Directed Reading Program Spring 2024: Difference between revisions

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(Created page with "'''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 de...")
 
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!Required Background
!Required Background
|-
|-
|Analysis On Graphs
|Polynomial Methods
|We will study analysis on graphs. More precisely, we will read the book ''Introduction to Analysis on Graphs'' by Alexander Grigor'yan. The purpose of the book is to provide an introduction to the subject of the discrete Laplace operator on locally finite graphs. Analysis on graphs is a subject studies classical calculus and differential equations on discrete objects such as graphs. It has connections with many other important areas in math and outside of math.  
|We will read Larry Guth’s book Polynomial Methods in Combinatorics and maybe some related short papers to understand how polynomials and their properties are applied to combinatorics, incidence geometry and harmonic analysis.
|Students should be familiar with basic calculus (221, 222 and 234) and linear algebra. Analysis courses such as 521 and 522 or basic probability theory are helpful.
|The material is accessible to students with linear algebra background. Corequisite of MATH 542 and 522 is recommended.
|-
|-
|Analysis, Geometry, and Combinatorics
|Algebraic Geometry
|I am very flexible and happy to work with your backgrounds and interests. A few things I've thought about are:
|This project introduces scheme theory, a fundamental part of modern algebraic geometry, from a geometrically focused point of view. Topics to be covered will include affine schemes and their topology, an introduction to sheaves, structure sheaves, general schemes, and some examples. We will use Eisenbud and Harris' "The Geometry of Schemes."
 
|The only strictly necessary requirement is familiarity with basic point set topology and commutative algebra at the level of a one-semester course. Rings, ideals, modules, localization, tensor products, etc. Basic algebraic geometry can be introduced at the beginning, but it would help to know about affine varieties, affine coordinate rings, and the correspondence between them.
- Looking at Guth's book "Polynomial Method's in Combinatorics", hopefully culminating in understanding the proof of the finite field Kakeya conjecture.  
 
- Measure theory - we'll start with Stein & Shakarchi's book, depending on how that goes, we could move on to Frank Morgan's "Geometric Measure Theory: A Beginner’s Guide" for a bit of geometric measure theory.
 
- Laszlo Lovasz's book "Combinatorial Problems and Exercises". The name is pretty self-explanatory - this books balances more towards problems and less towards theory, but won't require much background.
 
- Spivak's book "Calculus on manifolds". This isn't your usual calculus book - it will introduce differential forms, which are very useful in geometry and analysis.  
|-
|-
|Fractal geometry
|Analysis
|In recent years, a surprising connection between two seemingly unrelated branches of math has emerged. Fractal dimension is a notion of size for sets which, unlike how we normally think about dimension, is not necessarily an integer (e.g. the rough shape of a coastline, the branching of a bolt of lightning, or the chaotic movement of a stock's price have non-integer dimension). Kolmogorov complexity, which measures how much information a string contains, can be applied to characterize some types of fractal dimension. This connection has already yielded impressive new results about fractal sets.
|We will read Polynomial Methods in Combinatorics by Larry Guth. This covers a variety of applications of polynomials to different fields in mathematics, including combinatorics, analysis, and geometry. While I am most interested in analysis, we can choose what among the applications in the book interest you and focus on that.
In this DRP, we will start by understanding notions of fractal dimension and the definition/properties of Kolmogorov complexity. Then, we'll bridge the gap between the two via "effective" dimension. Effective dimension gives rise to elegant characterizations of both Hausdorff and packing dimension via the recently proven point-to-set principle, which we will study. Finally, we'll see some applications of this alternative definition, which makes it one of the most powerful new tools in fractal geometry.  
|Linear algebra, MATH 521-22.
|I would like students who have taken 521-522. But I'm flexible and happy to find something which you feel ready for.
|-
|-
|Number theory and Partition Theory
|Algebra/Probability
|We will investigate modular forms with a view towards algebraic combinatorics and number theory. In this project, we will first read about the interesting connections of modular forms and number theory/arithmetic geometry, with a focus on the theory of theta functions and elliptic curves. We will then transition to a more concrete application of modular forms via partition theory. In particular, we'll cover both analytic and combinatorial techniques, such as the circle method and enumerative combinatorics. We may also dive into more modern applications such as mock theta functions. Depending on participant interest this may turn into a research project based on prior work of the mentor on generalizing the Rogers-Ramanujan identities in partition theory.  
|Free probability is a field which attempts to apply ideas from probability to more abstract settings, especially where the variables don't commute. In these settings we might not even be able to talk about 'probabilities', but thinking about expectations and distributions (suitably translated) can still tell us a lot. For example, it can tell us what the eigenvalues of a random matrix look like. Some funny things happen in the translation, like the role of the bell curve being taken by the semicircle (see Wigner's Semicircle Law for a related result).
|Students should have completed a semester of algebra (541) and be familiar with proofs. Familiarity with complex analysis will be useful though not required. Coding knowledge with python would also be useful.  
 
