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Mathematical modeling and problem solving: from fundamentals to applications

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• Published: 15 March 2024

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• Masahito Ohue 1 ,
• Kotoyu Sasayama 2 &
• Masami Takata 3  

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The rapidly advancing fields of machine learning and mathematical modeling, greatly enhanced by the recent growth in artificial intelligence, are the focus of this special issue. This issue compiles extensively revised and improved versions of the top papers from the workshop on Mathematical Modeling and Problem Solving at PDPTA'23, the 29th International Conference on Parallel and Distributed Processing Techniques and Applications. Covering fundamental research in matrix operations and heuristic searches to real-world applications in computer vision and drug discovery, the issue underscores the crucial role of supercomputing and parallel and distributed computing infrastructure in research. Featuring nine key studies, this issue pushes forward computational technologies in mathematical modeling, refines techniques for analyzing images and time-series data, and introduces new methods in pharmaceutical and materials science, making significant contributions to these areas.

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The field of machine learning and mathematical modeling is rapidly evolving, significantly impacting diverse research areas. The recent surge in artificial intelligence technologies has further accelerated this trend, highlighting the growing importance of “mathematical modeling and problem solving” in scientific endeavors [ 1 ]. Modeling natural phenomena and engineering systems not only deepens our understanding of fundamental principles but also drives the development of innovative technologies for effective control. These advancements have considerable implications for both industry and academia.

This special issue showcases the latest advancements in mathematical modeling and problem solving across various disciplines. The scope of topics is wide, encompassing everything from foundational research in new matrix operation methods, heuristic search, and constrained optimization techniques to practical research in computer vision, drug discovery, materials science, financial engineering, and mechanical processes.

A key aspect of contemporary mathematical modeling research is its integration with supercomputing, which involves extensive parallel and distributed computing. The sheer volume and augmented data often require rapid computational strategies. The infrastructure, including hardware and software, supporting parallel and distributed computing is thus vital for applied research. This issue includes a selection of research presented at the “Mathematical Modeling and Problem Solving” workshop during the 29th International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA’23). After a thorough selection process, nine significant studies were chosen as articles on this issue.

Four papers focus on computational technologies foundational to mathematical modeling. Chiyonobu and colleagues enhance the two-sided Jacobi method for singular value decomposition for complex matrices, previously effective only for real matrices [ 2 ]. They incorporate QR decomposition for complex matrix scenarios, offering two distinct implementations for both complex and real matrices. Zhong et al. introduce a novel hyper-heuristic algorithm, the evolutionary multi-mode slime mold optimization (EMSMO), inspired by slime mold behaviors [ 3 ]. This algorithm demonstrates superior performance in benchmarks and engineering problems, outperforming traditional evolutionary and hyper-heuristic algorithms. Zhang et al. unveil the meta-generative data augmentation optimization (MGDAO), a method that advances data augmentation in foundational machine learning for image and natural language processing [ 4 ]. This technique surpasses standard auto-augmentation methods in few-shot image and text classification benchmarks. Matsuzaki and colleagues propose a mixed-integer programming (MIP)-based method for scheduling machining operations in automated manufacturing, considering worker conditions [ 5 ]. They validate this method through computer experiments modeled on real-world machining tasks.

Two papers address applications involving image and time-series data, traditional targets of mathematical modeling. Ishikawa et al. enhance concrete crack detection by using strongly blurred images in training data, improving recognizer accuracy [ 6 ]. Takata et al. develop a method for recommending stock combinations by analyzing price change waveforms, showing potential for diversifying portfolios and minimizing risks [ 7 ].

Last but not least, three papers focus on pharmaceutical and materials science applications. Ueki and Ohue assess AlphaFold2 and binder hallucination techniques for improving antibody binding affinity, indicating a more efficient method than traditional experimental approaches [ 8 ]. Morikawa et al. introduce a machine learning method using graph kernels for predicting metal–organic frameworks (MOFs) combinations, demonstrating accurate MOF structure prediction without physical synthesis [ 9 ]. Furui and Ohue present an enhanced version of the lead optimization mapper (Lomap) algorithm for drug discovery [ 10 ]. This improved algorithm offers a faster approach to create free energy perturbation (FEP) graphs for numerous compounds, while maintaining the quality of the output.

In summary, this special issue represents a significant contribution to the fields of mathematical modeling and application, providing innovative methods to the community. As editors, we extend our gratitude to all researchers who contributed to this collection, paving the way for the next era of mathematical modeling and problem solving.

Yüksel N, Börklü HR, Sezer HK, Canyurt OE (2023) Review of artificial intelligence applications in engineering design perspective. Eng Appl Artif Intell 118:105697

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Chiyonobu M, Miyamae T, Takata M, Harayama J, Kimura K, Nakamura Y (2024) Singular value decomposition for complex matrices using two-sided Jacobi method. J Supercomput. https://doi.org/10.1007/s11227-024-05903-6

Zhong R, Zhang E, Munetomo M (2024) Evolutionary multi-mode slime mold optimization: a hyper-heuristic algorithm inspired by slime mold foraging behaviors. J Supercomput. https://doi.org/10.1007/s11227-024-05909-0

Zhang E, Dong B, Wahib M, Zhong R, Munetomo M (2024) Meta generative image and text data augmentation optimization. J Supercomput. https://doi.org/10.1007/s11227-024-05912-5

Matsuzaki J, Sakakibara K, Nakamura M, Watanabe S (2024) Large neighborhood local search method with MIP techniques for large-scale machining scheduling with many constraints. J Supercomput. https://doi.org/10.1007/s11227-024-05912-5

Ishikawa S, Chiyonobu M, Iida S, Takata M (2024) Improvement of recognition rate using data augmentation with blurred images. J Supercomput. https://doi.org/10.1007/s11227-024-05901-8

Takata M, Kidoguchi N, Chiyonobu M (2024) Stock recommendation methods for stability. J Supercomput. https://doi.org/10.1007/s11227-024-05902-7

Ueki T, Ohue M (2024) Antibody complementarity-determining region design using AlphaFold2 and DDG Predictor. J Supercomput. https://doi.org/10.1007/s11227-023-05887-9

Morikawa Y, Shin K, Ohshima H, Kubouchi M (2024) Prediction of specific surface area of metal–organic frameworks by graph kernels. J Supercomput. https://doi.org/10.1007/s11227-024-05914-3

Furui K, Ohue M (2024) FastLomap: faster lead optimization mapper algorithm for large-scale relative free energy perturbation. J Supercomput

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This study was partially supported by JSPS KAKENHI (23H04887) (M.O.).

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Kotoyu Sasayama

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Ten simple rules for tackling your first mathematical models: A guide for graduate students by graduate students

Roles Conceptualization, Investigation, Writing – original draft, Writing – review & editing

* E-mail: [email protected]

Affiliations Department of Biological Sciences, University of Toronto Scarborough, Toronto, Ontario, Canada, Department of Ecology and Evolution, University of Toronto, Toronto, Ontario, Canada

Affiliation Department of Ecology and Evolution, University of Toronto, Toronto, Ontario, Canada

Affiliation Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario, Canada

Affiliation Department of Biology, Memorial University of Newfoundland, St John’s, Newfoundland, Canada

• Korryn Bodner, 
• Chris Brimacombe, 
• Emily S. Chenery, 
• Ariel Greiner, 
• Anne M. McLeod, 
• Stephanie R. Penk, 
• Juan S. Vargas Soto

Published: January 14, 2021

• https://doi.org/10.1371/journal.pcbi.1008539
• Reader Comments

Citation: Bodner K, Brimacombe C, Chenery ES, Greiner A, McLeod AM, Penk SR, et al. (2021) Ten simple rules for tackling your first mathematical models: A guide for graduate students by graduate students. PLoS Comput Biol 17(1): e1008539. https://doi.org/10.1371/journal.pcbi.1008539

Editor: Scott Markel, Dassault Systemes BIOVIA, UNITED STATES

Copyright: © 2021 Bodner et al. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: The authors received no specific funding for this work.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Biologists spend their time studying the natural world, seeking to understand its various patterns and the processes that give rise to them. One way of furthering our understanding of natural phenomena is through laboratory or field experiments, examining the effects of changing one, or several, variables on a measured response. Alternatively, one may conduct an observational study, collecting field data and comparing a measured response along natural gradients. A third and complementary way of understanding natural phenomena is through mathematical models. In the life sciences, more scientists are incorporating these quantitative methods into their research. Given the vast utility of mathematical models, ranging from providing qualitative predictions to helping disentangle multiple causation (see Hurford [ 1 ] for a more complete list), their increased adoption is unsurprising. However, getting started with mathematical models may be quite daunting for those with traditional biological training, as in addition to understanding new terminology (e.g., “Jacobian matrix,” “Markov chain”), one may also have to adopt a different way of thinking and master a new set of skills.

