduate studies; I. Peeva, director of graduate studies; M. Aguiar, D. Barbasch, Y. Berest, R. Connelly, K. Delp, B. Dozier, D. Freund, R. Griffiths, D. Halpern-Leistner, T. Healey, J. Hubbard, M. Huntley, S. Jeong, M. Kassabov, A. Knutson, L. Levine, Y. Luo, M. MacDonald, K. Mann, J. Manning, L. Mazurowski, K. Meszaros, M. Mirek, J. Moore, C. Muscalu, A. Nerode, M. Nussbaum, K. Pohland, M. Poór, R. Ramakrishna, R. Ramkumar, T. Riley, L. Saloff-Coste, R. Sjamaar, S. Solecki, P. Sosoe, B. Speh, D. Stern, M.E. Stillman, S. Strogatz, E. Swartz, N. Templier, A. Townsend, A. Vladimirsky, M. Wegkamp, J. West, Y. Yang, I. Zakharevich, X. Zhou, D. Zywina

Advanced Placement and Transfer Credit:

Students who have had some calculus should carefully read “ Advanced Placement   ,” and those who have not taken an advanced placement exam should take a placement test at Cornell during fall or spring orientation.

The linear algebra and multivariable calculus courses that we offer ( MATH 2210   , MATH 2220   , MATH 1920   , and MATH 2940   ) cover considerably more material and in considerably greater depth than that which is covered in high school courses in these subjects. Students who have completed coursework in linear algebra and/or multivariable calculus and have a strong interest in challenging theoretical mathematics may consider enrolling in MATH 2230   – MATH 2240   . College courses may be eligible for transfer credit. Students who have completed a rigorous course in multivariable calculus that is not transferable may take the Engineering Math Advanced Standing Exam. There is no placement test for linear algebra, and it should be noted that 4000-level linear algebra courses are generally not regarded as meeting the prerequisites for the math major or minor.

Visit the Math Department web site for more information on advanced placement , including dates, times, and locations for Cornell math placement exams, and guidelines for transferring credit from another institution .

Course Selection Guidance:

For guidance in selecting an appropriate course, including how to factor advanced placement or transfer credit into that decision, please consult First Steps in Math . New students will have an opportunity to ask questions about math placement during fall orientation at the Arts & Sciences Open House; however, it should be noted that the Cornell placement tests are often held before the open house. Students who are unsure if they need a calculus placement test should ask the director of undergraduate studies for advice in advance of the exam.

Precalculus:

Students who need to take Calculus I ( MATH 1106    or MATH 1110   ) but are lacking the necessary prerequisites may take MATH 1101    to prepare. Courses labeled “college algebra” or “precalculus” at other universities, while not eligible for transfer credit to Cornell, may be used to satisfy the prerequisites for Calculus I.

Calculus and Linear Algebra:

Students should consult their advisors and keep major prerequisites in mind when planning a suitable program. The following are general recommendations. Consult First Steps in Math for more detail. The director of undergraduate studies will gladly meet with students to offer further advice.

• Students who expect to major in mathematics should take MATH 1110   – MATH 1120    and continue with MATH 2210   – MATH 2220   . This sequence is also a good choice for those studying economics or a science for which a strong math background is recommended. Students with a 5 on the Calculus BC exam and a strong interest in challenging theoretical mathematics may consider MATH 2230   – MATH 2240   , especially if they have had some prior exposure to linear algebra and multivariable calculus.
• MATH 1910   – MATH 1920   – MATH 2930   – MATH 2940    is the core sequence for engineering students. It is also recommended by some advisors in fields strongly related to the mathematical and physical sciences, such as astronomy, computer science, physics, and physical chemistry. MATH 1910    assumes students have already taken a good first course in calculus, which can be satisfied with a 5 on the Calculus AB exam or a course like MATH 1110   . Students in this sequence who plan to take more math should consider a 3000-level course to gain experience with proofs before attempting a 4000-level course.
• MATH 1110   – MATH 1120    followed by MATH 2130    or MATH 2310    is a good choice for students who need to master the basic techniques of calculus but whose majors will not require a substantial amount of mathematics. MATH 1110   – MATH 2310    is an option for students who need some linear algebra but not a full year of calculus.
• MATH 1106    is an option for students whose major requires only one semester of calculus.  Some topics are covered in less depth than in MATH 1110   , while more advanced topics are introduced. MATH 1106    focuses on modeling using examples from the life sciences. It introduces some fundamental concepts of calculus and provides a brief introduction to differential equations. Students who may take more than one semester of calculus should take MATH 1110    rather than MATH 1106   .
• Students who are undecided about their future studies at Cornell but think they may involve a substantial amount of math can keep their options open by taking Calculus I ( MATH 1110    or AP credit), Calculus II ( MATH 1120    or AP credit), and Linear Algebra ( MATH 2210   ). Multivariable Calculus ( MATH 2220   ) would be the next step for students who are still leaning in the direction of a math-related major and may wish to take more advanced mathematics.

Some switching between sequences (1) and (2) is possible provided classes are taken in the right order. For example,  MATH 2220    must be preceded by a semester of linear algebra. In general,  MATH 2130    and  MATH 2310    are not the best preparation for further study in mathematics. Students who have taken these courses should consult the director of undergraduate studies for advice before continuing.

Special-Purpose Sequences:

Students who will take no more than two semesters of mathematics can gain a broader view of the subject by taking one semester of calculus and one non-calculus mathematics course. The following options are particularly useful for students in the life and social sciences and will satisfy the mathematics requirement for most medical schools.

• MATH 1105   – MATH 1106    provides a one-year introduction to the mathematical topics that are most useful to biologists and social scientists. ( MATH 1110    may be substituted for MATH 1106   .)
• An introductory statistics course ( MATH 1710   , for example), taken before or after a semester of calculus ( MATH 1106    or MATH 1110   ), teaches students how to work with data and can be more useful in some disciplines than a second semester of calculus.

Students who want two semesters of calculus are advised to take the first two semesters of one of the calculus sequences, but students with excellent performance in MATH 1106    may follow that course with MATH 1120   .

Minor in Mathematics:

The mathematics minor is available to undergraduates majoring in other disciplines across the university who have an interest in studying mathematics. Students planning a minor in mathematics may seek advice on course selection from the director of undergraduate studies . Information is also available at math.cornell.edu/minor , including how to apply for the minor.

