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An A-to-Z History of Mathematics

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Mathematics is the science of numbers. To be precise, the Merriam-Webster dictionary defines mathematics as:

The science of numbers and their operations, interrelations, combinations, generalizations, abstractions and of space configurations and their structure, measurement, transformations and generalizations.

There are several different branches of mathematical science, which include algebra, geometry and calculus.

Mathematics is not an invention . Discoveries and laws of science are not considered inventions since inventions are material things and processes. However, there is a history of mathematics, a relationship between mathematics and inventions and mathematical instruments themselves are considered inventions.

According to the book "Mathematical Thought from Ancient to Modern Times," mathematics as an organized science did not exist until the classical Greek period from 600 to 300 B.C. There were, however, prior civilizations in which the beginnings or rudiments of mathematics were formed.

For example, when civilization began to trade, a need to count was created. When humans traded goods, they needed a way to count the goods and to calculate the cost of those goods. The very first device for counting numbers was, of course, the human hand and fingers represented quantities. And to count beyond ten fingers, mankind used natural markers, rocks or shells. From that point, tools such as counting boards and the abacus were invented. 

Here's a quick tally of important developments introduced throughout the ages, beginning from A to Z. 

One of the first tools for counting invented, the abacus was invented around 1200 B.C. in China and was used in many ancient civilizations, including Persia and Egypt.

The innovative Italians of the Renaissance (14th through 16th century) are widely acknowledged to be the fathers of modern accounting .

The first treatise on algebra was written by Diophantus of Alexandria in the 3rd century B.C. Algebra comes from the Arabic word al-jabr, an ancient medical term meaning "the reunion of broken parts." Al-Khawarizmi is another early algebra scholar and was the first to teach the formal discipline.

Archimedes was a mathematician and inventor from ancient Greece best known for his discovery of the relationship between the surface and volume of a sphere and its circumscribing cylinder for his formulation of a hydrostatic principle (Archimedes' principle) and for inventing the Archimedes screw (a device for raising water).

Differential

Gottfried Wilhelm Leibniz (1646-1716) was a German philosopher, mathematician and logician who is probably most well known for having invented differential and integral calculus. He did this independently of Sir Isaac Newton .

A graph is a pictorial representation of statistical data or of a functional relationship between variables. William Playfair (1759-1823) is generally viewed as the inventor of most graphical forms used to display data, including line plots, the bar chart, and the pie chart.

Math Symbol

In 1557, the "=" sign was first used by Robert Record. In 1631, came the  ">" sign.

Pythagoreanism

Pythagoreanism is a school of philosophy and a religious brotherhood believed to have been founded by Pythagoras of Samos, who settled in Croton in southern Italy about 525 B.C. The group had a profound effect on the development of mathematics.

The simple protractor is an ancient device. As an instrument used to construct and measure plane angles, the simple protractor looks like a semicircular disk marked with degrees, beginning with 0º to 180º.

The first complex protractor was created for plotting the position of a boat on navigational charts. Called a three-arm protractor or station pointer, it was invented in 1801 by Joseph Huddart, a U.S. naval captain. The center arm is fixed, while the outer two are rotatable and capable of being set at any angle relative to the center one.

Slide Rulers

Circular and rectangular slide rules, an instrument used for mathematical calculations, were both invented by mathematician William Oughtred .

Zero was invented by the Hindu mathematicians Aryabhata and Varamihara in India around or shortly after the year 520 A.D.