Here, we present 10 simple rules for tackling your first mathematical models. While many of these rules are applicable to basic scientific research, our discussion relates explicitly to the process of model-building within ecological and epidemiological contexts using dynamical models. However, many of the suggestions outlined below generalize beyond these disciplines and are applicable to nondynamic models such as statistical models and machine-learning algorithms. As graduate students ourselves, we have created rules we wish we had internalized before beginning our model-building journey—a guide by graduate students, for graduate students—and we hope they prove insightful for anyone seeking to begin their own adventures in mathematical modelling.

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Boxes represent susceptible, infected, and recovered compartments, and directed arrows represent the flow of individuals between these compartments with the rate of flow being controlled by the contact rate, c , the probability of infection, γ , and the recovery rate, θ .

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Rule 1: Know your question

“All models are wrong, some are useful” is a common aphorism, generally attributed to statistician George Box, but determining which models are useful is dependent upon the question being asked. The practice of clearly defining a research question is often drilled into aspiring researchers in the context of selecting an appropriate research design, interpreting statistical results, or when outlining a research paper. Similarly, the practice of defining a clear research question is important for mathematical models as their results are only as interesting as the questions that motivate them [ 5 ]. The question defines the model’s main purpose and, in all cases, should extend past the goal of merely building a model for a system (the question can even answer whether a model is even necessary). Ultimately, the model should provide an answer to the research question that has been proposed.

When the research question is used to inform the purpose of the model, it also informs the model’s structure. Given that models can be modified in countless ways, providing a purpose to the model can highlight why certain aspects of reality were included in the structure while others were ignored [ 6 ]. For example, when deciding whether we should adopt a more realistic model (i.e., add more complexity), we can ask whether we are trying to inform general theory or whether we are trying to model a response in a specific system. For example, perhaps we are trying to predict how fast an epidemic will grow based on different age-dependent mixing patterns. In this case, we may wish to adapt our basic SIR model to have age-structured compartments if we suspect this factor is important for the disease dynamics. However, if we are exploring a different question, such as how stochasticity influences general SIR dynamics, the age-structured approach would likely be unnecessary. We suggest that one of the first steps in any modelling journey is to choose the processes most relevant to your question (i.e., your hypothesis) and the direct and indirect causal relationships among them: Are the relationships linear, nonlinear, additive, or multiplicative? This challenge can be aided with a good literature review. Depending on your model purpose, you may also need to spend extra time getting to know your system and/or the data before progressing forward. Indeed, the more background knowledge acquired when forming your research question, the more informed your decision-making when selecting the structure, parameters, and data for your model.

Rule 2: Define multiple appropriate models

Natural phenomena are complicated to study and often impossible to model in their entirety. We are often unsure about the variables or processes required to fully answer our research question(s). For example, we may not know how the possibility of reinfection influences the dynamics of a disease system. In cases such as these, our advice is to produce and sketch out a set of candidate models that consider alternative terms/variables which may be relevant for the phenomena under investigation. As in Fig 2 , we construct 2 models, one that includes the ability for recovered individuals to become infected again, and one that does not. When creating multiple models, our general objective may be to explore how different processes, inputs, or drivers affect an outcome of interest or it may be to find a model or models that best explain a given set of data for an outcome of interest. In our example, if the objective is to determine whether reinfection plays an important role in explaining the patterns of a disease, we can test our SIR candidate models using incidence data to determine which model receives the most empirical support. Here we consider our candidate models to be alternative hypotheses, where the candidate model with the least support is discarded. While our perspective of models as hypotheses is a view shared by researchers such as Hilborn and Mangel [ 7 ], and Penk and colleagues [ 8 ], please note that others such as Oreskes and colleagues [ 9 ] believe that models are not subject to proof and hence disagree with this notion. We encourage modellers who are interested in this debate to read the provided citations.

(A) A susceptible/infected/recovered model where individuals remain immune (gold) and (B) a susceptible/infected/recovered model where individuals can become susceptible again (blue). Arrows indicate the direction of movement between compartments, c is the contact rate, γ is the infection rate given contact, and θ is the recovery rate. The text below each conceptual model are the hypotheses ( H1 and H2 ) that represent the differences between these 2 SIR models.

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Finally, we recognize that time and resource constraints may limit the ability to build multiple models simultaneously; however, even writing down alternative models on paper can be helpful as you can always revisit them if your primary model does not perform as expected. Of course, some candidate models may not be feasible or relevant for your system, but by engaging in the activity of creating multiple models, you will likely have a broader perspective of the potential factors and processes that fundamentally shape your system.

Rule 3: Determine the skills you will need (and how to get them)

Equipping yourself with the necessary analytical tools that form the basis of all quantitative techniques is essential. As Darwin said, those that have knowledge of mathematics seem to be endowed with an extra sense [ 10 ], and having a background in calculus, linear algebra, and statistics can go a long way. Thus, make it a habit to set time for yourself to learn these mathematical skills, and do not treat all your methods like a black box. For instance, if you plan to use ODEs, consider brushing up on your calculus, e.g., using Stewart [ 11 ]. If you are working with a system of ODEs, also read up on linear algebra, e.g., using Poole [ 12 ]. Some universities also offer specialized math biology courses that combine topics from different math courses to teach the essentials of mathematical modelling. Taking these courses can help save time, and if they are not available, their syllabi can help focus your studying. Also note that while narrowing down a useful skillset in the early stages of model-building will likely spare you from some future headaches, as you progress in your project, it is inevitable that new skills will be required. Therefore, we advise you to check in at different stages of your modelling journey to assess the skills that would be most relevant for your next steps and how best to acquire them. Hopefully, these decisions can also be made with the help of your supervisor and/or a modelling mentor. Building these extra skills can at first seem daunting but think of it as an investment that will pay dividends in improving your future modelling work.

When first attempting to tackle a specific problem, find relevant research that accomplishes the same tasks and determine if you understand the processes and techniques that are used in that study. If you do, then you can implement similar techniques and methods, and perhaps introduce new methods. If not, then determine which tools you need to add to your toolbox. For instance, if the problem involves a system of ODEs (e.g., SIR models, see above), can you use existing symbolic software (e.g., Maple, Matlab, Mathematica) to determine the systems dynamics via a general solution, or is the complexity too great that you will need to create simulations to infer the dynamics? Figuring out questions like these is key to understanding what skills you will need to work with the model you develop. While there is a time and a place for involving collaborators to help facilitate methods that are beyond your current reach, we strongly advocate that you approach any potential collaborator only after you have gained some knowledge of the methods first. Understanding the methodology, or at least its foundation, is not only crucial for making a fruitful collaboration, but also important for your development as a scientist.

Rule 4: Do not reinvent the wheel

While we encourage a thorough understanding of the methods researchers employ, we simultaneously discourage unnecessary effort redoing work that has already been done. Preventing duplication can be ensured by a thorough review of the literature (but note that reproducing original model results can advance your knowledge of how a model functions and lead to new insights in the system). Often, we are working from established theory that provides an existing framework that can be applied to different systems. Adapting these frameworks can help advance your own research while also saving precious time. When digging through articles, bear in mind that most modelling frameworks are not system-specific. Do not be discouraged if you cannot immediately find a model in your field, as the perfect model for your question may have been applied in a different system or be published only as a conceptual model. These models are still useful! Also, do not be shy about reaching out to authors of models that you think may be applicable to your system. Finally, remember that you can be critical of what you find, as some models can be deceptively simple or involve assumptions that you are not comfortable making. You should not reinvent the wheel, but you can always strive to build a better one.

