Projects for Spring 2024

Mentor: Teressa Chambers (Fifth-year Graduate Student, Applied Mathematics)

Project Description: Networks are a highly versatile and widely applicable mathematical structure for modeling complex systems from food webs to shipping lanes to online social connections. This project is intended to introduce the theoretical foundations of network science, beginning with the graph theory underlying our understanding of networks and moving on to the practical question of how to construct and analyze network models. The initial phase will be facilitated through selected readings from “Networks: An Introduction” by M. E. J. Newman, which provides consistent practical context alongside the theoretical foundations. Once these tools have been established, the student will have the opportunity to construct a network model for a system of interest to them, and we will work together to ask and answer interesting questions about their network. A solid background in linear algebra will be necessary for this project, but otherwise there is very little prerequisite knowledge. The reading for this project will involve both textbook selections and papers in network science, so students are advised to have some experience with mathematical writing as well as proof structures. Basic familiarity with MATLAB is also strongly recommended.

Sample Text: Networks: An Introduction (Newman)

Suggested Prerequisites: MATH0520 Linear Algebra and ideally an exposure to an upper level APMA or MATH class for example: APMA1210 Operations Research, MATH1010 Real Analysis or MATH1530 Abstract Algebra.

Mentors: Pratyush Potu (Second-year Graduate Student, Applied Mathematics)

Project Description: When learning calculus, one learns to take derivatives and integrals and explores the effect of these operations on various types of functions. However, one interesting question one might consider is "Can I take half a derivative of a function?". What about pi'th derivatives? Imaginary order derivatives? From these questions the theory of fractional calculus was developed. At the time of development, very few applications were found for the theory. However, in more recent years fractional order models of physical phenomena have become more widespread. Applications have been found in physics, chemistry, biology, and economics for instance. In this project, the student will learn the definitions of fractional derivatives and fractional integrals and how to compute them.

Sample Text: The Fractional Calculus (Oldham and Spanier).

Suggested Prerequisites: Strong calculus experience e.g. MATH0180 or equivalent, APMA0350/AMPA0360 would also be helpful

Mentor: Phuc Lam (Second-year Graduate Student, Applied Mathematics)

Project Description: How do we show that a mathematical object with certain properties exists without constructing an explicit example? For example, loosely speaking, without picking, can we know if it is possible to pick out a very small group of people in a party so that everyone else in the party knows at least one person in the group? With the "constructive approach", we either explicitly construct an example or devise a method to do so. The probabilistic method, on the other hand, is a "non-constructive approach". With this method, one shows that if we randomly choose objects from a specified class, we can choose the desired object with (strictly) positive probability, thus there must be an instance where this desired object appears, and so it exists.

Though the idea is simple, the method is actually quite powerful. Developed by Paul Erdos and initially used in combinatorics and graph theory, the method has since yielded non-trivial and rich applications in computer science, real analysis, number theory, and even in fields as far removed from probability as algebraic number theory.

We will begin by reading the first 5 chapters of "The Probabilistic Method" by Noga Alon and Joel H. Spencer, introducing motivations and examples of this method, and a few basic techniques. Afterwards, depending on student's interests, we can either learn other techniques beyond the first 5 chapters or explore a few specific applications of this method. Students are assumed to be comfortable with calculus, probability, and enumeration (i.e. counting permutations, combinations, partitions, etc). Familiarity with graph theory, number theory, and/or algorithms is not required, but would be useful.

Sample Text: The Probabilistic Method (Alon and Spencer)

Suggested Prerequisites: APMA1650 Statistical Inference I or MATH1610 Probability or equivalent

Mentor: Wenjun Zhao (LFZ Assistant Professor, Applied Mathematics)

Project Description: Approximation theory is an established field that is concerned with how functions like exp(x), sin(x) can be approximated simpler functions such as polynomial or rational (ratio of polynomial) functions. The objective is to characterize the errors quantitatively, and make the approximation as close as possible to the actual function. Through the program, we will learn about the classical ideas in this field, illustrated by corresponding numerical examples.

Sample Text: Approximation Theory and Approximation Practice (Trefethen)

Suggested Prerequisites: Programming experience and some experience with numerical methods would be helpful e.g. APMA0160 Introduction to scientific computing

Mentor: Victoria Antonetti (Third-year Graduate Student, Applied Mathematics)

Project Description: Black body radiation confounded classical physicists. A black body is an object that absorbs all energy that is transferred to it. In the 1850s, Gustav Kirchhoff challenged physicists of his time to find the form of the energy of such an object knowing that its form depended only on temperature and frequency. Solutions to the problem worked in certain frequency (energy) regimes but not on the whole until the breakthrough of Max Plank, in which he assumed that energy could be thought of as a discrete variable made up of what he then labeled quanta. Here begins the story of quantum mechanics. The notion that science appears to behave differently at different scales is a profound one throughout the different fields of science from pattern scaling and formation in living systems to the mechanics of matter. In this reading project, we will focus on developing some of the mathematical tools that allowed quantum physicists in the second wave of quantum mechanics, the quantum we learn today, to address the scaling problem and understand phenomena that the formalism of classical mechanics failed to capture. As we develop some of the mathematics of quantum mechanics, which ranges from functional analysis to linear algebra to differential equations, we will consider some of the foundational triumphs of the theory like the Heisenberg uncertainty principle and solutions to the Schrödinger equation.

Sample Text: Quantum Mechanics, Concepts and Applications (Zettili)

Suggested Prerequisites: Familiarity with classical mechanics (e.g. PHYS0030 or 0050) and Linear algebra (MATH0520)

Mentor: Ezra Seidel (Second-year Graduate Student, Applied Mathematics)

Project Description: Game theory studies interactions between agents in which each agent can gain or lose depending on their own actions and those of their opponent, such as the famous Prisoner's Dilemma. It studies how to find strategies which are optimal in the sense that, if adopted by all participants, no agent can improve their outcome by switching to a different strategy. Evolutionary game theory applies this framework to evolutionary biology by studying the population dynamics which arise from different groups within a population having different strategies for interactions such as competition for resources, and how such dynamics can lead to the emergence of a single dominant strategy. A key concept in evolutionary game theory is the notion of an evolutionarily stable strategy, which roughly is a strategy which if adopted by most individuals in a population, cannot be overtaken by a small mutant population with a different strategy. For example, a strategy of always retreating from combat over resources would be unstable because an aggressive mutant population would quickly be able to gain an advantage. Evolutionary game theory has been able to provide explanations for puzzling phenomena in biology, such as the evolution of altruism. The majority of this project will be reading from a textbook on the subject; depending on the interest of the student we may also work on a coding project or look at recent papers. Sample Text:

• Evolutionary Game Theory (Alexander)
• Evolutionary Game Theory (Weibull)