Student Grade Option

Courses must be taken for a letter grade in order to count toward admission to the math minor or to satisfy any math minor requirement. This requirement is waived for all classes taken in spring 2020, and an “S” grade will be accepted regardless of the usual minimum grade requirements.

Transfer Credit

Courses taken at another institution may be used to satisfy the math minor prerequisites and to replace at most one course toward the minor requirements. These courses must be approved for transfer credit and appear on the Cornell transcript with Cornell course equivalents.

Visit the Math Department web site for more information about transferring credit from another institution .

Prerequisites:

Students are admitted to the minor after successfully completing a semester of linear algebra — MATH 2210   , MATH 2230   , or MATH 2940    with a grade of B– or better — and a semester of  multivariable calculus — MATH 2220   , MATH 2240   , or MATH 1920    with a grade of B– or better. The department recommends MATH 2210   – MATH 2220    or MATH 2230   – MATH 2240   . MATH 2130    and MATH 2310    are not recommended for students planning a math minor; however, MATH 2130    with a grade of B+ or better may be accepted as a substitute for MATH 2220   , and MATH 2310    with a grade of B+ or better may be accepted as a substitute for MATH 2210   .

Credit for  MATH 1920    may be obtained by passing a placement exam during orientation; however, a score equivalent to a B- or better is required to satisfy the prerequisite for the math minor. Students who score below a B- and wish to join the minor may not attempt the exam a second time but should instead enroll in a multivariable calculus course.

Students who receive below the minimum grade in one of these prerequisite courses should contact the undergraduate coordinator immediately.

Requirements:

Students must complete four 3000- or 4000-level MATH courses. Only courses with a MATH prefix or cross-listed as such are allowed; no substitutions. At least one course must be in algebra and one in analysis. Eligible algebra and analysis courses are the same as those listed for the math major requirements (1) and (2) below. At least one of the four courses must be at the 4000-level or above. A course may be counted toward the minor only if it is taken for a letter grade and a grade of C– or better is received for the course.

Major in Mathematics:

The mathematics major adapts to a number of purposes. It can emphasize the theoretical or the applied. It can be appropriate for professionals and nonprofessionals alike, and can be broad or narrow. It can also be combined easily with serious study in another subject in the physical, biological, or social sciences by means of a double major and/or concentration. (See “ Double Majors ” below for more information.)

Questions concerning the major should be brought to the undergraduate coordinator. Information is also available at math.cornell.edu/major , including how to apply for the major.

Note: In addition to the major requirements outlined below, all students must meet the college graduation requirements   . 

Courses must be taken for a letter grade in order to count toward admission to the math major or to satisfy any math major requirement. This requirement is waived for all classes taken in spring 2020, including courses taken for an outside concentration, and an “S” grade will be accepted regardless of the usual minimum grade requirements.

Courses taken at another institution may be used to satisfy the math major prerequisites and to replace at most two courses toward the major requirements. These courses must be approved for transfer credit and appear on the Cornell transcript with Cornell course equivalents.

Students are admitted to the major after successfully completing a semester of linear algebra — MATH 2210   , MATH 2230   , or MATH 2940    with a grade of B– or better — and a semester of  multivariable calculus — MATH 2220   , MATH 2240   , or MATH 1920    with a grade of B– or better. The department recommends MATH 2210   – MATH 2220    or MATH 2230   – MATH 2240   . MATH 2130    and MATH 2310    are not recommended for students planning a math major; however, MATH 2130    with a grade of B+ or better may be accepted as a substitute for MATH 2220   , and MATH 2310    with a grade of B+ or better may be accepted as a substitute for MATH 2210   . A 3- or 4-credit computer programming course is also required with a letter grade of C– or better. Eligible courses include: CS 1110   , CS 1112   , CS 2110   , and CS 2112   .

Credit for  MATH 1920    may be obtained by passing a placement exam during orientation; however, a score equivalent to a B- or better is required to satisfy the prerequisite for the math major. Students who score below a B- and wish to join the major may not attempt the exam a second time but should instead enroll in a multivariable calculus course.

Students who have taken a course in linear algebra and/or multivariable calculus during high school should consider taking MATH 2230   – MATH 2240   .  This sequence gives a more abstract, proof-oriented treatment of the material. Students with an advanced background in linear algebra and/or multivariable calculus should contact the undergraduate coordinator for advice as soon as possible. Note that 4000-level linear algebra courses are generally not regarded as meeting the prerequisites for the math major.

Students who receive below the minimum grade in one of these prerequisite courses should contact the undergraduate coordinator immediately. Any repeated attempt to fulfill a math major prerequisite requires pre-approval from the math majors committee.

The minimum number of required credits to complete the major is 27. However, typical students accrue between 32 and 36 credits.

Students must complete nine courses, as described in items 1–3 below, under the following constraints:

• At least 5 courses with a MATH prefix numbered 3000 or above must appear on the student’s transcript. (Double majors enrolling in cross-listed courses should pay particular attention to this constraint.)
• At least two of the MATH courses taken must be at the 4000-level (or above).
• A course may be counted toward the major only if it is taken for a letter grade and a grade of C– or better is received for the course.
• No course may be used to satisfy more than one requirement for the major.
• 2-credit courses count as half courses.
• MATH courses numbered between 4980 and 5999 do not count toward the major.

Major advisors may make adjustments to the major requirements upon request from an advisee, provided the intent of the requirements is met. In particular, many suitable graduate courses are not listed here, but are available for undergraduates who are well prepared.

1. Two courses in algebra:

Eligible courses are:

• MATH 3320 - Introduction to Number Theory
• MATH 3340 - Abstract Algebra
• MATH 3360 - Applicable Algebra
• MATH 4310 - Linear Algebra
• MATH 4330 - Honors Linear Algebra
• MATH 4340 - Honors Introduction to Algebra
• MATH 4370 - Computational Algebra
• MATH 4500 - Matrix Groups
• MATH 4560 - [Geometry of Discrete Groups]

2. Two courses in analysis:

• MATH 3110 - Introduction to Analysis
• MATH 3210 - Manifolds and Differential Forms
• MATH 3270 - Introduction to Ordinary Differential Equations
• MATH 4130 - Honors Introduction to Analysis I
• MATH 4140 - Honors Introduction to Analysis II
• MATH 4180 - Complex Analysis
• MATH 4200 - Differential Equations and Dynamical Systems
• MATH 4210 - Nonlinear Dynamics and Chaos
• MATH 4220 - Applied Complex Analysis
• MATH 4250 - Numerical Analysis and Differential Equations (crosslisted)
• MATH 4260 - Numerical Analysis: Linear and Nonlinear Problems (crosslisted)
• MATH 4280 - Introduction to Partial Differential Equations