Rule 5: Study and apply good coding practices

The modelling process will inevitably require some degree of programming, and this can quickly become a challenge for some biologists. However, learning to program in languages commonly adopted by the scientific community (e.g., R, Python) can increase the transparency, accessibility, and reproducibility of your models. Even if you only wish to adopt preprogrammed models, you will likely still need to create code of your own that reads in data, applies functions from a collection of packages to analyze the data, and creates some visual output. Programming can be highly rewarding—you are creating something after all—but it can also be one of the most frustrating parts of your research. What follows are 3 suggestions to avoid some of the frustration.

Organization is key, both in your workflow and your written code. Take advantage of existing software and tools that facilitate keeping things organized. For example, computational notebooks like Jupyter notebooks or R-Markdown documents allow you to combine text, commands, and outputs in an easily readable and shareable format. Version control software like Git makes it simple to both keep track of changes as well as to safely explore different model variants via branches without worrying that the original model has been altered. Additionally, integrating with hosting services such as Github allows you to keep your changes safely stored in the cloud. For more details on learning to program, creating reproducible research, programming with Jupyter notebooks, and using Git and Github, see the 10 simple rules by Carey and Papin [ 13 ], Sandve and colleagues [ 14 ], Rule and colleagues [ 15 ], and Perez-Riverol and colleagues [ 16 ], respectively.

Comment your code and comment it well (see Fig 3 ). These comments can be the pseudocode you have written on paper prior to coding. Assume that when you revisit your code weeks, months, or years later, you will have forgotten most of what you did and why you did it. Good commenting can also help others read and use your code, making it a critical part of increasing scientific transparency. It is always good practice to write your comments before you write the code, explaining what the code should do. When coding a function, include a description of its inputs and outputs. We also encourage you to publish your commented model code in repositories such that they are easily accessible to others—not only to get useful feedback for yourself but to provide the modelling foundation for others to build on.

Two functionally identical codes in R [ 17 ] can look very different without comments (left) and with descriptive comments (right). Writing detailed comments will help you and others understand, adapt, and use your code.

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When writing long code, test portions of it separately. If you are writing code that will require a lot of processing power or memory to run, use a simple example first, both to estimate how long the project will take, and to avoid waiting 12 hours to see if it works. Additionally, when writing code, try to avoid too many packages and “tricks” as it can make your code more difficult to understand. Do not be afraid of writing 2 separate functions if it will make your code more intuitive. As with writing, your skill as a writer is not dependent on your ability to use big words, but instead about making sure your reader understands what you are trying to communicate.

Rule 6: Sweat the “right” small stuff

By “sweat the ‘right’ small stuff,” we mean considering the details and assumptions that can potentially make or break a mathematical model. A good start would be to ensure your model follows the rules of mass and energy conservation. In a closed system, mass and energy cannot be created nor destroyed, and thus, the left side of the mathematical equation must equal the right under all circumstances. For example, in Eq 2 , if the number of susceptible individuals decreases due to infection, we must include a negative term in this equation (− cγIS ) to indicate that loss and its conjugate (+ cγIS ) to the infected individuals equation, Eq 3 , to represent that gain. Similarly, units of all terms must also be balanced on both sides of the equation. For example, if we wish to add or subtract 2 values, we must ensure their units are equivalent (e.g., cannot add day −1 and year −1 ). Simple oversights in units can lead to major setbacks and create bizarre dynamics, so it is worth taking the time to ensure the units match up.

Modellers should also consider the fundamental boundary conditions of each parameter to determine if there are some values that are illogical. Logical constraints and boundaries can be developed for each parameter using prior knowledge and assumptions (e.g., Huntley [ 18 ]). For example, when considering an SIR model, there are 2 parameters that comprise the transmission rate—the contact rate, c , and the probability of infection given contact, γ . Using our intuition, we can establish some basic rules: (1) the contact rate cannot be negative; (2) the number of susceptible, infected, and recovered individuals cannot be below 0; and (3) the probability of infection given contact must fall between 0 and 1. Keeping these in mind as you test your model’s dynamics can alert you to problems in your model’s structure. Finally, simulating your model is an excellent method to obtain more reasonable bounds for inputs and parameters and ensure behavior is as expected. See Otto and Day [ 5 ] for more information on the “basic ingredients” of model-building.

Rule 7: Simulate, simulate, simulate

Even though there is a lot to be learned from analyzing simple models and their general solutions, modelling a complex world sometimes requires complex equations. Unfortunately, the cost of this complexity is often the loss of general solutions [ 19 ]. Instead, many biologists must calculate a numerical solution, an approximate solution, and simulate the dynamics of these models [ 20 ]. Simulations allow us to explore model behavior, given different structures, initial conditions, and parameters ( Fig 4 ). Importantly, they allow us to understand the dynamics of complex systems that may otherwise not be ethical, feasible, or economically viable to explore in natural systems [ 21 ].

Gold lines represent the SIR structure ( Fig 2A ) where lifelong immunity of individuals is inferred after infection, and blue lines represent an SIRS structure ( Fig 2B ) where immunity is lost over time. The solid lines represent model dynamics assuming a recovery rate ( θ ) of 0.05, while dotted lines represent dynamics assuming a recovery rate of 0.1. All model runs assume a transmission rate, cγ , of 0.2 and an immunity loss rate, ψ , of 0.01. By using simulations, we can explore how different processes and rates change the system’s dynamics and furthermore determine at what point in time these differences are detectable. SIR, Susceptible-Infected-Recovered; SIRS, Susceptible-Infected-Recovered-Susceptible.

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One common method of exploring the dynamics of complex systems is through sensitivity analysis (SA). We can use this simulation-based technique to ascertain how changes in parameters and initial conditions will influence the behavior of a system. For example, if simulated model outputs remain relatively similar despite large changes in a parameter value, we can expect the natural system represented by that model to be robust to similar perturbations. If instead, simulations are very sensitive to parameter values, we can expect the natural system to be sensitive to its variation. Here in Fig 4 , we can see that both SIR models are very sensitive to the recovery rate parameter ( θ ) suggesting that the natural system would also be sensitive to individuals’ recovery rates. We can therefore use SA to help inform which parameters are most important and to determine which are distinguishable (i.e., identifiable). Additionally, if observed system data are available, we can use SA to help us establish what are the reasonable boundaries for our initial conditions and parameters. When adopting SA, we can either vary parameters or initial conditions one at a time (local sensitivity) or preferably, vary multiple of them in tandem (global sensitivity). We recognize this topic may be overwhelming to those new to modelling so we recommend reading Marino and colleagues [ 22 ] and Saltelli and colleagues [ 23 ] for details on implementing different SA methods.

Simulations are also a useful tool for testing how accurately different model fitting approaches (e.g., Maximum Likelihood Estimation versus Bayesian Estimation) can recover parameters. Given that we know the parameter values for simulated model outputs (i.e., simulated data), we can properly evaluate the fitting procedures of methods when used on that simulated data. If your fitting approach cannot even recover simulated data with known parameters, it is highly unlikely your procedure will be successful given real, noisy data. If a procedure performs well under these conditions, try refitting your model to simulated data that more closely resembles your own dataset (i.e., imperfect data). If you know that there was limited sampling and/or imprecise tools used to collect your data, consider adding noise, reducing sample sizes, and adding temporal and spatial gaps to see if the fitting procedure continues to return reasonably correct estimates. Remember, even if your fitting procedures continue to perform well given these additional complexities, issues may still arise when fitting to empirical data. Models are approximations and consequently their simulations are imperfect representations of your measured outcome of interest. However, by evaluating procedures on perfectly known imperfect data, we are one step closer to having a fitting procedure that works for us even when it seems like our data are against us.

Rule 8: Expect model fitting to be a lengthy, arduous but creative task

Model fitting requires an understanding of both the assumptions and limitations of your model, as well as the specifics of the data to be used in the fitting. The latter can be challenging, particularly if you did not collect the data yourself, as there may be additional uncertainties regarding the sampling procedure, or the variables being measured. For example, the incidence data commonly adopted to fit SIR models often contain biases related to underreporting, selective reporting, and reporting delays [ 24 ]. Taking the time to understand the nuances of the data is critical to prevent mismatches between the model and the data. In a bad case, a mismatch may lead to a poor-fitting model. In the worst case, a model may appear well-fit, but will lead to incorrect inferences and predictions.