The material involves an interplay between things like operator algebras, combinatorics and probability, and would be interesting mostly to students who enjoy probability and didn't mind taking abstract algebra.
|Students should be comfortable with linear algebra and have taken some abstract algebra (341, 541, and maybe 540 would be good). Some background in probability (not necessarily measure-theoretic) would help provide context. The material in the book doesn't require much heavy theory, so it should be possible to fill in any blanks along the way.
|-
|-
|Measure Theory
|Real Analysis
|Euclidean space may seem simple enough, but it contains many complicated subsets with highly counterintuitive properties. Throughout the semester, we'll consider a number of examples of such subsets, including the standard Cantor set, fat and thin Cantor sets, Vitali sets, "large" meager sets, purely unrectifiable sets, Kakeya sets, Nikodym sets, and many more. Readings will be provided and will come from a variety of sources,  
|Wavelets are a fun topic if you're interested in learning about a very useful application of both real analysis and linear algebra! They are used in lots of different areas of engineering (e.g. signal processing) and are interesting in their own right as a pure math tool! If you are interested in learning a more advanced topic of math, sign up for this DRP!
|Students should either have taken 521 or currently be enrolled in it.  
|Real analysis (Math 521) and linear algebra (math 341 or 540) are prerequisites. If you have studied, Fourier series that's a plus but not a prerequisite! If you are motivated to learn something new, don't hesitate to sign up! We're all here to learn so questions are always and will always be welcome! I'm planning on using "An introduction to wavelets through linear algebra" by Michael Frazier.
|-
|-
|Basics of D-modules
|History of Mathematics
|In this project, students will be introduced to D-modules, which allow us to think about ideas from calculus, such as derivatives and differential equations, in an algebraic context. The topics covered will include the Weyl algebra, the Jacobian conjecture, rings of differential operators, modules over the Weyl algebra, and differential equations. The text we will use is "A Primer of Algebraic D-modules" by S.C. Coutinho.
|We will read and discuss sections from John Stillwell's Mathematics and Its History. Which particular sections we read can be selected by any students in the group based on their interests. Our goal is to gain an appreciation for some of how mathematics became what it is today. What problems motivated its development, and how do our modern conceptions of things align (or not!) with what mathematicians were doing historically? What can we learn for doing math and other kinds of problem solving in the present?
|The prerequisite is math 541 or an equivalent: rings, ideals, homomorphisms, polynomial rings, and abstract vector spaces. A basic knowledge of modules is necessary, but this can be introduced at the beginning of the project.
|We think interested students should have taken at least one proof-based math course. We think this project is unique in that it can accommodate students of widely varying pre-existing mathematical knowledge. Having little math knowledge means this project can give you perspective and intuition for math you will learn as you progress through the curriculum. Having a lot of math knowledge means this project can help you see the "big picture" of how facts you have learned connect to each other, sometimes in surprising ways.
|-
|-
|Commutative Algebra
|Dynamical Systems (Reaction Networks)
|Ideals, Varieties, and Algorithms (Chapters 1-2) with a view toward algebraic geometry.Students are expected to give "mini-talks" on the board for the section we read that week.  
|Ever since Edward Lorenz found chaos emerging from his simple meteorological model, we have noticed that it is quite hard to try to understand the behavior of even very simple non-linear systems of differential equations. But these systems show up time and time again in nature, with a variety of very particular non-chaotic dynamics that we can make sense of. The study of Reaction Networks surprisingly manages to identify and classify these dynamics in systems that are very common in nature; from the enzymatic reactions on cells, to epidemiological models and even ecological population models. Using simple graph theory and linear algebra, we can uncover amazing results for the dynamics of these types of systems by abstracting some ideas. We will follow Martin Feinberg's book "Foundations of Chemical Reaction Network Theory".
|Some knowledge of abstract algebra but not much else.
 