• Popular Math Terms and Definitions
• The History of Algebra
• What Did Archimedes Invent?
• The History of Computers
• Women in Mathematics History
• 17th Century Timeline, 1600 Through 1699
• Biography of Blaise Pascal, 17th Century Inventor of the Calculator
• The Life of Pythagoras
• What Is Calculus? Definition and Practical Applications
• Algebra Definition
• Biography of Joseph Louis Lagrange, Mathematician
• The First Computer
• John Napier - Napier's Bones
• Algorithms in Mathematics and Beyond
• Biography of Sophie Germain, Mathematical Pioneer Woman
• Biography of Christiaan Huygens, Prolific Scientist

The Role of the History of Mathematics in the Teaching/Learning Process

A CIEAEM Sourcebook

• © 2023
• Sixto Romero Sanchez   ORCID: https://orcid.org/0000-0002-0673-6196 0 ,
• Ana Serradó Bayés 1 ,
• Peter Appelbaum 2 ,
• Gilles Aldon   ORCID: https://orcid.org/0000-0001-5712-4977 3

University of Huelva, Huelva, Spain

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La Salle-Buen Consejo, Cadiz, Spain

School of education, arcadia university, philadelphia, usa, institut francais de l’éducation, université lyon, lyon cedex 07, france.

• Offers new ways of thinking for mathematics teachers and teacher educators
• Brings mathematics histories into relation with learning mathematics
• Includes contemporary history of technology as central to history of mathematics

Part of the book series: Advances in Mathematics Education (AME)

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About this book

This volume presents multiple perspectives on the uses of the history of mathematics for teaching and learning, including the value of historical topics in challenging mathematics tasks, for provoking teachers’ reflection on the nature of mathematics, curriculum development questions that mirror earlier pedagogical choices in the history of mathematics education, and the history of technological innovations in the teaching and learning of mathematics. An ethnomathematical perspective on the history of mathematics challenges readers to appreciate the role of mathematics in perpetuating consequences of colonialism. Histories of the textbook and its uses offer interesting insights into how technology has changed the fundamental role of curriculum materials and classroom pedagogies. History is explored as a source for the training of teachers, for good puzzles and problems, and for a broad understanding of mathematics education policy.

Third in a series of sourcebooks fromthe International Commission for the Study and Improvement of Mathematics Teaching, this collection of cutting-edge research, stories from the field, and policy implications is a contemporary and global perspective on current possibilities for the history of mathematics for mathematics education. This latest volume integrates discussions regarding history of mathematics, history of mathematics education and history of technology for education that have taken place at the Commission's recent annual conferences.

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The Philosophy of Mathematics Education: Stephen Lerman’s Contributions

• History of Mathematics
• Teaching mathematics
• history of mathematics education
• Improvement of Mathematics Teaching
• technology in mathematics education

Table of contents (16 chapters)

Front matter, the exploration of inaugural understandings in the history of mathematics and its potential for didactic and pedagogical reflection.

• David Guillemette

The Value of Historical Knowledge Through Challenging Mathematical Tasks

• Luís Menezes, Ana Maria Costa

An Historic Approach to Modelling: Enriching High School Student’s Capacities

• Sixto Romero Sánchez

The Introduction of the Algebraic Thought in Spain: The Resolution of the Second Degree Equation

• María José Madrid, Carmen León-Mantero, Alexander Maz-Machado

Mathematics Education in Different Times and Cultures

Integrating the history of mathematics in mathematics education: examples and reflections from the classroom.

• Sonia Kafoussi, Christina Margaritidou

Re-constructing the Image of Mathematics Through the Diversity of the Historical Journeys of Famous Mathematicians

• Fragkiskos Kalavasis, Andreas Moutsios-Rentzos

History of Ethnomathematics: Recent Developments

• Peter Appelbaum, Charoula Stathopoulou

Problems and Puzzles in History of Mathematics

• Pedro Palhares

The Potential in Teaching the History of Mathematics to Pre-service Secondary School Teachers

• Susan Gerofsky

The Role of History in Enriching Mathematics Teachers’ Training for Primary Education (6–12 Years Old Students)

• Yuly Vanegas, Joaquín Giménez, Montserrat Prat

Recent Trends of History of Mathematics Teacher Education: The Iberic American Tradition

• Joaquín Giménez, Javier Díez-Palomar

Tools and Technologies in a Sociocultural Approach of Learning Mathematical Modelling