Model fitting, like all aspects of modelling, is easier with the appropriate set of skills (see Rule 2). In particular, being proficient at constructing and analyzing mathematical models does not mean you are prepared to fit them. Fitting models typically requires additional in-depth statistical knowledge related to the characteristics of probability distributions, deriving statistical moments, and selecting appropriate optimization procedures. Luckily, a substantial portion of this knowledge can be gleaned from textbooks and methods-based research articles. These resources can range from covering basic model fitting, such as determining an appropriate distribution for your data and constructing a likelihood for that distribution (e.g., Hilborn and Mangel [ 7 ]), to more advanced topics, such as accounting for uncertainties in parameters, inputs, and structures during model fitting (e.g., Dietze [ 25 ]). We find these sources among others (e.g., Hobbs and Hooten [ 26 ] for Bayesian methods; e.g., Adams and colleagues [ 27 ] for fitting noisy and sparse datasets; e.g., Sirén and colleagues [ 28 ] for fitting individual-based models; and Williams and Kendall [ 29 ] for multiobject optimization—to name a few) are not only useful when starting to fit your first models, but are also useful when switching from one technique or model to another.

After you have learned about your data and brushed up on your statistical knowledge, you may still run into issues when model fitting. If you are like us, you will have incomplete data, small sample sizes, and strange data idiosyncrasies that do not seem to be replicated anywhere else. At this point, we suggest you be explorative in the resources you use and accept that you may have to combine multiple techniques and/or data sources before it is feasible to achieve an adequate model fit (see Rosenbaum and colleagues [ 30 ] for parameter estimation with multiple datasets). Evaluating the strength of different techniques can be aided by using simulated data to test these techniques, while SA can be used to identify insensitive parameters which can often be ignored in the fitting process (see Rule 7).

Model accuracy is an important metric but “good” models are also precise (i.e., reliable). During model fitting, to make models more reliable, the uncertainties in their inputs, drivers, parameters, and structures, arising due to natural variability (i.e., aleatory uncertainty) or imperfect knowledge (i.e., epistemic uncertainty), should be identified, accounted for, and reduced where feasible [ 31 ]. Accounting for uncertainty may entail measurements of uncertainties being propagated through a model (a simple example being a confidence interval), while reducing uncertainty may require building new models or acquiring additional data that minimize the prioritized uncertainties (see Dietze [ 25 ] and Tsigkinopoulou and colleagues [ 32 ] for a more thorough review on the topic). Just remember that although the steps outlined in this rule may take a while to complete, when you do achieve a well-fitted reliable model, it is truly something to be celebrated.

Rule 9: Give yourself time (and then add more)

Experienced modellers know that it often takes considerable time to build a model and that even more time may be required when fitting to real data. However, the pervasive caricature of modelling as being “a few lines of code here and there” or “a couple of equations” can lead graduate students to hold unrealistic expectations of how long finishing a model may take (or when to consider a model “finished”). Given the multiple considerations that go into selecting and implementing models (see previous rules), it should be unsurprising that the modelling process may take weeks, months, or even years. Remembering that a published model is the final product of long and hard work may help reduce some of your time-based anxieties. In reality, the finished product is just the tip of the iceberg and often unseen is the set of failed or alternative models providing its foundation. Note that taking time early on to establish what is “good enough” given your objective, and to instill good modelling practices, such as developing multiple models, simulating your models, and creating well-documented code, can save you considerable time and stress.

Rule 10: Care about the process, not just the endpoint

As a graduate student, hours of labor coupled with relative inexperience may lead to an unwillingness to change to a new model later down the line. But being married to one model can restrict its efficacy, or worse, lead to incorrect conclusions. Early planning may mitigate some modelling problems, but many issues will only become apparent as time goes on. For example, perhaps model parameters cannot be estimated as you previously thought, or assumptions made during model formulation have since proven false. Modelling is a dynamic process, and some steps will need to be revisited many times as you correct, refine, and improve your model. It is also important to bear in mind that the process of model-building is worth the effort. The process of translating biological dynamics into mathematical equations typically forces us to question our assumptions, while a misspecified model often leads to novel insights. While we may wish we had the option to skip to a final finished product, in the words of Drake, “sometimes it’s the journey that teaches you a lot about your destination”.

There is no such thing as a failed model. With every new error message or wonky output, we learn something useful about modelling (mostly begrudgingly) and, if we are lucky, perhaps also about the study system. It is easy to cave in to the ever-present pressure to perform, but as graduate students, we are still learning. Luckily, you are likely surrounded by other graduate students, often facing similar challenges who can be an invaluable resource for learning and support. Finally, remember that it does not matter if this was your first or your 100th mathematical model, challenges will always present themselves. However, with practice and determination, you will become more skilled at overcoming them, allowing you to grow and take on even greater challenges.

Acknowledgments

We thank Marie-Josée Fortin, Martin Krkošek, Péter K. Molnár, Shawn Leroux, Carina Rauen Firkowski, Cole Brookson, Gracie F.Z. Wild, Cedric B. Hunter, and Philip E. Bourne for their helpful input on the manuscript.

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• 25. Dietze MC. Ecological forecasting. Princeton, NJ: Princeton University Press; 2017.
• 26. Hobbs NT, Hooten MB. Bayesian models: A statistical primer for ecologists. Princeton, NJ: Princeton University Press

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• Language: English
• Publisher: Princeton University Press
• Copyright year: 2007
• Audience: College/higher education;Professional and scholarly;
• Main content: 336
• Other: 31 halftones. 52 line illus. 5 tables.
• Keywords: Ordinary differential equation ; Initial condition ; Differential equation ; Equation ; Mathematics ; Proportionality (mathematics) ; Quantity ; Partial differential equation ; Year ; Linear differential equation ; Calculation ; Chaos theory ; Mathematician ; Convection ; Logistic function ; Nonlinear system ; Acceleration ; Probability ; Fishery ; Temperature ; Linear stability ; Billion years ; Periodic function ; Lorenz system ; Population dynamics ; Mathematical model ; Parameter ; Textbook ; Newton's laws of motion ; Newton's law of universal gravitation ; Population density ; Logistic map ; Age of the Earth ; Age of the universe ; Climate ; Depensation ; El Niño–Southern Oscillation ; Herbivore ; Irrational number ; Kepler's laws of planetary motion ; Natural frequency ; Payment ; Power law ; Spouse ; Exponential growth ; Prediction ; Fibonacci number ; Clockwise ; Climate model ; Trade winds ; Logarithmic spiral ; Earth ; Cartesian coordinate system ; Snowball Earth ; Coefficient ; Rule of 72 ; Time series ; Interest rate ; Carbon dioxide ; Radiocarbon dating ; Fibonacci ; Atmosphere of Earth ; Temperature gradient ; Iteration ; Eigenvalues and eigenvectors ; Insect ; Geometric series ; Elliptic orbit ; Approximation ; Thermocline
• Published: June 14, 2016
• ISBN: 9781400884056

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Topics in Mathematical Modeling is an introductory textbook on mathematical modeling. The book teaches how simple mathematics can help formulate and solve real problems of current research interest in a wide range of fields, including biology, ecology, computer science, geophysics, engineering, and the social sciences. Yet the prerequisites are minimal: calculus and elementary differential equations. Among the many topics addressed are HIV; plant phyllotaxis; global warming; the World Wide Web; plant and animal vascular networks; social networks; chaos and fractals; marriage and divorce; and El Niño. Traditional modeling topics such as predator-prey interaction, harvesting, and wars of attrition are also included. Most chapters begin with the history of a problem, follow with a demonstration of how it can be modeled using various mathematical tools, and close with a discussion of its remaining unsolved aspects. Designed for a one-semester course, the book progresses from problems that can be solved with relatively simple mathematics to ones that require more sophisticated methods. The math techniques are taught as needed to solve the problem being addressed, and each chapter is designed to be largely independent to give teachers flexibility. The book, which can be used as an overview and introduction to applied mathematics, is particularly suitable for sophomore, junior, and senior students in math, science, and engineering.