3. Five further high-level mathematical courses:

A. concentration in mathematics:, i. four additional math courses numbered 3000 or above:.

At least one of the four courses must be among the geometry/topology courses. Eligible courses include:

• MATH 4520 - Classical Geometries and Modern Applications
• MATH 4530 - Introduction to Topology
• MATH 4540 - Introduction to Differential Geometry
• MATH 4550 - [Applicable Geometry]

MATH 3210    is eligible only if not used for the analysis requirement; MATH 4500    and MATH 4560    are eligible only if not used toward the algebra requirement.

ii. One course dealing with mathematical models:

Eligible courses include  MATH 3610    and any course from outside mathematics with serious mathematical content that deals with scientific matters. Serious mathematical content includes, but is not limited to, extensive use of calculus or linear algebra. Any course from another department that would satisfy one of the concentrations may be used, as well as:

• CS 2110 - Object-Oriented Programming and Data Structures (crosslisted)
• PHYS 1116 - Physics I: Mechanics and Special Relativity
• PHYS 2208 - Fundamentals of Physics II
• PHYS 2213 - Physics II: Electromagnetism
• PHYS 2217 - Physics II: Electricity and Magnetism (crosslisted)

Other 1000-level physics courses and PHYS 2207    may not be used, but some courses in other fields may be accepted. AP credit may not be used.

b. Concentration in Applied Mathematics:

Five additional courses from (iii) and (iv) below, of which at least three are from (iii) and one is from (iv). Of the 9 courses used to fulfill requirements (1), (2), (3 iii), and (3 iv) of the math major with an applied mathematics concentration, at least one course must be taken from three of the four Groups A, B, C, and D below. Non-MATH courses in these groups may be used toward the math modeling requirement (3 iv).

iii. MATH courses numbered 3000 or above:

Iv. courses dealing with mathematical models:.

Eligible courses include  MATH 3610    and any course outside mathematics with serious mathematical content that deals with scientific matters.  Serious mathematical content includes, but is not limited to, extensive use of calculus or linear algebra. Any course from another department that would satisfy one of the concentrations may be used. At most one of the following may be used:

• CS 2110 - Object-Oriented Programming and Data Structures     
• PHYS 1116 - Physics I: Mechanics and Special Relativity    
• PHYS 2208 - Fundamentals of Physics II    
• PHYS 2213 - Physics II: Electromagnetism    
• PHYS 2217 - Physics II: Electricity and Magnetism    

Other 1000-level physics courses and  PHYS 2207    may not be used, but some courses in other fields may be accepted.  AP credit may not be used.

Group A. Differential equations

• MATH 3270 - Introduction to Ordinary Differential Equations  
• MATH 4200 - Differential Equations and Dynamical Systems    
• MATH 4210 - Nonlinear Dynamics and Chaos    
• MATH 4280 - Introduction to Partial Differential Equations    

Group B. Discrete mathematics and combinatorics

• MATH 3360 - Applicable Algebra    
• MATH 4370 - Computational Algebra    
• MATH 4410 - Introduction to Combinatorics I    
• MATH 4420 - Introduction to Combinatorics II    
• MATH 4550 - [Applicable Geometry]    
• CS 4820 - Introduction to Analysis of Algorithms    
• ECON 4020 - [Game Theory I]    
• ECON 4022 - [Game Theory II]    
• ORIE 3300 - Optimization I    
• ORIE 4350 - Introduction to Game Theory    

Group C. Numerical and computational methods

• MATH 4250 - Numerical Analysis and Differential Equations    
• MATH 4260 - Numerical Analysis: Linear and Nonlinear Problems     
• CS 4620 - Introduction to Computer Graphics    
• CS 4670 - Introduction to Computer Vision     
• MAE 4700 - Finite Element Analysis for Mechanical and Aerospace Design    

Group D. Probability and statistics

• MATH 4710 - Basic Probability    
• MATH 4720 - Statistics    
• MATH 4740 - Stochastic Processes    
• ECON 3130 - Statistics and Probability    
• ECON 4130 - Statistical Decision Theory    
• ORIE 3500 - Engineering Probability and Statistics II    
• STSCI 3080 - Probability Models and Inference    
• STSCI 3100 - Statistical Sampling    
• STSCI 4030 - Linear Models with Matrices    

c. Concentration in Computer Science:

Five additional courses from (v) and (vi) below, of which at least one is from (v) and three are from (vi).

v. MATH courses numbered 3000 or above:

Vi. computer science courses with significant mathematical content:.

• CS 3220 - [Computational Mathematics for Computer Science]
• CS 4110 - [Programming Languages and Logics]
• CS 4160 - [Formal Verification]
• CS 4210 - Numerical Analysis and Differential Equations (crosslisted)
• CS 4220 - Numerical Analysis: Linear and Nonlinear Problems (crosslisted)
• CS 4620 - Introduction to Computer Graphics
• CS 4670 - Introduction to Computer Vision
• CS 4700 - Foundations of Artificial Intelligence
• CS 4740 - Natural Language Processing (crosslisted)
• CS 4744 - Computational Linguistics I (crosslisted)
• CS 4775 - Computational Genetics and Genomics (crosslisted)
• CS 4780 - Introduction to Machine Learning
• CS 4783 - Mathematical Foundations of Machine Learning
• CS 4786 - [Machine Learning for Data Science]
• CS 4787 - Principles of Large-Scale Machine Learning Systems
• CS 4789 - Introduction to Reinforcement Learning
• CS 4810 - [Introduction to Theory of Computing]
• CS 4812 - Quantum Information Processing (crosslisted)
• CS 4814 - Introduction to Computational Complexity
• CS 4820 - Introduction to Analysis of Algorithms
• CS 4830 - Introduction to Cryptography
• CS 4850 - [Mathematical Foundations for the Information Age]
• CS 4852 - Networks II: Market Design (crosslisted)
• CS 4860 - Applied Logic (crosslisted)

There are also many CS graduate courses with significant mathematical content that may be used. Interested students should discuss these options with their math faculty advisor (after being admitted to the math major).  

d. Concentration in Economics:

Five additional courses from (vii), (viii), and (ix) below, as follows: one course from (vii), three courses from (viii), and a fifth course from any of (vii), (viii), or (ix).

vii. MATH courses numbered 3000 or above:

Viii. economics courses with significant mathematical content:.