• Fernando Hitt, José-Luis Soto-Munguía, José-Luis Lupiáñez-Gómez

Technology in Primary and Secondary School to Teach and Learn Mathematics in the Last Decades

• Giulia Bini, Monica Panero, Carlotta Soldano

A Trajectory of Digital Technologies Integration in Mathematical Education in Brazil: Challenges and Opportunities

• Maria Elisabette Brisola Brito Prado, Nielce Meneguelo Lobo da Costa, José Armando Valente

Editors and Affiliations

Sixto Romero Sanchez

Ana Serradó Bayés

Peter Appelbaum

Gilles Aldon

About the editors

Sixto Romero Sánchez is a past Vice President and current member of the International Commission for the Study and Improvement of Mathematics Education (CIEAEM) and the president of the Iberio-American Academy of La Rabida. He is the Editor in Chief of Review MSEL (Modeling in Science Education and Learning) and a member of the Advisory Council of the Union Magazine -FISEM-Ibero-American Federation of Mathematics Teachers Societies.

Ana Serradó Bayés is currently the Vice President of the International Commission for the Study and Improvement of Mathematics Education (CIEAEM), Vice President of the Education Commission of the Royal Spanish Mathematics Society (RSME) and a member of the Education Commission of the European Mathematical Society (EMS). She has an extensive record of publications in mathematics teacher professional development, statistics education, evaluation instruments, reading in mathematics classrooms, and technology integration in mathematics education.

Peter Appelbaum is a past Vice President of the International Commission for the Study and Improvement of Mathematics Education (CIEAEM) and Director of the Youth Mathematician Laureate Project. He has an extensive record of publications in curriculum theory, post-colonial and critical ethnomathematics education, and creative approaches to teaching and learning mathematics including, Embracing Mathematics: On Becoming a Teacher and Changing with Mathematics (Routledge, 2007), and Popular Culture, Educational Discourse, and Mathematics (SUNY Press, 1995).

Gilles Aldon is the current President of the International Commission for the Study and Improvement of Mathematics Education (CIEAEM), past Director of the EducTice interdisciplinary research team working on the use of technology in education, and former Assistant Director of IREM of Lyon (Research Institute on Mathematics Education). He is a faculty member in mathematics and mathematics education in the Ecole Normale Supérieure de Lyon, and has an extensive publication history on the uses of technology in mathematics education, assessment, and augmented reality in education. Professor Aldon is one of the editors of Mathematics and Technology: A C.I.E.A.E.M. Sourcebook (Springer, 2017).

Bibliographic Information

Book Title : The Role of the History of Mathematics in the Teaching/Learning Process

Book Subtitle : A CIEAEM Sourcebook

Editors : Sixto Romero Sanchez, Ana Serradó Bayés, Peter Appelbaum, Gilles Aldon

Series Title : Advances in Mathematics Education

DOI : https://doi.org/10.1007/978-3-031-29900-1

Publisher : Springer Cham

eBook Packages : Education , Education (R0)

Copyright Information : The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

Hardcover ISBN : 978-3-031-29899-8 Published: 16 June 2023

Softcover ISBN : 978-3-031-29902-5 Due: 17 July 2023

eBook ISBN : 978-3-031-29900-1 Published: 15 June 2023

Series ISSN : 1869-4918

Series E-ISSN : 1869-4926

Edition Number : 1

Number of Pages : XX, 460

Number of Illustrations : 1 b/w illustrations

Topics : Mathematics Education , Teaching and Teacher Education , History of Mathematical Sciences

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Intellectual Mathematics

How to write a history of mathematics essay

This is a guide for students writing a substantial course essay or bachelors thesis in the history of mathematics.

The essence of a good essay is that it shows independent and critical thought. You do not want to write yet another account of some topic that has already been covered many times before. Your goal should not be to write an encyclopaedia-style article that strings together various facts that one can find in standard sources. Your goal should not be to simply retell in your own words a story that has already been told many times before in various books. Such essays do not demonstrate thought, and therefore it is impossible to earn a good grade this way.