My goal in this DRP is to showcase this beautiful field of mathematics that lies in the intersection of both Pure and Applied Math, showing that these are more related that they usually seem, and how with abstraction and math we can uncover deeper truths about the structure of nature.
|Students should be familiar with the language of elementary qualitative theory of ordinary differential equations (e. g., the meaning of asymptotic stability), good knowledge of modern linear algebra and calculus. For differential equations, Math 415 or Math 519 will suffice. For linear algebra, any course which dictates it is good enough.
|-
|-
|Computability Theory, or, Algebra
|Linear Representations of Finite Groups
|1, Computability Theory: we will focus on computability theory, complexity theory, and cryptography. There is a book called Computational Complexity: A Moder Approach, and we can pay more attention to Non-determinism and NP Completeness.
|We’ll be reading Serre’s Linear Representation of Finite Groups.  
 
|Abstract Algebra (Math 541) is expected. I expect the student to be somewhat independent due to time.
2, Algebra: we will focus on Linear Representations of Finite Groups. It is also the name of a book which describes the correspondence, due to Frobenius, between linear representations and charac­ters. The book says this is a fundamental result, of constant use in mathematics as well as in quantum chemistry or physics and have tried to give proofs as elementary as possible, using only the definition of a group and the rudiments of linear algebra.
|Strong abilities of Analysis and Discrete Mathematics (including Algebra, etc)
|-
|-
|Machine Learning
|Geometric Measure Theory
| In this project, students will read/watch parts of the "Practical Deep Learning for Coders" course of fast.ai, a deep learning library that aims to make deep learning as accessible as possible, with the aim of training a basic machine learning model of their own design. In parallel, the students will also develop the skills for some software development practices, such as unit testing, documentation, and key bindings.
|Starting from the basics, the goal is to cover a solid amount of geometric measure theory, which is a field used to understand the geometry of complicated (but common in the real world) sets. We'll begin by discussing measures, and move into rectifiable curves/sets. We'll also talk about how projections affect the geometry of sets, in particular purely unrectifiable sets. Then, I hope to get into other topics like weak tangents and Plateau's problem generalized beyond smooth curves.
|Calculus, Linear Algebra, and basic programming experience (preferably including Python) are required. I would also like the students to be mostly independent as designing their own machine learning model is a goal for this project (although this goal is not a hard requirement) and as I might be away for some weeks during the semester.
|Students should have taken 521. Additional experience in proof-based math courses, especially analysis classes, will be very helpful. Students don't necessarily need to have seen measure theory, but it wouldn't hurt.
|-
|-
|Dynamics (Analysis/Geometry)
|Abstract Algebra
|Ergodic theory studies dynamical systems (we can think of this as a repeated action) and their statistical properties (i.e. time averages). For example, consider a circle and pick a point and an angle. Now our action will be repeatedly rotating the point by the chosen angle. Now if we do this action repeatedly for a long time, what will the resulting orbit of the point be? Will it miss part of the circle? Go everywhere? Go everywhere equally often? If we choose an irrational angle the last is true and we call the system ergodic. Ergodic theory is concerned with the study of these systems. While we won’t directly study them, ergodic theory has applications in many other fields -- including number theory, harmonic analysis, probability, and mathematical physics -- making it an incredibly exciting multi-disciplinary area to study. Our plan will be to first spend some time reviewing measure theory, then look at some examples and define ergodicity rigorously. After that we will spend some time studying the various ergodic theorems. If time allows we will extend our study to mixing and entropy.
|We plan to follow the book Ideals, Varieties, and Algorithms by Cox-Little-O'Shea, roughly aiming to cover the first two chapters. This means we'll explore the relationship between algebraic objects (like polynomials) and geometric objects (like curves). We'll read about an important computational tool called Gröbner bases, which are used to find solutions to systems of polynomial equations (motivated by methods from linear algebra). This project, and the book we've chosen, will assume no prior knowledge of abstract algebra, and we'll learn any necessary concepts along the way! If you like a mix of theory and explicit examples, this project is for you!
|The Real Analysis sequence (or equivalent) is required. Having seen some measure theory will be extremely helpful though we will spend the first week or two reviewing this topic. Any geometry/topology is useful but not at all required.
|This project is intended for students who have taken linear algebra and are interested in abstract algebra, but don't have much (or any) background in abstract algebra.
|}
|}