• ECON 3130 - Statistics and Probability or
• ECON 6190 - Econometrics I
• ECON 3140 - Econometrics or
• ECON 6200 - Econometrics II
• ECON 3810 - [Decision Theory I]
• ECON 3825 - Networks II: Market Design (crosslisted)
• ECON 4020 - [Game Theory I]
• ECON 4022 - [Game Theory II]
• ECON 4110 - Cross Section and Panel Econometrics
• ECON 4130 - Statistical Decision Theory
• ECON 4907 - The Economics of Asymmetric Information and Contracts
• ECON 6090 - Microeconomic Theory I
• ECON 6100 - Microeconomic Theory II
• ECON 6130 - Macroeconomics I
• ECON 6140 - Macroeconomics II

Undergraduate enrollment in ECON graduate courses requires permission of instructor.

ix. Courses in operations research with significant mathematical content and dealing with material of interest in economics:

• ORIE 3300 - Optimization I
• ORIE 3310 - Optimization II
• ORIE 4350 - Introduction to Game Theory
• ORIE 4580 - Simulation Modeling and Analysis
• ORIE 4600 - Introduction to Financial Engineering
• ORIE 4740 - Statistical Data Mining I
• ORIE 4741 - Learning with Big Messy Data
• ORIE 5600 - Financial Engineering with Stochastic Calculus I
• ORIE 5610 - Financial Engineering with Stochastic Calculus II

e. Concentration in Mathematical Biology:

Five additional courses from (x) and (xi) below, with three courses from (x) and two courses from (xi).

x. Biology courses that have mathematical content and provide background necessary for work at the interface between biology and mathematics:

• BIOEE 3620 - Dynamic Models in Biology (crosslisted)
• BIONB 4220 - [Modeling Behavioral Evolution]
• BME 3110 - Cellular Systems Biology
• BTRY 3080 - Probability Models and Inference (crosslisted)
• BTRY 4090 - Theory of Statistics (crosslisted)
• BIOCB 4830 - Quantitative Genomics and Genetics
• NTRES 4120 - Wildlife Population Analysis: Techniques and Models

xi. MATH courses numbered 3000 or above:

Particularly appropriate are:

• MATH 4710 - Basic Probability

f. Concentration in Mathematical Physics:

Five additional courses from (xii) and (xiii) below, of which at least one is from (xii) and three are from (xiii).

xii. MATH courses numbered 3000 or above:

Xiii. physics courses that make significant use of advanced mathematics:.

• PHYS 3316 - Basics of Quantum Mechanics
• PHYS 3317 - Applications of Quantum Mechanics
• PHYS 3318 - Analytical Mechanics
• PHYS 3327 - Advanced Electricity and Magnetism
• PHYS 4230 - Statistical Thermodynamics (crosslisted)
• PHYS 4443 - Intermediate Quantum Mechanics
• PHYS 4444 - Introduction to Particle Physics
• PHYS 4445 - Introduction to General Relativity (crosslisted)
• PHYS 4454 - Introductory Solid State Physics (crosslisted)
• PHYS 4481 - Quantum Information Processing (crosslisted)
• PHYS 4488 - Statistical Mechanics
• AEP 4340 - Fluid and Continuum Mechanics
• AEP 4400 - [Nonlinear and Quantum Optics]

g. Concentration in Operations Research:

Five additional courses from (xiv) and (xv) below, of which at least one is from (xiv) and three are from (xv).

xiv. MATH courses numbered 3000 or above:

Xv. courses in operations research in which the primary focus involves mathematical techniques:.

• ORIE 3500 - Engineering Probability and Statistics II
• ORIE 3510 - Introduction to Engineering Stochastic Processes I (crosslisted)
• ORIE 4630 - Operations Research Tools for Financial Engineering (crosslisted)
• ORIE 5640 - Statistics for Financial Engineering (crosslisted)

h. Concentration in Statistics:

Five additional courses from (xvi), (xvii), and (xviii) below. No substitutions are allowed for MATH 4710    or MATH 4720   . Students who have already taken a course with overlapping content should contact the undergraduate coordinator . (For students who have not had experience with real-world data, MATH 1710    is recommended before or concurrent with MATH 4710   . It will not, however, count toward any of the math major requirements.)

• MATH 4720 - Statistics

xvii. One additional MATH course numbered 3000 or above:

Xviii. two courses in other departments with significant content in statistics, complementing (xvii):.

• ECON 3140 - Econometrics
• STSCI 3100 - Statistical Sampling (crosslisted)
• STSCI 3510 - Introduction to Engineering Stochastic Processes I (crosslisted)
• STSCI 4030 - Linear Models with Matrices (crosslisted)
• STSCI 4060 - Python Programming and its Applications in Statistics
• STSCI 4100 - Multivariate Analysis (crosslisted)
• STSCI 4110 - Categorical Data (crosslisted)
• STSCI 4140 - Applied Design (crosslisted)
• STSCI 4520 - Statistical Computing
• STSCI 4550 - Applied Time Series Analysis (crosslisted)
• STSCI 4740 - Data Mining and Machine Learning
• STSCI 4780 - Bayesian Data Analysis: Principles and Practice

STSCI 3510   / ORIE 3510    may not be counted toward (xviii) if MATH 4740    is used for (xvii). At most one regression course ( ECON 3140    or STSCI 4030   / BTRY 4030   ) is allowed for (xviii). At most one of CS 4780   , CS 4786   , ORIE 4740   , or STSCI 4740    is allowed for (xviii).

Double Majors:

A double major with computer science, economics, or physics can be facilitated by the corresponding concentrations described above. The Departments of Computer Science and Economics permit double majors to use courses in the corresponding concentrations to satisfy the requirements of both majors.

Double majors with physics may count eligible physics courses toward both the physics major and the math major’s math physics concentration; however, the Physics Department will not approve courses for an outside concentration if they are being used toward another major or minor.

When enrolling in cross-listed courses, double majors must take care that at least 5 courses with a MATH prefix numbered 3000 or above will appear on their transcript. Students should consult other major departments about any further conditions they may have.

Senior Thesis:

A senior thesis can form a valuable part of a student’s experience in the mathematics major. It is intended to allow students to conduct an in-depth investigation not possible in regular course work. The work should be independent and creative. It can involve the solution of a serious mathematics problem, or it can be an expository work, or variants of these. Conducting independent research, paying careful attention to exposition in the finished written product, and the delivery of an optional oral presentation can have a lasting positive impact on a student’s educational and professional future.