So you want to look for ways of framing your essay that give you opportunity for thought. The following is a basic taxonomy of some typical ways in which this can be done.

Critique. A good rule of thumb is: if you want a good grade you should, in your essay, disagree with and argue against at least one statement in the secondary literature. This is probably easier than you might think; errors and inaccuracies are very common, especially in general and popular books on the history of mathematics. When doing research for your essay, it is a good idea to focus on a small question and try to find out what many different secondary sources say about it. Once you have understood the topic well, you will most likely find that some of the weaker secondary sources are very superficial and quite possibly downright wrong. You want to make note of such shortcomings in the literature and cite and explain what is wrong about them in your essay, and why their errors are significant in terms of a proper understanding of the matter.

The point, of course, is not that finding errors in other people’s work is an end in itself. The point, rather, is that if you want to get anywhere in history it is essential to read all texts with a critical eye. It is therefore a good exercise to train yourself to look for errors in the literature, not because collecting errors is interesting in itself but because if you believe everything you read you will never get anywhere in this world, especially as far as history is concerned.

Maybe what you really wanted to do was simply to learn some nice things about the topic and write them up in your essay as a way of organising what you learned when reading about it. That is a fine goal, and certainly history is largely about satisfying our curiosities in this way. However, when it comes to grading it is difficult to tell whether you have truly thought something through and understood it, or whether you are simply paraphrasing someone else who has done so. Therefore such essays cannot generally earn a very good grade. But if you do this kind of work it will not be difficult for you to use the understanding you develop to find flaws in the secondary literature, and this will give a much more concrete demonstration of your understanding. So while developing your understanding was the true goal, critiquing other works will often be the best way to make your understanding evident to the person grading your essay.

For many examples of how one might write a critique, see my book reviews categorised as “critical.”

Debate. A simple way of putting yourself in a critical mindset is to engage with an existing debate in the secondary literature. There are many instances where historians disagree and offer competing interpretations, often in quite heated debates. Picking such a topic will steer you away from the temptation to simply accumulate information and facts. Instead you will be forced to critically weigh the evidence and the arguments on both sides. Probably you will find yourself on one side or the other, and it will hopefully come quite naturally to you to contribute your own argument for your favoured side and your own replies to the arguments of the opposing side.

Some sample “debate” topics are: Did Euclid know “algebra”? Did Copernicus secretly borrow from Islamic predecessors? “Myths” in the historiography of Egyptian mathematics? Was Galileo a product of his social context? How did Leibniz view the foundations of infinitesimals?

Compare & contrast. The compare & contrast essay is a less confrontational sibling of the debate essay. It too deals with divergent interpretations in the secondary literature, but instead of trying to “pick the winner” it celebrates the diversity of approaches. By thoughtfully comparing different points of view, it raises new questions and illuminates new angles that were not evident when each standpoint was considered in isolation. In this way, it brings out more clearly the strengths and weaknesses, and the assumptions and implications, of each point of view.

When you are writing a compare & contrast essay you are wearing two (or more) “hats.” One moment you empathise with one viewpoint, the next moment with the other. You play out a dialog in your mind: How would one side reply to the arguments and evidence that are key from the other point of view, and vice versa? What can the two learn from each other? In what ways, if any, are they irreconcilable? Can their differences be accounted for in terms of the authors’ motivations and goals, their social context, or some other way?

Following the compare & contrast model is a relatively straightforward recipe for generating reflections of your own. It is almost always applicable: all you need is two alternate accounts of the same historical development. It could be for instance two different mathematical interpretations, two perspectives emphasising different contexts, or two biographies of the same person.