"This beautifully produced book should provide a joyful and stimulating reading experience for any layman who is curious about real-life events in the context of mathematical modelling, and it provides an excellent entry point to more advanced areas such as mathematical biology or climate modelling."—Z. Q. John Lu, Significance

"What do global warming, predator-prey interactions, and the World Wide Web have in common? All of these disparate phenomena can be modeled using mathematics. In Topics in Mathematical Modeling , K. K. Tung demonstrates math¹s relevance to problems of current research interest in biology, ecology, computer science, geophysics, engineering, and the social sciences."— Scientific American Book Club

"[T]his is a good introductory book about the nature and purpose of mathematical modeling. The topics chosen and the way in which they have been motivated and presented will help a wide range of students to 'see the point' and thereby arouse and stimulate their confidence about their mathematical problem solving skills."—Bob Anderssen, Australian Mathematics Society

"I was so impressed by the breadth of examples contained in its 336 pages that I immediately set about using it to update one of my own undergraduate courses. . . . A wonderful source book for all kinds of undergraduate mathematical activities. . . . Extremely clear. . . . It is highly recommended."—Chris Howls, Times Higher Education

"Tung's preface shows that he is a dyed-in-the-wool teacher of considerable talent whose only mission is to show the student how to take raw empirical data and turn it into a mathematical paradigm that can be analyzed. His prerequisites are solid but minimal: calculus and a smattering of ordinary differential equations (ODEs). He is wise to provide an appendix with a quick treatment of ODEs for those whose background is deficient. Tung also describes in the preface a clear path for those who wish to avoid the differential equations altogether. Tung covers some of the usual modeling topics but also many others that are surprising and refreshing."—Steven G. Krantz, UMAP Journal

"This book has a refreshing style that should appeal to undergraduates. Indeed, the author has produced a textbook that might well achieve his goal of teaching applied mathematics without those being taught noticing!"—Andrew Wathen, University of Oxford

"With courses in mathematical modeling getting ever more popular, this book will make a valuable addition to the subject. It deals with topics that should be appealing even to students not majoring in math or science, and the level of mathematical sophistication is carefully increased throughout the book."—Henrik Kalisch, University of Bergen, Norway

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251+ Math Research Topics [2024 Updated]

Mathematics, often dubbed as the language of the universe, holds immense significance in shaping our understanding of the world around us. It’s not just about crunching numbers or solving equations; it’s about unraveling mysteries, making predictions, and creating innovative solutions to complex problems. In this blog, we embark on a journey into the realm of math research topics, exploring various branches of mathematics and their real-world applications.

How Do You Write A Math Research Topic?

Writing a math research topic involves several steps to ensure clarity, relevance, and feasibility. Here’s a guide to help you craft a compelling math research topic:

• Identify Your Interests: Start by exploring areas of mathematics that interest you. Whether it’s pure mathematics, applied mathematics, or interdisciplinary topics, choose a field that aligns with your passion and expertise.
• Narrow Down Your Focus: Mathematics is a broad field, so it’s essential to narrow down your focus to a specific area or problem. Consider the scope of your research and choose a topic that is manageable within your resources and time frame.
• Review Existing Literature: Conduct a thorough literature review to understand the current state of research in your chosen area. Identify gaps, controversies, or unanswered questions that could form the basis of your research topic.
• Formulate a Research Question: Based on your exploration and literature review, formulate a clear and concise research question. Your research question should be specific, measurable, achievable, relevant, and time-bound (SMART).
• Consider Feasibility: Assess the feasibility of your research topic in terms of available resources, data availability, and research methodologies. Ensure that your topic is realistic and achievable within the constraints of your project.
• Consult with Experts: Seek feedback from mentors, advisors, or experts in the field to validate your research topic and refine your ideas. Their insights can help you identify potential challenges and opportunities for improvement.
• Refine and Iterate: Refine your research topic based on feedback and further reflection. Iterate on your ideas to ensure clarity, coherence, and relevance to the broader context of mathematics research.
• Craft a Title: Once you have finalized your research topic, craft a compelling title that succinctly summarizes the essence of your research. Your title should be descriptive, engaging, and reflective of the key themes of your study.
• Write a Research Proposal: Develop a comprehensive research proposal outlining the background, objectives, methodology, and expected outcomes of your research. Your research proposal should provide a clear roadmap for your study and justify the significance of your research topic.

By following these steps, you can effectively write a math research topic that is well-defined, relevant, and poised to make a meaningful contribution to the field of mathematics.