Graduate Courses:

Some exceptional undergraduates, upon completing a rigorous foundation of 4000-level math courses, may wish to further develop their understanding of the material in subsequent graduate courses that the math department offers. The core courses from the mathematics graduate program — MATH 6110   , MATH 6120   , MATH 6310   , MATH 6320   , MATH 6510   , and MATH 6520    — represent a good first exposure to graduate-level mathematics. MATH 6150   , MATH 6160   , MATH 6210   , MATH 6220   , MATH 6710   , and MATH 6720    cover some additional material in a manner suitable to advanced undergraduates.

Undergraduates taking graduate courses should have completed advanced undergraduate courses on the same topic with a grade of A– or better. Interested students should discuss the possibility of taking graduate courses with their faculty advisor in the Math Department prior to enrolling in the course.

The Department of Mathematics awards honors ( cum laude ) and high honors ( magna cum laude and summa cum laude ) to graduating mathematics majors who have performed outstandingly in the major program.

The awards are determined by the Mathematics Major Committee in the latter part of the semester before graduation. The committee will primarily be looking for excellent performance in mathematics courses, particularly in challenging courses at the 4000-level or beyond. Independent study at a high performance level can also contribute to honors. Students interested in any level of honors should consult their major advisors or a member of the Mathematics Major Committee concerning suitable courses. Outstanding performance in the core graduate classes ( MATH 6110   – MATH 6120   , MATH 6310   – MATH 6320   , MATH 6510   - MATH 6520   ) or an excellent senior thesis can contribute to high honors.

Senior Theses

• Categories: Strategies for Learning

Doing a senior thesis is an exciting enterprise. It’s often the first time students are engaging in truly original research and trying to develop a significant contribution to a field of inquiry. But as joyful as an independent research process can be, you don’t have to go it alone. It’s important to have support as you navigate such a large endeavor, and the ARC is here to offer one of those layers of support. 

Whether or not to write a senior thesis is just the first in a long line of questions thesis writers need to consider. In addition to questions about the topic and scope of your thesis, there are questions about timing, schedule, and support. For example, if you are collecting data, when should data collection start and when should it be completed? What kind of schedule will you write on? How will you work with your adviser? Do you want to meet with your adviser about your progress once a month? Once a week? What other resources can you turn to for information, feedback, and support? 

Even though there is a lot to think about and a lot to do, doing a thesis really can be an enjoyable experience! Keep reminding yourself why you chose this topic and why you care about it. 

Tips for Tackling Big Projects:  

• When you’re approaching a big project, it can seem overwhelming to look at the whole thing at once, so it’s essential to identify the smaller steps that will move you towards the completed project. 
• Your advisor is best suited to help you break down the thesis process with field-specific advice. 
• If you need to refine the breakdown further so it makes sense for you, schedule an appointment with an Academic Coach . An academic coach can help you think through the steps in a way that works for you. 
• Pre-determine the time, place, and duration. 
• Keep it short (15 to 60 minutes). 
• Have a clear and reasonable goal for each writing session. 
• Make it a regular event (every day, every other day, MWF). 
• time is not wasted deciding to write if it’s already in your calendar; 
• keeping sessions short reduces the competition from other tasks that are not getting done; 
• having an achievable goal for each session provides a sense of accomplishment (a reward for your work); 
• writing regularly can turn into a productive habit. 
• In addition to having a clear goal for each writing session, it’s important to have clear goals for each week and to find someone to communicate these goals to, such as your adviser, a “thesis buddy,” your roommate, etc. Communicating your goals and progress to someone else creates a useful sense of accountability. 
• If your adviser is not the person you are communicating your progress to on a weekly basis, then request to set up a structure with your adviser that requires you to check in at less frequent but regular intervals. 
• Commit to attending Accountability Hours at the ARC on the same day every week. Making that commitment will add both social support and structure to your week. Use the ARC Scheduler to register for Accountability Hours. 
• Set up an accountability group in your department or with thesis writers from different departments. 
• It’s important to have a means for getting consistent feedback on your work and to get that feedback early. Work on large projects often lacks the feeling of completeness, so don’t wait for a whole section (and certainly not the whole thesis) to feel “done” before you get feedback on it! 
• Your thesis adviser is typically the person best positioned to give you feedback on your research and writing, so communicate with your adviser about how and how often you would like to get feedback. 
• If your adviser isn’t able to give you feedback with the frequency you’d like, then fill in the gaps by creating a thesis writing group or exploring if there is already a writing group in your department or lab. 
• The Harvard College Writing Center is a great resource for thesis feedback. Writing Center Senior Thesis Tutors can provide feedback on the structure, argument, and clarity of your writing and help with mapping out your writing plan. Visit the Writing Center website to schedule an appointment with a thesis tutor . 
• Working on a big project can be anxiety provoking because it’s hard to keep all the pieces in your head and you might feel like you are losing track of your argument. 
• To reduce this source of anxiety, try keeping a separate document where you jot down ideas on how your research questions or central argument might be clarifying or changing as you research and write. Doing this will enable you to stay focused on the section you are working on and to stop worrying about forgetting the new ideas that are emerging. 
• You might feel anxious when you realize that you need to update your argument in response to the evidence you have gathered or the new thinking your writing has unleashed. Know that that is OK. Research and writing are iterative processes – new ideas and new ways of thinking are what makes progress possible. 
• It’s also anxiety provoking to feel like you can’t “see” from the beginning to the end of your project in the way that you are used to with smaller projects. 
• Breaking down big projects into manageable chunks and mapping out a schedule for working through each chunk is one way to reduce this source of anxiety. It’s reassuring to know you are working towards the end even if you cannot quite see how it will turn out. 
• It may be that your thesis or dissertation never truly feels “done” to you, but that’s okay. Academic inquiry is an ongoing endeavor. 
• Thesis work is not a time for social comparison; each project is different and, as a result, each thesis writer is going to work differently. 
• Just because your roommate wrote 10 pages in a day doesn’t mean that’s the right pace or strategy for you. 
• If you are having trouble figuring out what works for you, use the ARC Scheduler to make an appointment with an Academic Coach , who can help you come up with daily, weekly, and semester-long plans. 
• If you’re having trouble finding a source, email your question or set up a research consult via Ask a Librarian . 
• If you’re looking for additional feedback or help with any aspect of writing, contact the Harvard College Writing Center . The Writing Center has Senior Thesis Tutors who will read drafts of your thesis (more typically, parts of your thesis) in advance and meet with you individually to talk about structure, argument, clear writing, and mapping out your writing plan. 
• If you need help with breaking down your project or setting up a schedule for the week, the semester, or until the deadline, use the ARC Scheduler to make an appointment with an Academic Coach . 
• If you would like an accountability structure for social support and to keep yourself on track, come to Accountability Hours at the ARC. 