The compare & contrast approach is therefore a great choice if you want to spend most of your research time reading and learning fairly broadly about a particular topic. Unlike the critique or debate approaches, which requires you to survey the literature for weak spots and zero in for pinpoint attacks, it allows you to take in and engage with the latest and best works of scholarship in a big-picture way. The potential danger, on the other hand, is that it may come dangerously close to merely survey or summarise the works of others. They way to avoid this danger is to always emphasise the dialog between the different points of view, rather than the views themselves. Nevertheless, if you are very ambitious you may want to complement a compare & contrast essay with elements of critique or debate.

Verify or disprove. People often appeal to history to justify certain conclusions. They give arguments of the form: “History works like this, so therefore [important conclusions].” Often such accounts allude briefly to specific historical examples without discussing them in any detail. Do the historical facts of the matter bear out the author’s point, or did he distort and misrepresent history to serve his own ends? Such a question is a good starting point for an essay. It leads you to focus your essay on a specific question and to structure your essay as an analytical argument. It also affords you ample opportunity for independent thought without unreasonable demands on originality: your own contribution lies not in new discoveries but in comparing established scholarly works from a new point of view. Thus it is similar to a compare & contrast essay, with the two works being compared being on the one hand the theoretical work making general claims about history, and on the other hand detailed studies of the historical episodes in question.

Sample topics of this type are: Are there revolutions in mathematics in the sense of Kuhn ? Or does mathematics work according to the model of Kitcher ? Or that of Lakatos or Crowe ? Does the historical development of mathematical concepts mirror the stages of the learning process of students learning the subject today, in the manner suggested by Sfard or Sierpinska ? Was Kant’s account of the nature of geometrical knowledge discredited by the discovery of non-Euclidean geometry?

Cross-section. Another way of combining existing scholarship in such a way as to afford scope for independent thought is to ask “cross-sectional” questions, such as comparing different approaches to a particular mathematical idea in different cultures or different time periods. Again, a compare & contrast type of analysis gives you the opportunity to show that you have engaged with the material at a deeper and more reflective level than merely recounting existing scholarship.

Dig. There are still many sources and issues in the history of mathematics that have yet to be investigated thoroughly by anyone. In such cases you can make valuable and original contributions without any of the above bells and whistles by simply being the first to really study something in depth. It is of course splendid if you can do this, but there are a number of downsides: (1) you will be studying something small and obscure, while the above approaches allow you to tackle any big and fascinating question you are interested in; (2) it often requires foreign language skills; (3) finding a suitable topic is hard, since you must locate an obscure work and master all the related secondary literature so that you can make a case that it has been insufficiently studied.

In practice you may need someone to do (3) for you. I have some suggestions which go with the themes of 17th-century mathematics covered in my history of mathematics book . It would be interesting to study for instance 18th-century calculus textbooks (see e.g. the bibliography in this paper ) in light of these issues, especially the conflict between geometric and analytic approaches. If you know Latin there are many more neglected works, such as the first book on integral calculus, Gabriele Manfredi’s De constructione aequationum differentialium primi gradus (1707), or Henry Savile’s Praelectiones tresdecim in principium Elementorum Euclidis , 1621, or many other works listed in a bibliography by Schüling .

Expose. A variant of the dig essay is to look into certain mathematical details and write a clear exposition of them. Since historical mathematics is often hard to read, being able to explain its essence in a clear and insightful way is often an accomplishment in itself that shows considerable independent thought. This shares some of the drawbacks of the dig essay, except it is much easier to find a topic, even an important one. History is full of important mathematics in need of clear exposition. But the reason for this points to another drawback of this essay type: it’s hard. You need to know your mathematics very well to pull this off, but the rewards are great if you do.

Whichever of the above approaches you take you want to make it very clear and explicit in your essay what parts of it reflect your own thinking and how your discussion goes beyond existing literature. If this is not completely clear from the essay itself, consider adding a note to the grader detailing these things. If you do not make it clear when something is your own contribution the grader will have to assume that it is not, which will not be good for your grade.

Here’s another way of looking at it. This table is a schematic overview of different ways in which your essay can add something to the literature:

The table shows the state of the literature before and after your research project has been carried out.