251+ Math Research Topics: Beginners To Advanced

• Prime Number Distribution in Arithmetic Progressions
• Diophantine Equations and their Solutions
• Applications of Modular Arithmetic in Cryptography
• The Riemann Hypothesis and its Implications
• Graph Theory: Exploring Connectivity and Coloring Problems
• Knot Theory: Unraveling the Mathematics of Knots and Links
• Fractal Geometry: Understanding Self-Similarity and Dimensionality
• Differential Equations: Modeling Physical Phenomena and Dynamical Systems
• Chaos Theory: Investigating Deterministic Chaos and Strange Attractors
• Combinatorial Optimization: Algorithms for Solving Optimization Problems
• Computational Complexity: Analyzing the Complexity of Algorithms
• Game Theory: Mathematical Models of Strategic Interactions
• Number Theory: Exploring Properties of Integers and Primes
• Algebraic Topology: Studying Topological Invariants and Homotopy Theory
• Analytic Number Theory: Investigating Properties of Prime Numbers
• Algebraic Geometry: Geometry Arising from Algebraic Equations
• Galois Theory: Understanding Field Extensions and Solvability of Equations
• Representation Theory: Studying Symmetry in Linear Spaces
• Harmonic Analysis: Analyzing Functions on Groups and Manifolds
• Mathematical Logic: Foundations of Mathematics and Formal Systems
• Set Theory: Exploring Infinite Sets and Cardinal Numbers
• Real Analysis: Rigorous Study of Real Numbers and Functions
• Complex Analysis: Analytic Functions and Complex Integration
• Measure Theory: Foundations of Lebesgue Integration and Probability
• Topological Groups: Investigating Topological Structures on Groups
• Lie Groups and Lie Algebras: Geometry of Continuous Symmetry
• Differential Geometry: Curvature and Topology of Smooth Manifolds
• Algebraic Combinatorics: Enumerative and Algebraic Aspects of Combinatorics
• Ramsey Theory: Investigating Structure in Large Discrete Structures
• Analytic Geometry: Studying Geometry Using Analytic Methods
• Hyperbolic Geometry: Non-Euclidean Geometry of Curved Spaces
• Nonlinear Dynamics: Chaos, Bifurcations, and Strange Attractors
• Homological Algebra: Studying Homology and Cohomology of Algebraic Structures
• Topological Vector Spaces: Vector Spaces with Topological Structure
• Representation Theory of Finite Groups: Decomposition of Group Representations
• Category Theory: Abstract Structures and Universal Properties
• Operator Theory: Spectral Theory and Functional Analysis of Operators
• Algebraic Number Theory: Study of Algebraic Structures in Number Fields
• Cryptanalysis: Breaking Cryptographic Systems Using Mathematical Methods
• Discrete Mathematics: Combinatorics, Graph Theory, and Number Theory
• Mathematical Biology: Modeling Biological Systems Using Mathematical Tools
• Population Dynamics: Mathematical Models of Population Growth and Interaction
• Epidemiology: Mathematical Modeling of Disease Spread and Control
• Mathematical Ecology: Dynamics of Ecological Systems and Food Webs
• Evolutionary Game Theory: Evolutionary Dynamics and Strategic Behavior
• Mathematical Neuroscience: Modeling Brain Dynamics and Neural Networks
• Mathematical Physics: Mathematical Models in Physical Sciences
• Quantum Mechanics: Foundations and Applications of Quantum Theory
• Statistical Mechanics: Statistical Methods in Physics and Thermodynamics
• Fluid Dynamics: Modeling Flow of Fluids Using Partial Differential Equations
• Mathematical Finance: Stochastic Models in Finance and Risk Management
• Option Pricing Models: Black-Scholes Model and Beyond
• Portfolio Optimization: Maximizing Returns and Minimizing Risk
• Stochastic Calculus: Calculus of Stochastic Processes and Itô Calculus
• Financial Time Series Analysis: Modeling and Forecasting Financial Data
• Operations Research: Optimization of Decision-Making Processes
• Linear Programming: Optimization Problems with Linear Constraints
• Integer Programming: Optimization Problems with Integer Solutions
• Network Flow Optimization: Modeling and Solving Flow Network Problems
• Combinatorial Game Theory: Analysis of Games with Perfect Information
• Algorithmic Game Theory: Computational Aspects of Game-Theoretic Problems
• Fair Division: Methods for Fairly Allocating Resources Among Parties
• Auction Theory: Modeling Auction Mechanisms and Bidding Strategies
• Voting Theory: Mathematical Models of Voting Systems and Social Choice
• Social Network Analysis: Mathematical Analysis of Social Networks
• Algorithm Analysis: Complexity Analysis of Algorithms and Data Structures
• Machine Learning: Statistical Learning Algorithms and Data Mining
• Deep Learning: Neural Network Models with Multiple Layers
• Reinforcement Learning: Learning by Interaction and Feedback
• Natural Language Processing: Statistical and Computational Analysis of Language
• Computer Vision: Mathematical Models for Image Analysis and Recognition
• Computational Geometry: Algorithms for Geometric Problems
• Symbolic Computation: Manipulation of Mathematical Expressions
• Numerical Analysis: Algorithms for Solving Numerical Problems
• Finite Element Method: Numerical Solution of Partial Differential Equations
• Monte Carlo Methods: Statistical Simulation Techniques
• High-Performance Computing: Parallel and Distributed Computing Techniques
• Quantum Computing: Quantum Algorithms and Quantum Information Theory
• Quantum Information Theory: Study of Quantum Communication and Computation
• Quantum Error Correction: Methods for Protecting Quantum Information from Errors
• Topological Quantum Computing: Using Topological Properties for Quantum Computation
• Quantum Algorithms: Efficient Algorithms for Quantum Computers
• Quantum Cryptography: Secure Communication Using Quantum Key Distribution
• Topological Data Analysis: Analyzing Shape and Structure of Data Sets
• Persistent Homology: Topological Invariants for Data Analysis
• Mapper Algorithm: Method for Visualization and Analysis of High-Dimensional Data
• Algebraic Statistics: Statistical Methods Based on Algebraic Geometry
• Tropical Geometry: Geometric Methods for Studying Polynomial Equations
• Model Theory: Study of Mathematical Structures and Their Interpretations
• Descriptive Set Theory: Study of Borel and Analytic Sets
• Ergodic Theory: Study of Measure-Preserving Transformations
• Combinatorial Number Theory: Intersection of Combinatorics and Number Theory
• Additive Combinatorics: Study of Additive Properties of Sets
• Arithmetic Geometry: Interplay Between Number Theory and Algebraic Geometry
• Proof Theory: Study of Formal Proofs and Logical Inference
• Reverse Mathematics: Study of Logical Strength of Mathematical Theorems
• Nonstandard Analysis: Alternative Approach to Analysis Using Infinitesimals
• Computable Analysis: Study of Computable Functions and Real Numbers
• Graph Theory: Study of Graphs and Networks
• Random Graphs: Probabilistic Models of Graphs and Connectivity
• Spectral Graph Theory: Analysis of Graphs Using Eigenvalues and Eigenvectors
• Algebraic Graph Theory: Study of Algebraic Structures in Graphs
• Metric Geometry: Study of Geometric Structures Using Metrics
• Geometric Measure Theory: Study of Measures on Geometric Spaces
• Discrete Differential Geometry: Study of Differential Geometry on Discrete Spaces
• Algebraic Coding Theory: Study of Error-Correcting Codes
• Information Theory: Study of Information and Communication
• Coding Theory: Study of Error-Correcting Codes
• Cryptography: Study of Secure Communication and Encryption
• Finite Fields: Study of Fields with Finite Number of Elements
• Elliptic Curves: Study of Curves Defined by Cubic Equations
• Hyperelliptic Curves: Study of Curves Defined by Higher-Degree Equations
• Modular Forms: Analytic Functions with Certain Transformation Properties
• L-functions: Analytic Functions Associated with Number Theory
• Zeta Functions: Analytic Functions with Special Properties
• Analytic Number Theory: Study of Number Theoretic Functions Using Analysis
• Dirichlet Series: Analytic Functions Represented by Infinite Series
• Euler Products: Product Representations of Analytic Functions
• Arithmetic Dynamics: Study of Iterative Processes on Algebraic Structures
• Dynamics of Rational Maps: Study of Dynamical Systems Defined by Rational Functions
• Julia Sets: Fractal Sets Associated with Dynamical Systems
• Mandelbrot Set: Fractal Set Associated with Iterations of Complex Quadratic Polynomials
• Arithmetic Geometry: Study of Algebraic Geometry Over Number Fields
• Diophantine Geometry: Study of Solutions of Diophantine Equations Using Geometry
• Arithmetic of Elliptic Curves: Study of Elliptic Curves Over Number Fields
• Rational Points on Curves: Study of Rational Solutions of Algebraic Equations
• Galois Representations: Study of Representations of Galois Groups
• Automorphic Forms: Analytic Functions with Certain Transformation Properties
• L-functions: Analytic Functions Associated with Automorphic Forms
• Selberg Trace Formula: Tool for Studying Spectral Theory and Automorphic Forms
• Langlands Program: Program to Unify Number Theory and Representation Theory
• Hodge Theory: Study of Harmonic Forms on Complex Manifolds
• Riemann Surfaces: One-dimensional Complex Manifolds
• Shimura Varieties: Algebraic Varieties Associated with Automorphic Forms
• Modular Curves: Algebraic Curves Associated with Modular Forms
• Hyperbolic Manifolds: Manifolds with Constant Negative Curvature
• Teichmüller Theory: Study of Moduli Spaces of Riemann Surfaces
• Mirror Symmetry: Duality Between Calabi-Yau Manifolds
• Kähler Geometry: Study of Hermitian Manifolds with Special Symmetries
• Algebraic Groups: Linear Algebraic Groups and Their Representations
• Lie Algebras: Study of Algebraic Structures Arising from Lie Groups
• Representation Theory of Lie Algebras: Study of Representations of Lie Algebras
• Quantum Groups: Deformation of Lie Groups and Lie Algebras
• Algebraic Topology: Study of Topological Spaces Using Algebraic Methods
• Homotopy Theory: Study of Continuous Deformations of Spaces
• Homology Theory: Study of Algebraic Invariants of Topological Spaces
• Cohomology Theory: Study of Dual Concepts to Homology Theory
• Singular Homology: Homology Theory Defined Using Simplicial Complexes
• Sheaf Theory: Study of Sheaves and Their Cohomology
• Differential Forms: Study of Multilinear Differential Forms
• De Rham Cohomology: Cohomology Theory Defined Using Differential Forms
• Morse Theory: Study of Critical Points of Smooth Functions
• Symplectic Geometry: Study of Symplectic Manifolds and Their Geometry
• Floer Homology: Study of Symplectic Manifolds Using Pseudoholomorphic Curves
• Gromov-Witten Invariants: Invariants of Symplectic Manifolds Associated with Pseudoholomorphic Curves
• Mirror Symmetry: Duality Between Symplectic and Complex Geometry
• Calabi-Yau Manifolds: Ricci-Flat Complex Manifolds
• Moduli Spaces: Spaces Parameterizing Geometric Objects
• Donaldson-Thomas Invariants: Invariants Counting Sheaves on Calabi-Yau Manifolds
• Algebraic K-Theory: Study of Algebraic Invariants of Rings and Modules
• Homological Algebra: Study of Homology and Cohomology of Algebraic Structures
• Derived Categories: Categories Arising from Homological Algebra
• Stable Homotopy Theory: Homotopy Theory with Stable Homotopy Groups
• Model Categories: Categories with Certain Homotopical Properties
• Higher Category Theory: Study of Higher Categories and Homotopy Theory
• Higher Topos Theory: Study of Higher Categorical Structures
• Higher Algebra: Study of Higher Categorical Structures in Algebra
• Higher Algebraic Geometry: Study of Higher Categorical Structures in Algebraic Geometry
• Higher Representation Theory: Study of Higher Categorical Structures in Representation Theory
• Higher Category Theory: Study of Higher Categorical Structures
• Homotopical Algebra: Study of Algebraic Structures in Homotopy Theory
• Homotopical Groups: Study of Groups with Homotopical Structure
• Homotopical Categories: Study of Categories with Homotopical Structure
• Homotopy Groups: Algebraic Invariants of Topological Spaces
• Homotopy Type Theory: Study of Foundations of Mathematics Using Homotopy Theory

In conclusion, the world of mathematics is vast and multifaceted, offering endless opportunities for exploration and discovery. Whether delving into the abstract realms of pure mathematics or applying mathematical principles to solve real-world problems, mathematicians play a vital role in advancing human knowledge and shaping the future of our world.