MATH 4399 - Senior Honors Thesis

Guidelines for Senior Theses in the Mathematics Department

A senior thesis can form a valuable part of a student's experience in the Mathematics Major . It is intended to allow students to cover significant areas of mathematics not covered in course work, or not covered there in sufficient depth. The work should be independent and creative. It can involve the solution of a serious mathematics problem, or it can be an expository work, or variants of these. Both the process of doing independent research and mathematics exposition, as well as the finished written product and optional oral presentation, can have a lasting positive impact on a student's educational and professional future.

Recent Senior Theses

Supervision

Supervision by a qualified member of the field of mathematics at Cornell is the normal requirement for a senior thesis. Other arrangements are possible, however, provided they are made with the assistance of the student's major advisor, and with the approval of the Mathematics Major Committee .

Finding a supervisor/Encouraging students. It should be emphasized that both the writing and the supervising of a senior thesis are optional activities, both for students and faculty. Students interested in doing this will need to find a suitable supervisor — perhaps with the aid of their major advisor or another faculty member whom they know. Advisors and other faculty who encounter students whom they think would benefit from this activity are invited to mention this option to them and assist them in finding a supervisor.

Standard venues for senior theses. One obvious way in which a senior thesis can be produced is through an independent reading course (MATH 490); another way is through an REU experience, either at Cornell or elsewhere. (If the REU work was accomplished or initiated elsewhere, a "local expert" will still be needed to supervise or "vouch for" the work as a senior thesis.) In yet a third way, a student may present a faculty member with a solution or partial solution to an interesting problem. In such cases, this could form the core of a senior thesis. Faculty are invited to encourage such work from their students.

Public Lecture

A public lecture in which the results of the senior thesis are presented is welcome but optional. This should be arranged by the thesis supervisor in conjunction with the undergraduate coordinator and adequately advertised. Department faculty and graduate students are encouraged to attend these presentations.

Submitting the Completed Thesis

The supervisor must approve the student's thesis. No later than April 15th, the student must submit a completed thesis to the thesis supervisor. If the supervisor asks the student to make changes, the student has until April 30th to do so. By April 30th, the student must give the supervisor two paper copies and an electronic copy of the thesis in final form. The electronic copy will be posted on the department's web site. [Students who expect to graduate in January must submit a completed thesis by November 15th and the final form by November 30th.]

Form of the Thesis

Ideally, the final document should be TeXed or prepared in some equivalent technical document preparation system. The document must have large left margins (one and one-half inches or slightly larger). The title page should contain:

• The student's name and graduating class.
• The title of the senior thesis.
• The name of the faculty supervisor. (If there is more than one supervisor, list both. If one of the supervisors is not in the Mathematics Department, list the department and institution.)
• The date of completion of the thesis.

This information will be used to produce a standard frontispiece page, which will be added to the document in its library copies.

Merit of the Senior Thesis

Judgment as to the merit of a senior thesis will be based largely on the recommendation of the faculty member supervising the thesis. The Mathematics Major Committee will use this recommendation both in its determination of honors and in its decision on whether to place the thesis in our permanent library collection.

Honors Consideration

The senior thesis will automatically be considered by the Mathematics Major Committee as one of the ingredients for deciding on an honors designation for the student. Students may receive honors without a thesis and are not guaranteed honors with one. However, an excellent senior thesis combined with an otherwise excellent record can elevate the level of honors awarded.

Mathematics Library Collection of Senior Theses

Meritorious senior theses will be catalogued, bound, and stored in the Mathematics Library.

Last modified: June 20, 2008

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Ten with MIT connections win 2024 Hertz Foundation Fellowships

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The Fannie and John Hertz Foundation  announced that it has awarded fellowships to 10 PhD students with ties to MIT. The prestigious award provides each recipient with five years of doctoral-level research funding (up to a total of $250,000), which allows them the flexibility and autonomy to pursue their own innovative ideas.

Fellows also receive lifelong access to Hertz Foundation programs, such as events, mentoring, and networking. They join the ranks of over 1,300 former Hertz Fellows who are leaders and scholars in a range of fields in science, engineering, and technology. Connections among fellows over the years have sparked collaborations in startups, research, and technology commercialization.

The 10 MIT recipients are among a total of 18 Hertz Foundation Fellows scholars selected this year from across the country. Five of them received their undergraduate degrees at the Institute and will pursue their PhDs at other schools. Two are current MIT graduate students, and four will begin their studies here in the fall.

“For more than 60 years, Hertz Fellows have led scientific and technical innovation in national security, applied biological sciences, materials research, artificial intelligence, space exploration, and more. Their contributions have been essential in advancing U.S. competitiveness,” says Stephen Fantone, chair of the Hertz Foundation board of directors and founder and president of Optikos Corp. “I’m excited to watch our newest Hertz Fellows as they pursue challenging research and continue the strong tradition of applying their work for the greater good.”

This year’s MIT-affiliated awardees are:

Owen Dugan ’24 graduated from MIT in just two-and-a-half years with a degree in physics, and he plans to pursue a PhD in computer science at Stanford University. His research interests lie at the intersection of AI and physics. As an undergraduate, he conducted research in a broad range of areas, including using physics concepts to enhance the speed of large language models and developing machine learning algorithms that automatically discover scientific theories. He was recognized with MIT’s Outstanding Undergraduate Research Award and is a U.S. Presidential Scholar, a Neo Scholar, and a Knight-Hennessy Scholar. Dugan holds multiple patents, co-developed an app to reduce food waste, and co-founded a startup that builds tools to verify the authenticity of digital images.

Kaylie Hausknecht will begin her physics doctorate at MIT in the fall, having completing her undergraduate degree in physics and astrophysics at Harvard University. While there, her undergraduate research focused on developing new machine learning techniques to solve problems in a range of fields, such as fluid dynamics, astrophysics, and condensed matter physics. She received the Hoopes Prize for her senior thesis, was inducted into Phi Beta Kappa as a junior, and won two major writing awards. In addition, she completed five NASA internships. As an intern, she helped identify 301 new exoplanets using archival data from the Kepler Space Telescope. Hausknecht served as the co-president of Harvard’s chapter of Science Club for Girls, which works to encourage girls from underrepresented backgrounds to pursue STEM.

