Directed Reading Program Fall 2024

What is it? The Directed Reading Program (DRP) in the UW Madison Department of Mathematics pairs undergraduate students with graduate mentors for semester-long independent studies. During the semester, the student will work through a mathematical text and meet weekly to discuss it with their mentor. The original DRP was started by graduate students at the University of Chicago over a decade ago, and has had immense success. It has since spread to many other math departments who are members of the DRP Network.

Why be a student?

• Learn about exciting math from outside the mainstream curriculum!
• Prepare for future reading and research, including REUs!
• Meet other students interested in math!

Why be a mentor?

• Practice your mentorship skills!
• It strengthens our math community!
• Solidify your knowledge in a subject!

Current Organizers: Ivan Aidun, Allison Byars, Jake Fiedler, John Spoerl

Requirements

At least one hour per week spent in a mentor/mentee setting. Students spend about two hours a week on individual study, outside of mentor/mentee meetings. At the end, students give a 10-12 minute presentation at the end of the semester introducing their topic. This semester, presentations will be Wednesday, December 4th.

Applications

You can find examples of past projects from the DRP main page.

Mentors: https://docs.google.com/forms/d/e/1FAIpQLScMQDojBKdIAFj9sMzeyYr4pn2JQ8buWPx_8gtMRFPq9P2D2A/viewform?usp=sf_link

Students:

https://forms.gle/kQC9vApGEB7USNnU8

Questions?

Contact us at drp-organizers@g-groups.wisc.edu

Projects

Title	Branch of Math	Abstract	Required Background
Ergodic theory and entropy	Probability	To start, we will review/introduce basic measure theory as needed. Then, we will define measure-preserving transformations and discuss the Poincare recurrence theorem. We will introduce notions of mixing along with ergodicity and study Birkhoff's ergodic theorem. Finally, we will introduce topological and measure-theoretic entropy, and possibly close with some applications to physics. We will likely use Walters' Introduction to Ergodic Theory at least in part.	Ideally, students should have completed 521, but if need be they can be currently enrolled in it.
Quadratic Number Fields	Number Theory	The goal of this DRP is to study quadratic number fields and their rings of integers and see how they can help us understand the properties of the integers that number theorists study. Along the way, we may learn how to find solutions to Pell's Equations (an equation of the form x^2-ny^2=1), and if we have time, we may also learn about quadratic reciprocity.	This DRP will use a lot of abstract algebra so you should have taken Math 541. Math 542 will might be helpful but certainly not required.
Model Theory	Logic	This directed reading program will serve as an elementary introduction to model theory, with an emphasis on pure model theory. We will start by introducing basic concepts and techniques, and aim to cover Morley’s theorem by the end. Along the way, we will cover many well-known results, such as Erdős-Rado Theorem and Henkin’s Omitting Types Theorem. I have several reference materials in mind that we can decide on together. The plan is flexible and can be adjusted to accommodate your interests. If you are already comfortable with elementary concepts in model theory, we can dedicate this directed reading program to Classification Theory for Abstract Elementary Classes (AEC), which studies generalizations of Morley’s theorem in uncountable languages.	A course in logic, preferably set theory
Magical Modality: The Logic of Possibility and Necessity	Logic	Come along on the Modal Magical Mystery Tour, where we will start with the “language” of logic that you have probably seen at some point in your introductory courses and add two new symbols. The addition of these two symbols, called box and diamond, opens the world of logic to being able to model philosophical, ethical, computational, linguistic, and cognitive phenomena. We will journey from the syntax and semantics of the two symbols in terms of Kripke models and frames, to bisimulations that act like isomorphisms between these structures, to applications that could range as far as you are interested, from the philosophy of time to the structure of shared information in a group. Answering questions like “If something is necessary, is it necessary that it is necessary?” will feature prominently as we explore different choices of axioms on our structures, and how those axioms can give us insight into the First-order language we began with. I hope that by the end of the semester, we can discuss the correspondence between Modal logic and First-order logic, through the lenses of van Benthem’s Theorem and the Salqvist-van Benthem algorithm.	Students should be comfortable with propositional and first-order logic and their notation. Working knowledge of universal and existential quantifiers (\forall and \exists) will be very helpful. Previous work with mathematical proofs will be assumed.
Harmonic Analysis of Boolean Functions	Discrete Mathematics	This DRP is about the analysis of Boolean Functions, and the properties of such functions that can be determined through discrete analogues of methods of analysis, e.g. The Fourier transform, Lp norm bounds, etc. Our goal will be to learn the basic theory of such methods, and then to see how these results can be applied to various problems in discrete mathematics. Boolean functions can encode many different mathematical structures, so depending on your interest, we can study various topics. For example: applications to graph theory (expansion / threshold properties of graphs), to voting theory (why making decisions with > 2 options is hard), to computational complexity (why random proofs can be small), or to statistics / machine learning (relations between Fourier analytic properties of data and why learning certain phenomena can be difficult).	Basic familiarity with concepts in discrete mathematics will be helpful, as well as familiarity with Lp spaces, e.g. using Cauchy-Schwartz and Holder's inequality to bound such quantities (though all our vector spaces will be finite dimensional).
Solving the Inverse Scattering Problem: Classical Methods, Machine Learning, and Generative AI	Applied Mathematics	This DRP explores the inverse scattering problem through three approaches: classical methods, deterministic machine learning, and generative AI. Students will begin with an introduction to the problem and classical methods. The course then transitions to a study of a deep neural network architectures that approximates the inverse map. Finally, the course covers score-based generative model, which approximates the posterior distribution of the inverse scattering map, yielding sharp and computationally efficient reconstructions even in challenging scenarios.	MATH 341, MATH 521, Programming Experience.
An Introduction to Proof Theory	Logic	We will read An Introduction to Proof Theory: Normalization, Cut-elimination and Consistency Proofs by Mancosu, Galvan, and Zach, with an eye towards the consistency proofs for sequent calculi and Peano Arithmetic. The proofs of these results from classical proof theory provide a natural and gentle introduction to the more complicated forms of mathematical induction frequently encountered in the broader field of mathematical logic.	We hope for students to have familiarity with the basics of propositional and first order logic as encountered in courses such as MATH 240 or PHILOS 211, though the reading we have selected is largely self-contained.
The Beauty of Math	Calculus , Mathematical analysis, algebra	We will read the book 'Linear Algebra' by Steven J. Leon. This book comprehensively covers various topics related to Linear Algebra, such as matrices, linear equation systems, determinants, linear spaces, inner product spaces, etc. I will be focusing on the theoretical aspects of Linear Algebra. We can select a topic based on your interest and delve deeper into it.	A strong desire to learn math!

Presentation Schedule

Presentations will be held Wednesday, December 4th. Precise details will be given in November.