By embracing diverse math research topics and interdisciplinary collaborations, we can unlock new possibilities and harness the power of mathematics to address the challenges of today and tomorrow. So, let’s embark on this journey together as we unravel the mysteries of numbers and explore the boundless horizons of mathematical inquiry.

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Introduction to Mathematical Modelling

What is a Mathematical Model? #

A mathematical model is a mathematical representation of a system used to make predictions and provide insight about a real-world scenario, and mathematical modelling is the process of constructing, simulating and evaluating mathematical models.

Why do we construct mathematical models? It can often be costly (or impossible!) to conduct experiments to study a real-world problem and so a mathematical model is a way to describe the behaviour of a system and predict outcomes using mathematical equations and computer simulations .

Check out the following resources to get started with mathematical modelling:

Chapter 1: What is Mathematical Modelling? in Principles of Mathematical Modeling

What is Math Modeling?

Wikipedia: Mathematical Model

Outline of the Modelling Process #

Mathematical modelling involves observing some real-world phenomenon and formulating a mathematical representation of the system. But how do we even know where to start? Or how to find a solution? The modelling process is a systematic approach:

Clearly state the problem

Identify variables and parameters

Make assumptions and identify constraints

Build solutions

Analyze and assess

Report the results

Models can have a wide range of complexity ! More complex does not necessarily mean better and we can sometimes work with more simplistic models to achieve good results. In many instances, we often start with a simple model and then build-up the complexity by iterating through the steps in modelling process until the model accurately describes the real-world application.

Check out Math Modeling: Getting Started and Getting Solutions to read more about the modelling process.

Types of Models #

There are many different types of mathematical models! In this course we focus on the following:

Deterministic models predict future based on current information and do not include randomness. These kinds of models often take the from of systems of differential equations which describe the evolution of a system over time.

Stochastic models include randomness and are based on probability distributions and stochastic processes .

Data-driven models look for patterns in observed data to predict the output of a system. These kinds of models often take the form of functions with parameters computed to fit observed data.

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181 Mathematics Research Topics From PhD Experts

If you are reading this blog post, it means you are looking for some exceptional math research topics. You want them to be original, unique even. If you manage to find topics like this, you can be sure your professor will give you a top grade (if you write a decent paper, that is). The good news is that you have arrived at just the right place – at the right time. We have just finished updating our list of topics, so you will find plenty of original ideas right on this page. All our topics are 100 percent free to use as you see fit. You can reword them and you don’t need to give us any credit.

And remember: if you need assistance from a professional, don’t hesitate to reach out to us. We are not just the best place for math research topics for high school students; we are also the number one choice for students looking for top-notch research paper writing services.

Our Newest Research Topics in Math

We know you probably want the best and most recent research topics in math. You want your paper to stand out from all the rest. After all, this is the best way to get some bonus points from your professor. On top of this, finding some great topics for your next paper makes it easier for you to write the essay. As long as you know at least something about the topic, you’ll find that writing a great paper or buy phd thesis isn’t as difficult as you previously thought.

So, without further ado, here are the 181 brand new topics for your next math research paper:

Cool Math Topics to Research

Are you looking for some cool math topics to research? We have a list of original topics for your right here. Pick the one you like and start writing now:

• Roll two dice and calculate a probability
• Discuss ancient Greek mathematics
• Is math really important in school?
• Discuss the binomial theorem
• The math behind encryption
• Game theory and its real-life applications
• Analyze the Bernoulli scheme
• What are holomorphic functions and how do they work?
• Describe big numbers
• Solving the Tower of Hanoi problem

Undergraduate Math Research Topics

If you are an undergraduate looking for some research topics for your next math paper, you will surely appreciate our list of interesting undergraduate math research topics:

• Methods to count discrete objects
• The origins of Greek symbols in mathematics
• Methods to solve simultaneous equations
• Real-world applications of the theorem of Pythagoras
• Discuss the limits of diffusion
• Use math to analyze the abortion data in the UK over the last 100 years
• Discuss the Knot theory
• Analyze predictive models (take meteorology as an example)
• In-depth analysis of the Monte Carlo methods for inverse problems
• Squares vs. rectangles (compare and contrast)

Number Theory Topics to Research

Interested in writing about number theory? It is not an easy subject to discuss, we know. However, we are sure you will appreciate these number theory topics:

• Discuss the greatest common divisor
• Explain the extended Euclidean algorithm
• What are RSA numbers?
• Discuss Bézout’s lemma
• In-depth analysis of the square-free polynomial
• Discuss the Stern-Brocot tree
• Analyze Fermat’s little theorem
• What is a discrete logarithm?
• Gauss’s lemma in number theory
• Analyze the Pentagonal number theorem

Math Research Topics for High School

High school students shouldn’t be too worried about their math papers because we have some unique, and quite interesting, math research topics for high school right here:

• Discuss Brun’s constant
• An in-depth look at the Brahmagupta–Fibonacci identity
• What is derivative algebra?
• Describe the Symmetric Boolean function
• Discuss orders of approximation in limits
• Solving Regiomontanus’ angle maximization problem
• What is a Quadratic integral?
• Define and describe complementary angles
• Analyze the incircle and excircles of a triangle
• Analyze the Bolyai–Gerwien theorem in geometry
• Math in our everyday life

Complex Math Topics

If you want to give some complex math topics a try, we have the best examples below. Remember, these topics should only be attempted by students who are proficient in mathematics:

• Mathematics and its appliance in Artificial Intelligence
• Try to solve an unsolved problem in math
• Discuss Kolmogorov’s zero-one law
• What is a discrete random variable?
• Analyze the Hewitt–Savage zero-one law
• What is a transferable belief model?
• Discuss 3 major mathematical theorems
• Describe and analyze the Dempster-Shafer theory
• An in-depth analysis of a continuous stochastic process
• Identify and analyze Gauss-Markov processes

Easy Math Research Paper Topics

Perhaps you don’t want to spend too much time working on your next research paper. Who can blame you? Check out these easy math research paper topics:

• Define the hyperbola
• Do we need to use a calculator during math class?
• The binomial theorem and its real-world applications
• What is a parabola in geometry?
• How do you calculate the slope of a curve?
• Define the Jacobian matrix
• Solving matrix problems effectively
• Why do we need differential equations?
• Should math be mandatory in all schools?
• What is a Hessian matrix?

Logic Topics to Research

We have some interesting logical topics for research papers. These are perfect for students interested in writing about math logic. Pick one right now:

• Discuss the reductio ad absurdum approach
• Discuss Boolean algebra
• What is consistency proof?
• Analyze Trakhtenbrot’s theorem (the finite model theory)
• Discuss the Gödel completeness theorem
• An in-depth analysis of Morley’s categoricity theorem
• How does the Back-and-forth method work?
• Discuss the Ehrenfeucht–Fraïssé game technique
• Discuss Aleph numbers (Aleph-null and Aleph-one)
• Solving the Suslin problem

Algebra Topics for a Research Paper

Would you like to write about an algebra topic? No problem, our seasoned writers have compiled a list of the best algebra topics for a research paper:

• Discuss the differential equation
• Analyze the Jacobson density theorem
• The 4 properties of a binary operation in algebra
• Analyze the unary operator in depth
• Analyze the Abel–Ruffini theorem
• Epimorphisms vs. monomorphisms: compare and contrast
• Discuss the Morita duality in algebraic structures
• Idempotent vs. nilpotent in Ring theory
• Discuss the Artin-Wedderburn theorem
• What is a commutative ring in algebra?
• Analyze and describe the Noetherian ring

Math Education Research Topics

There is nothing wrong with writing about math education, especially if your professor did not give you writing prompts. Here are some very nice math education research topics:

• What are the goals a mathematics professor should have?
• What is math anxiety in the classroom?
• Teaching math in UK schools: the difficulties
• Computer programming or math in high school?
• Is math education in Europe at a high enough level?
• Common Core Standards and their effects on math education
• Culture and math education in Africa
• What is dyscalculia and how does it manifest itself?
• When was algebra first thought in schools?
• Math education in the United States versus the United Kingdom