Elijah Lew-Smith majored in physics at Brown University and plans to pursue a doctoral degree in physics at MIT. He is a theoretical physicist with broad intellectual interests in effective field theory (EFT), which is the study of systems with many interacting degrees of freedom. EFT reveals how to extract the relevant, long-distance behavior from complicated microscopic rules. In 2023, he received a national award to work on applying EFT systematically to non-equilibrium and active systems such as fluctuating hydrodynamics or flocking birds. In addition, Lew-Smith received a scholarship from the U.S. State Department to live for a year in Dakar, Senegal, and later studied at ’École Polytechnique in Paris, France.

Rupert Li ’24 earned his bachelor’s and master’s degrees at MIT in mathematics as well as computer science, data science, and economics, with a minor in business analytics.He was named a 2024 Marshall Scholar and will study abroad for a year at Cambridge University before matriculating at Stanford University for a mathematics doctorate. As an undergraduate, Li authored 12 math research articles, primarily in combinatorics, but also including discrete geometry, probability, and harmonic analysis. He was recognized for his work with a Barry Goldwater Scholarship and an honorable mention for the Morgan Prize, one of the highest undergraduate honors in mathematics.

Amani Maina-Kilaas is a first-year doctoral student at MIT in the Department of Brain and Cognitive Sciences, where he studies computational psycholinguistics. In particular, he is interested in using artificial intelligence as a scientific tool to study how the mind works, and using what we know about the mind to develop more cognitively realistic models. Maina-Kilaas earned his bachelor’s degree in computer science and mathematics from Harvey Mudd College. There, he conducted research regarding intention perception and theoretical machine learning, earning the Astronaut Scholarship and Computing Research Association’s Outstanding Undergraduate Researcher Award.

Zoë Marschner ’23 is a doctoral student at Carnegie Mellon University working on geometry processing, a subfield of computer graphics focused on how to represent and work with geometric data digitally; in her research, she aims to make these representations capable of enabling fundamentally better algorithms for solving geometric problems across science and engineering. As an undergraduate at MIT, she earned a bachelor’s degree in computer science and math and pursued research in geometry processing, including repairing hexahedral meshes and detecting intersections between high-order surfaces. She also interned at Walt Disney Animation Studios, where she worked on collision detection algorithms for simulation. Marschner is a recipient of the National Science Foundation’s Graduate Research Fellowship and the Goldwater Scholarship.

Zijian (William) Niu will start a doctoral program in computational and systems biology at MIT in the fall. He has a particular interest in developing new methods for imaging proteins and other biomolecules in their native cellular environments and using those data to build computational models for predicting their dynamics and molecular interactions. Niu received his bachelor’s degree in biochemistry, biophysics, and physics from the University of Pennsylvania. His undergraduate research involved developing novel computational methods for biological image analysis. He was awarded the Barry M. Goldwater Scholarship for creating a deep-learning algorithm for accurately detecting tiny diffraction-limited spots in fluorescence microscopy images that outperformed existing methods in quantifying spatial transcriptomics data.

James Roney received his bachelor’s and master’s degrees from Harvard University in computer science and statistics, respectively. He is currently working as a machine learning research engineer at D.E. Shaw Research. His past research has focused on interpreting the internal workings of AlphaFold and modeling cancer evolution. Roney plans to pursue a PhD in computational biology at MIT, with a specific interest in developing computational models of protein structure, function, and evolution and using those models to engineer novel proteins for applications in biotechnology.

Anna Sappington ’19 is a student in the Harvard University-MIT MD-PhD Program, currently in the first year of her doctoral program at MIT in electrical engineering and computer science. She is interested in building methods to predict evolutionary events, especially connections among machine learning, biology, and chemistry to develop reinforcement learning models inspired by evolutionary biology. Sappington graduated from MIT with a bachelor’s degree in computer science and molecular biology. As an undergraduate, she was awarded a 2018 Barry M. Goldwater Scholarship and selected as a Burchard Scholar and an Amgen Scholar. After graduating, she earned a master’s degree in genomic medicine from the University of Cambridge, where she studied as a Marshall Scholar, as well as a master’s degree in machine learning from University College London.

Jason Yang ’22  received his bachelor’s degree in biology with a minor in computer science from MIT and is currently a doctoral student in genetics at Stanford University. He is interested in understanding the biological processes that underlie human health and disease. At MIT, and subsequently at Massachusetts General Hospital, Yang worked on the mechanisms involved in neurodegeneration in repeat expansion diseases, uncovering a novel molecular consequence of repeat protein aggregation.

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635th Anti-Aircraft Missile Regiment

635-й зенитно-ракетный полк

Military Unit: 86646

Activated 1953 in Stepanshchino, Moscow Oblast - initially as the 1945th Anti-Aircraft Artillery Regiment for Special Use and from 1955 as the 635th Anti-Aircraft Missile Regiment for Special Use.

1953 to 1984 equipped with 60 S-25 (SA-1) launchers:

• Launch area: 55 15 43N, 38 32 13E (US designation: Moscow SAM site E14-1)
• Support area: 55 16 50N, 38 32 28E
• Guidance area: 55 16 31N, 38 30 38E

1984 converted to the S-300PT (SA-10) with three independent battalions:

• 1st independent Anti-Aircraft Missile Battalion (Bessonovo, Moscow Oblast) - 55 09 34N, 38 22 26E
• 2nd independent Anti-Aircraft Missile Battalion and HQ (Stepanshchino, Moscow Oblast) - 55 15 31N, 38 32 23E
• 3rd independent Anti-Aircraft Missile Battalion (Shcherbovo, Moscow Oblast) - 55 22 32N, 38 43 33E

Disbanded 1.5.98.

Subordination:

• 1st Special Air Defence Corps , 1953 - 1.6.88
• 86th Air Defence Division , 1.6.88 - 1.10.94
• 86th Air Defence Brigade , 1.10.94 - 1.10.95
• 86th Air Defence Division , 1.10.95 - 1.5.98

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Out of the Centre

Savvino-storozhevsky monastery and museum.