A Describe project starts from a chaos of isolated bits of information and analyses it so as to impose order and organisation on it. You are like an explorer going into unknown jungles. You find exotic, unknown things. You record the riches of this strange new world and start organise it into a systematic taxonomy.

You need an exotic “jungle” for this project to work. In the history of mathematics, this could mean obscure works or sources that have virtually never been studied, or mathematical arguments that have never been elucidated or explained in accessible form.

An Explain project is suitable when others have done the exploration and descriptions of fact, but left why-questions unanswered. First Darwin and other naturalists went to all the corners of the world and gathered and recorded all the exotic species they could find. That was the Describe phase. Darwin then used that mass of information to formulate and test his hypothesis of the origin of species. That was the Explain phase.

Many areas of the history of mathematics have been thoroughly Described but never Explained.

What if you find that someone has done the Explain already? If you think the Explain is incomplete, you can Critique it. If you think the Explain is great you can Extend it: do the same thing but to a different but similar body of data. That way you get to work with the stimulating work that appealed to you, but you also add something of your own.

Likewise if you find two or more Explains that are all above Critique in your opinion. Then you can do a Compare & Contrast, or a Synthesise. This way you get to work with the interesting works but also show your independent contribution by drawing out aspects and connections that were not prominent in the originals.

See also History of mathematics literature guide .

A Brief History of the Department

Early years.

Mathematics has played an important part at MIT since the founding of the Institute. In the early years of the Institute, the teaching of mathematics courses was overseen by John D. Runkle, president of MIT during the absence of William Barton Rogers. Runkle viewed mathematics as a "service subject" for engineers. Following Runkle, Harry W. Tyler headed the Mathematics Department until 1930 and fostered growth by hiring top mathematicians. Finally, in 1933, under Department Head Frederick Woods, Mathematics left Course IX, General Studies, and became its own entity as Course XVIII.

The Sputnik Years

The Mathematics Department grew into a top-ranked center for mathematical research under the leadership of Department Head William (Ted) Martin in the 1950s and 1960s. The book Recountings: Conversations with MIT Mathematicians (A K Peters, 2009) weaves together some of the storied history of the Mathematics Department through interviews with some of our faculty who were here during that period.

Notable Faculty

Clarence L. E. Moore first came to MIT in 1904 and mentored a generation of mathematicians, including Norbert Wiener. Wiener is known as the founder of cybernetics as well as for leading research in pure and applied mathematics, artificial intelligence, and computer science. Wiener's eccentricities are also a part of his legacy at MIT.

Norman Levinson, who received his PhD from MIT in 1935, was a student of Wiener's. Levinson initially was a student in electrical engineering but moved to mathematics and later served as head of the Department. His research included nonlinear differential equations and number theory.

Dirk Struik was a faculty member during this period of growth in the department. He joined the faculty in 1928 and described the 1930s as, "a lively time, and a time in which the mathematics department was greatly strengthened, due to new appointments, more than once from the ranks of excellent graduate students." His recollections can be found in A Century of Mathematics in America (AMS 1989).

Claude Shannon was another notable member of the Mathematics faculty. During World War II, Shannon developed information theory at Bell Telephone Laboratories. Upon returning to MIT in 1956, he was a professor in both the Mathematics and Electrical Engineering Departments. For his own amusement, Shannon built mechanical toys and games that can be found at the MIT Museum .

• Struik, Dirk J. " The MIT Mathematics Department During Its First Seventy-Five Years: Some Recollections ." A Century of Mathematics in America. Ed. Peter Duran. Providence: American Mathematical Society, 1989. 163-178.
• Wylie, Francis E. M.I.T. in Perspective: A Pictorial History of the Massachusetts Institute of Technology . Boston: Little, Brown & Company, 1975.
• Mannix, Loretta H., and Julius A. Stratton. Mind and Hand: The Birth of MIT . Cambridge: The MIT Press, 2005.