Revision as of 19:54, 5 February 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, it is scheduled for Wednesday, December 6th.

Applications

Check out our main page for examples of past projects.

Students: https://docs.google.com/forms/d/e/1FAIpQLSf2lm8Geuc6jwznBgGP5JjSJZuMITOw252e9qPOZCEQuFGIQw/viewform

Mentors: Applications are closed.

Questions?

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

Projects

Spring 2024 Projects
Title Abstract Required Background
Polynomial Methods We will read Larry Guth’s book Polynomial Methods in Combinatorics and maybe some related short papers to understand how polynomials and their properties are applied to combinatorics, incidence geometry and harmonic analysis. The material is accessible to students with linear algebra background. Corequisite of MATH 542 and 522 is recommended.
Algebraic Geometry This project introduces scheme theory, a fundamental part of modern algebraic geometry, from a geometrically focused point of view. Topics to be covered will include affine schemes and their topology, an introduction to sheaves, structure sheaves, general schemes, and some examples. We will use Eisenbud and Harris' "The Geometry of Schemes." The only strictly necessary requirement is familiarity with basic point set topology and commutative algebra at the level of a one-semester course. Rings, ideals, modules, localization, tensor products, etc. Basic algebraic geometry can be introduced at the beginning, but it would help to know about affine varieties, affine coordinate rings, and the correspondence between them.
Analysis We will read Polynomial Methods in Combinatorics by Larry Guth. This covers a variety of applications of polynomials to different fields in mathematics, including combinatorics, analysis, and geometry. While I am most interested in analysis, we can choose what among the applications in the book interest you and focus on that. Linear algebra, MATH 521-22.
Algebra/Probability Free probability is a field which attempts to apply ideas from probability to more abstract settings, especially where the variables don't commute. In these settings we might not even be able to talk about 'probabilities', but thinking about expectations and distributions (suitably translated) can still tell us a lot. For example, it can tell us what the eigenvalues of a random matrix look like. Some funny things happen in the translation, like the role of the bell curve being taken by the semicircle (see Wigner's Semicircle Law for a related result).

The material involves an interplay between things like operator algebras, combinatorics and probability, and would be interesting mostly to students who enjoy probability and didn't mind taking abstract algebra.