Zvenigorod's most famous sight is the Savvino-Storozhevsky Monastery, which was founded in 1398 by the monk Savva from the Troitse-Sergieva Lavra, at the invitation and with the support of Prince Yury Dmitrievich of Zvenigorod. Savva was later canonised as St Sabbas (Savva) of Storozhev. The monastery late flourished under the reign of Tsar Alexis, who chose the monastery as his family church and often went on pilgrimage there and made lots of donations to it. Most of the monastery’s buildings date from this time. The monastery is heavily fortified with thick walls and six towers, the most impressive of which is the Krasny Tower which also serves as the eastern entrance. The monastery was closed in 1918 and only reopened in 1995. In 1998 Patriarch Alexius II took part in a service to return the relics of St Sabbas to the monastery. Today the monastery has the status of a stauropegic monastery, which is second in status to a lavra. In addition to being a working monastery, it also holds the Zvenigorod Historical, Architectural and Art Museum.

Belfry and Neighbouring Churches

Located near the main entrance is the monastery's belfry which is perhaps the calling card of the monastery due to its uniqueness. It was built in the 1650s and the St Sergius of Radonezh’s Church was opened on the middle tier in the mid-17th century, although it was originally dedicated to the Trinity. The belfry's 35-tonne Great Bladgovestny Bell fell in 1941 and was only restored and returned in 2003. Attached to the belfry is a large refectory and the Transfiguration Church, both of which were built on the orders of Tsar Alexis in the 1650s.  

To the left of the belfry is another, smaller, refectory which is attached to the Trinity Gate-Church, which was also constructed in the 1650s on the orders of Tsar Alexis who made it his own family church. The church is elaborately decorated with colourful trims and underneath the archway is a beautiful 19th century fresco.

Nativity of Virgin Mary Cathedral

The Nativity of Virgin Mary Cathedral is the oldest building in the monastery and among the oldest buildings in the Moscow Region. It was built between 1404 and 1405 during the lifetime of St Sabbas and using the funds of Prince Yury of Zvenigorod. The white-stone cathedral is a standard four-pillar design with a single golden dome. After the death of St Sabbas he was interred in the cathedral and a new altar dedicated to him was added.

Under the reign of Tsar Alexis the cathedral was decorated with frescoes by Stepan Ryazanets, some of which remain today. Tsar Alexis also presented the cathedral with a five-tier iconostasis, the top row of icons have been preserved.

Tsaritsa's Chambers

The Nativity of Virgin Mary Cathedral is located between the Tsaritsa's Chambers of the left and the Palace of Tsar Alexis on the right. The Tsaritsa's Chambers were built in the mid-17th century for the wife of Tsar Alexey - Tsaritsa Maria Ilinichna Miloskavskaya. The design of the building is influenced by the ancient Russian architectural style. Is prettier than the Tsar's chambers opposite, being red in colour with elaborately decorated window frames and entrance.

At present the Tsaritsa's Chambers houses the Zvenigorod Historical, Architectural and Art Museum. Among its displays is an accurate recreation of the interior of a noble lady's chambers including furniture, decorations and a decorated tiled oven, and an exhibition on the history of Zvenigorod and the monastery.

Palace of Tsar Alexis

The Palace of Tsar Alexis was built in the 1650s and is now one of the best surviving examples of non-religious architecture of that era. It was built especially for Tsar Alexis who often visited the monastery on religious pilgrimages. Its most striking feature is its pretty row of nine chimney spouts which resemble towers.

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The Unique Burial of a Child of Early Scythian Time at the Cemetery of Saryg-Bulun (Tuva)

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Pages:  379-406

In 1988, the Tuvan Archaeological Expedition (led by M. E. Kilunovskaya and V. A. Semenov) discovered a unique burial of the early Iron Age at Saryg-Bulun in Central Tuva. There are two burial mounds of the Aldy-Bel culture dated by 7th century BC. Within the barrows, which adjoined one another, forming a figure-of-eight, there were discovered 7 burials, from which a representative collection of artifacts was recovered. Burial 5 was the most unique, it was found in a coffin made of a larch trunk, with a tightly closed lid. Due to the preservative properties of larch and lack of air access, the coffin contained a well-preserved mummy of a child with an accompanying set of grave goods. The interred individual retained the skin on his face and had a leather headdress painted with red pigment and a coat, sewn from jerboa fur. The coat was belted with a leather belt with bronze ornaments and buckles. Besides that, a leather quiver with arrows with the shafts decorated with painted ornaments, fully preserved battle pick and a bow were buried in the coffin. Unexpectedly, the full-genomic analysis, showed that the individual was female. This fact opens a new aspect in the study of the social history of the Scythian society and perhaps brings us back to the myth of the Amazons, discussed by Herodotus. Of course, this discovery is unique in its preservation for the Scythian culture of Tuva and requires careful study and conservation.

Keywords: Tuva, Early Iron Age, early Scythian period, Aldy-Bel culture, barrow, burial in the coffin, mummy, full genome sequencing, aDNA

Information about authors: Marina Kilunovskaya (Saint Petersburg, Russian Federation). Candidate of Historical Sciences. Institute for the History of Material Culture of the Russian Academy of Sciences. Dvortsovaya Emb., 18, Saint Petersburg, 191186, Russian Federation E-mail: [email protected] Vladimir Semenov (Saint Petersburg, Russian Federation). Candidate of Historical Sciences. Institute for the History of Material Culture of the Russian Academy of Sciences. Dvortsovaya Emb., 18, Saint Petersburg, 191186, Russian Federation E-mail: [email protected] Varvara Busova  (Moscow, Russian Federation).  (Saint Petersburg, Russian Federation). Institute for the History of Material Culture of the Russian Academy of Sciences.  Dvortsovaya Emb., 18, Saint Petersburg, 191186, Russian Federation E-mail:  [email protected] Kharis Mustafin  (Moscow, Russian Federation). Candidate of Technical Sciences. Moscow Institute of Physics and Technology.  Institutsky Lane, 9, Dolgoprudny, 141701, Moscow Oblast, Russian Federation E-mail:  [email protected] Irina Alborova  (Moscow, Russian Federation). Candidate of Biological Sciences. Moscow Institute of Physics and Technology.  Institutsky Lane, 9, Dolgoprudny, 141701, Moscow Oblast, Russian Federation E-mail:  [email protected] Alina Matzvai  (Moscow, Russian Federation). Moscow Institute of Physics and Technology.  Institutsky Lane, 9, Dolgoprudny, 141701, Moscow Oblast, Russian Federation E-mail:  [email protected]

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