Home > College of Natural Sciences > Mathematics > Mathematics Theses, Projects, and Dissertations

Mathematics Theses, Projects, and Dissertations

Theses/projects/dissertations from 2024 2024.

On Cheeger Constants of Knots , Robert Lattimer

Information Based Approach for Detecting Change Points in Inverse Gaussian Model with Applications , Alexis Anne Wallace

Theses/Projects/Dissertations from 2023 2023

DNA SELF-ASSEMBLY OF TRAPEZOHEDRAL GRAPHS , Hytham Abdelkarim

An Exposition of the Curvature of Warped Product Manifolds , Angelina Bisson

Jackknife Empirical Likelihood Tests for Equality of Generalized Lorenz Curves , Anton Butenko

MATHEMATICS BEHIND MACHINE LEARNING , Rim Hammoud

Statistical Analysis of Health Habits for Incoming College Students , Wendy Isamara Lizarraga Noriega

Reverse Mathematics of Ramsey's Theorem , Nikolay Maslov

Distance Correlation Based Feature Selection in Random Forest , Jose Munoz-Lopez

Constructing Hyperbolic Polygons in the Poincaré Disk , Akram Zakaria Samweil

KNOT EQUIVALENCE , Jacob Trubey

Theses/Projects/Dissertations from 2022 2022

SYMMETRIC GENERATIONS AND AN ALGORITHM TO PROVE RELATIONS , Diddier Andrade

The Examination of the Arithmetic Surface (3, 5) Over Q , Rachel J. Arguelles

Error Terms for the Trapezoid, Midpoint, and Simpson's Rules , Jessica E. Coen

de Rham Cohomology, Homotopy Invariance and the Mayer-Vietoris Sequence , Stacey Elizabeth Cox

Symmetric Generation , Ana Gonzalez

SYMMETRIC PRESENTATIONS OF FINITE GROUPS AND RELATED TOPICS , Samar Mikhail Kasouha

Simple Groups and Related Topics , Simrandeep Kaur

Homomorphic Images and Related Topics , Alejandro Martinez

LATTICE REDUCTION ALGORITHMS , Juan Ortega

THE DECOMPOSITION OF THE SPACE OF ALGEBRAIC CURVATURE TENSORS , Katelyn Sage Risinger

Verifying Sudoku Puzzles , Chelsea Schweer

AN EXPOSITION OF ELLIPTIC CURVE CRYPTOGRAPHY , Travis Severns

Theses/Projects/Dissertations from 2021 2021

Non-Abelian Finite Simple Groups as Homomorphic Images , Sandra Bahena

Matroids Determinable by Two Partial Representations , Aurora Calderon Dojaquez

SYMMETRIC REPRESENTATIONS OF FINITE GROUPS AND RELATED TOPICS , Connie Corona

Symmetric Presentation of Finite Groups, and Related Topics , Marina Michelle Duchesne

MEASURE AND INTEGRATION , JeongHwan Lee

A Study in Applications of Continued Fractions , Karen Lynn Parrish

Partial Representations for Ternary Matroids , Ebony Perez

Theses/Projects/Dissertations from 2020 2020

Sum of Cubes of the First n Integers , Obiamaka L. Agu

Permutation and Monomial Progenitors , Crystal Diaz

Tile Based Self-Assembly of the Rook's Graph , Ernesto Gonzalez

Research In Short Term Actuarial Modeling , Elijah Howells

Hyperbolic Triangle Groups , Sergey Katykhin

Exploring Matroid Minors , Jonathan Lara Tejeda

DNA COMPLEXES OF ONE BOND-EDGE TYPE , Andrew Tyler Lavengood-Ryan

Modeling the Spread of Measles , Alexandria Le Beau

Symmetric Presentations and Related Topics , Mayra McGrath

Minimal Surfaces and The Weierstrass-Enneper Representation , Evan Snyder

ASSESSING STUDENT UNDERSTANDING WHILE SOLVING LINEAR EQUATIONS USING FLOWCHARTS AND ALGEBRAIC METHODS , Edima Umanah

Excluded minors for nearly-paving matroids , Vanessa Natalie Vega

Theses/Projects/Dissertations from 2019 2019

Fuchsian Groups , Bob Anaya

Tribonacci Convolution Triangle , Rosa Davila

VANISHING LOCAL SCALAR INVARIANTS ON GENERALIZED PLANE WAVE MANIFOLDS , Brian Matthew Friday

Analogues Between Leibniz's Harmonic Triangle and Pascal's Arithmetic Triangle , Lacey Taylor James