Students should be comfortable with linear algebra and have taken some abstract algebra (341, 541, and maybe 540 would be good). Some background in probability (not necessarily measure-theoretic) would help provide context. The material in the book doesn't require much heavy theory, so it should be possible to fill in any blanks along the way.
Real Analysis Wavelets are a fun topic if you're interested in learning about a very useful application of both real analysis and linear algebra! They are used in lots of different areas of engineering (e.g. signal processing) and are interesting in their own right as a pure math tool! If you are interested in learning a more advanced topic of math, sign up for this DRP! Real analysis (Math 521) and linear algebra (math 341 or 540) are prerequisites. If you have studied, Fourier series that's a plus but not a prerequisite! If you are motivated to learn something new, don't hesitate to sign up! We're all here to learn so questions are always and will always be welcome! I'm planning on using "An introduction to wavelets through linear algebra" by Michael Frazier.
History of Mathematics We will read and discuss sections from John Stillwell's Mathematics and Its History. Which particular sections we read can be selected by any students in the group based on their interests. Our goal is to gain an appreciation for some of how mathematics became what it is today. What problems motivated its development, and how do our modern conceptions of things align (or not!) with what mathematicians were doing historically? What can we learn for doing math and other kinds of problem solving in the present? We think interested students should have taken at least one proof-based math course. We think this project is unique in that it can accommodate students of widely varying pre-existing mathematical knowledge. Having little math knowledge means this project can give you perspective and intuition for math you will learn as you progress through the curriculum. Having a lot of math knowledge means this project can help you see the "big picture" of how facts you have learned connect to each other, sometimes in surprising ways.
Dynamical Systems (Reaction Networks) Ever since Edward Lorenz found chaos emerging from his simple meteorological model, we have noticed that it is quite hard to try to understand the behavior of even very simple non-linear systems of differential equations. But these systems show up time and time again in nature, with a variety of very particular non-chaotic dynamics that we can make sense of. The study of Reaction Networks surprisingly manages to identify and classify these dynamics in systems that are very common in nature; from the enzymatic reactions on cells, to epidemiological models and even ecological population models. Using simple graph theory and linear algebra, we can uncover amazing results for the dynamics of these types of systems by abstracting some ideas. We will follow Martin Feinberg's book "Foundations of Chemical Reaction Network Theory".

My goal in this DRP is to showcase this beautiful field of mathematics that lies in the intersection of both Pure and Applied Math, showing that these are more related that they usually seem, and how with abstraction and math we can uncover deeper truths about the structure of nature.

Students should be familiar with the language of elementary qualitative theory of ordinary differential equations (e. g., the meaning of asymptotic stability), good knowledge of modern linear algebra and calculus. For differential equations, Math 415 or Math 519 will suffice. For linear algebra, any course which dictates it is good enough.
Linear Representations of Finite Groups We’ll be reading Serre’s Linear Representation of Finite Groups. Abstract Algebra (Math 541) is expected. I expect the student to be somewhat independent due to time.
Geometric Measure Theory Starting from the basics, the goal is to cover a solid amount of geometric measure theory, which is a field used to understand the geometry of complicated (but common in the real world) sets. We'll begin by discussing measures, and move into rectifiable curves/sets. We'll also talk about how projections affect the geometry of sets, in particular purely unrectifiable sets. Then, I hope to get into other topics like weak tangents and Plateau's problem generalized beyond smooth curves. Students should have taken 521. Additional experience in proof-based math courses, especially analysis classes, will be very helpful. Students don't necessarily need to have seen measure theory, but it wouldn't hurt.
Abstract Algebra We plan to follow the book Ideals, Varieties, and Algorithms by Cox-Little-O'Shea, roughly aiming to cover the first two chapters. This means we'll explore the relationship between algebraic objects (like polynomials) and geometric objects (like curves). We'll read about an important computational tool called Gröbner bases, which are used to find solutions to systems of polynomial equations (motivated by methods from linear algebra). This project, and the book we've chosen, will assume no prior knowledge of abstract algebra, and we'll learn any necessary concepts along the way! If you like a mix of theory and explicit examples, this project is for you! This project is intended for students who have taken linear algebra and are interested in abstract algebra, but don't have much (or any) background in abstract algebra.

Presentation Schedule

Room 1 (Engineering Hall 3349)
Time Speakers Title
3:30-3:45 Erkin Delic Intro to D-modules
3:45-4:00 Yikai Zhang & Beining Mu Computability
4:00-4:45 Aidin Simkin, Yifan Yang, Sena Witzeling Several Strange Cantor Sets
4:45-5:00 Yancheng Zhu FastAi in NLP
5:00-5:15 Shi Kaiwen Machine learning application in stock index prediction
5:15-5:30 Hannah Wang Trained model for predicting the insurance price
Room 2 (Engineering Hall 3418)
Time Speakers Title
3:30-4:00 Aiden Styers, Rui Yu, Yizhou Qian & Anvith Aravati Laplace Operator on Graphs
4:30-5:00 Benjamin Braiman & Ruoyu Men Ergodic Theory
5:00-5:15 David Jiang Ramanujan Partition Identities
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