Computability Theory Topics to Research

Writing about computability theory can be a very interesting adventure. Give it a try! Here are some of our most interesting computability theory topics to research:

• What is a multiplication table?
• Analyze the Scholz conjecture
• Explain exponentiating by squaring
• Analyze the Myhill-Nerode theorem
• What is a tree automaton?
• Compare and contrast the Pushdown automaton and the Büchi automaton
• Discuss the Markov algorithm
• What is a Turing machine?
• Analyze the post correspondence problem
• Discuss the linear speedup theorem
• Discuss the Boolean satisfiability problem

Interesting Math Research Topics

We know you want topics that are interesting and relatively easy to write about. This is why we have a separate list of our most interesting math research topics:

• What is two-element Boolean algebra?
• The life of Gauss
• The life of Isaac Newton
• What is an orthodiagonal quadrilateral?
• Tessellation in Euclidean plane geometry
• Describe a hyperboloid in 3D geometry
• What is a sphericon?
• Discuss the peculiarities of Borel’s paradox
• Analyze the De Finetti theorem in statistics
• What are Martingales?
• The basics of stochastic calculus

Applied Math Research Topics

Interested in writing about applied mathematics? Our team managed to create a list of awesome applied math research topics from scratch for you:

• Discuss Newton’s laws of motion
• Analyze the perpendicular axes rule
• How is a Galilean transformation done?
• The conservation of energy and its applications
• Discuss Liouville’s theorem in Hamiltonian mechanics
• Analyze the quantum field theory
• Discuss the main components of the Lorentz symmetry
• An in-depth look at the uncertainty principle

Geometry Topics for a Research Paper

Geometry can be a very captivating subject, especially when you know plenty about it. Check out our list of geometry topics for a research paper and pick the best one today:

• Most useful trigonometry functions in math
• The life of Archimedes and his achievements
• Trigonometry in computer graphics
• Using Vincenty’s formulae in geodesy
• Define and describe the Heronian tetrahedron
• The math behind the parabolic microphone
• Discuss the Japanese theorem for concyclic polygons
• Analyze Euler’s theorem in geometry

Math Research Topics for Middle School

Yes, even middle school children can write about mathematics. We have some original math research topics for middle school right here:

• Finding critical points in a graph
• The basics of calculus
• What makes a graph ultrahomogeneous?
• How do you calculate the area of different shapes?
• What contributions did Euclid have to the field of mathematics?
• What is Diophantine geometry?
• What makes a graph regular?
• Analyze a full binary tree

Math Research Topics for College Students

As you’ve probably already figured out, college students should pick topics that are a bit more complex. We have some of the best math research topics for college students right here:

• What are extremal problems and how do you solve them?
• Discuss an unsolvable math problem
• How can supercomputers solve complex mathematical problems?
• An in-depth analysis of fractals
• Discuss the Boruvka’s algorithm (related to the minimum spanning tree)
• Discuss the Lorentz–FitzGerald contraction hypothesis in relativity
• An in-depth look at Einstein’s field equation
• The math behind computer vision and object recognition

Calculus Topics for a Research Paper

Let’s face it: calculus is not a very difficult field. So, why don’t you pick one of our excellent calculus topics for a research paper and start writing your essay right away:

• When do we need to apply the L’Hôpital rule?
• Discuss the Leibniz integral rule
• Calculus in ancient Egypt
• Discuss and analyze linear approximations
• The applications of calculus in real life
• The many uses of Stokes’ theorem
• Discuss the Borel regular measure
• An in-depth analysis of Lebesgue’s monotone convergence theorem

Simple Math Research Paper Topics for High School

This is the place where you can find some pretty simple topics if you are a high school student. Check out our simple math research paper topics for high school:

• The life and work of the famous Pierre de Fermat
• What are limits and why are they useful in calculus?
• Explain the concept of congruency
• The life and work of the famous Jakob Bernoulli
• Analyze the rhombicosidodecahedron and its applications
• Calculus and the Egyptian pyramids
• The life and work of the famous Jean d’Alembert
• Discuss the hyperplane arrangement in combinatorial computational geometry
• The smallest enclosing sphere method in combinatorics

Business Math Topics

If you want to surprise your professor, why don’t you write about business math? We have some exceptional topics that nobody has thought about right here:

• Is paying a loan with another loan a good approach?
• Discuss the major causes of a stock market crash
• Best debt amortization methods in the US
• How do bank loans work in the UK?
• Calculating interest rates the easy way
• Discuss the pros and cons of annuities
• Basic business math skills everyone should possess
• Business math in United States schools
• Analyze the discount factor

Probability and Statistics Topics for Research

Probability and statistics are not easy fields. However, you can impress your professor with one of our unique probability and statistics topics for research:

• What is the autoregressive conditional duration?
• Applying the ANOVA method to ranks
• Discuss the practical applications of the Bates distribution
• Explain the principle of maximum entropy
• Discuss Skorokhod’s representation theorem in random variables
• What is the Factorial moment in the Theory of Probability?
• Compare and contrast Cochran’s C test and his Q test
• Analyze the De Moivre-Laplace theorem
• What is a negative probability?

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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

An Introduction to Lie Algebra , Amanda Renee Talley

SIMPLE AND SEMI-SIMPLE ARTINIAN RINGS , Ulyses Velasco

CONSTRUCTION OF FINITE GROUP , Michelle SoYeong Yeo

Theses/Projects/Dissertations from 2016 2016

Upset Paths and 2-Majority Tournaments , Rana Ali Alshaikh

Regular Round Matroids , Svetlana Borissova

GEODESICS IN LORENTZIAN MANIFOLDS , Amir A. Botros

REALIZING TOURNAMENTS AS MODELS FOR K-MAJORITY VOTING , Gina Marie Cheney

Solving Absolute Value Equations and Inequalities on a Number Line , Melinda A. Curtis

BIO-MATHEMATICS: INTRODUCTION TO THE MATHEMATICAL MODEL OF THE HEPATITIS C VIRUS , Lucille J. Durfee

ANALYSIS AND SYNTHESIS OF THE LITERATURE REGARDING ACTIVE AND DIRECT INSTRUCTION AND THEIR PROMOTION OF FLEXIBLE THINKING IN MATHEMATICS , Genelle Elizabeth Gonzalez

LIFE EXPECTANCY , Ali R. Hassanzadah

PLANAR GRAPHS, BIPLANAR GRAPHS AND GRAPH THICKNESS , Sean M. Hearon

A Dual Fano, and Dual Non-Fano Matroidal Network , Stephen Lee Johnson

Mathematical Reasoning and the Inductive Process: An Examination of The Law of Quadratic Reciprocity , Nitish Mittal

The Kauffman Bracket and Genus of Alternating Links , Bryan M. Nguyen

Probabilistic Methods In Information Theory , Erik W. Pachas

THINKING POKER THROUGH GAME THEORY , Damian Palafox

Indicators of Future Mathematics Proficiency: Literature Review & Synthesis , Claudia Preciado

Ádám's Conjecture and Arc Reversal Problems , Claudio D. Salas

AN INTRODUCTION TO BOOLEAN ALGEBRAS , Amy Schardijn

The Evolution of Cryptology , Gwendolyn Rae Souza

Theses/Projects/Dissertations from 2015 2015

SYMMETRIC PRESENTATIONS AND RELATED TOPICS , Mashael U. Alharbi

Homomorphic Images And Related Topics , Kevin J. Baccari

Geometric Constructions from an Algebraic Perspective , Betzabe Bojorquez

Discovering and Applying Geometric Transformations: Transformations to Show Congruence and Similarity , Tamara V. Bonn

Symmetric Presentations and Generation , Dustin J. Grindstaff

HILBERT SPACES AND FOURIER SERIES , Terri Joan Harris Mrs.

SYMMETRIC PRESENTATIONS OF NON-ABELIAN SIMPLE GROUPS , Leonard B. Lamp

Simple Groups and Related Topics , Manal Abdulkarim Marouf Ms.

Elliptic Curves , Trinity Mecklenburg

A Fundamental Unit of O_K , Susana L. Munoz

CONSTRUCTIONS AND ISOMORPHISM TYPES OF IMAGES , Jessica Luna Ramirez

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