Geodesics on Generalized Plane Wave Manifolds , Moises Pena

Algebraic Methods for Proving Geometric Theorems , Lynn Redman

Pascal's Triangle, Pascal's Pyramid, and the Trinomial Triangle , Antonio Saucedo Jr.

THE EFFECTIVENESS OF DYNAMIC MATHEMATICAL SOFTWARE IN THE INSTRUCTION OF THE UNIT CIRCLE , Edward Simons

CALCULUS REMEDIATION AS AN INDICATOR FOR SUCCESS ON THE CALCULUS AP EXAM , Ty Stockham

Theses/Projects/Dissertations from 2018 2018

PROGENITORS, SYMMETRIC PRESENTATIONS AND CONSTRUCTIONS , Diana Aguirre

Monomial Progenitors and Related Topics , Madai Obaid Alnominy

Progenitors Involving Simple Groups , Nicholas R. Andujo

Simple Groups, Progenitors, and Related Topics , Angelica Baccari

Exploring Flag Matroids and Duality , Zachary Garcia

Images of Permutation and Monomial Progenitors , Shirley Marina Juan

MODERN CRYPTOGRAPHY , Samuel Lopez

Progenitors, Symmetric Presentations, and Related Topics , Joana Viridiana Luna

Symmetric Presentations, Representations, and Related Topics , Adam Manriquez

Toroidal Embeddings and Desingularization , LEON NGUYEN

THE STRUGGLE WITH INVERSE FUNCTIONS DOING AND UNDOING PROCESS , Jesus Nolasco

Tutte-Equivalent Matroids , Maria Margarita Rocha

Symmetric Presentations and Double Coset Enumeration , Charles Seager

MANUAL SYMMETRIC GENERATION , Joel Webster

Theses/Projects/Dissertations from 2017 2017

Investigation of Finite Groups Through Progenitors , Charles Baccari

CONSTRUCTION OF HOMOMORPHIC IMAGES , Erica Fernandez

Making Models with Bayes , Pilar Olid

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Mathematics > Representation Theory

Title: counting representations of quivers with multiplicities.

Abstract: In this thesis, we study counts of quiver representations over finite rings of truncated power series. We prove a plethystic formula relating counts of quiver representations over these rings and counts of jets on fibres of quiver moment maps. This solves a conjecture of Wyss and allows us to compute both counts on additional examples, using local zeta functions. The relation between counts of representations and counts of jets generalises the relation between Kac polynomials and counts of points on preprojective stacks. Pursuing this analogy, we establish further properties of our counts. We show that, for totally negative quivers, counts of jets converge to p-adic integrals on fibres of quiver moment maps. One expects a relation between these p-adic integrals and BPS invariants of preprojective algebras i.e. Kac polynomials. For small rank vectors, we also prove that the polynomials counting indecomposable quiver representations over finite rings have non-negative coefficients. Moreover, we show that jet schemes of fibres of quiver moment maps are cohomologically pure in that setting, so that their Poincaré polynomials are given by the former counts. This is reminiscent of the structure of cohomological Hall algebras, which are built from the cohomology of preprojective stacks. Finally, we compute the cohomology of jet spaces of preprojective stacks explicitly for the A2 quiver. Building on the structure of the preprojective cohomological Hall algebra of A2, we propose a candidate analogue of the BPS Lie algebra and conjecture the existence of a Hall product on the cohomology of these jet spaces.

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