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The AMS Student Chapter Seminar is an informal, graduate student seminar on a wide range of mathematical topics. Pastries (usually donuts) will be provided.
The AMS Student Chapter Seminar (aka Donut Seminar) is an informal, graduate student seminar on a wide range of mathematical topics. The goal of the seminar is to promote community building and give graduate students an opportunity to communicate fun, accessible math to their peers in a stress-free (but not sugar-free) environment. Pastries (usually donuts) will be provided.


* '''When:''' Wednesdays, 3:20 PM – 3:50 PM
* '''When:''' Wednesdays, 3:30 PM – 4:00 PM
* '''Where:''' Van Vleck, 9th floor lounge (unless otherwise announced)
* '''Where:''' Van Vleck, 9th floor lounge (unless otherwise announced)
* '''Organizers:''' [https://www.math.wisc.edu/~malexis/ Michel Alexis], [https://www.math.wisc.edu/~drwagner/ David Wagner], [http://www.math.wisc.edu/~nicodemus/ Patrick Nicodemus], [http://www.math.wisc.edu/~thaison/ Son Tu]
* '''Organizers:''' Ivan Aidun, Kaiyi Huang, Ethan Schondorf


Everyone is welcome to give a talk. To sign up, please contact one of the organizers with a title and abstract. Talks are 30 minutes long and should avoid assuming significant mathematical background beyond first-year graduate courses.
Everyone is welcome to give a talk. To sign up, please contact one of the organizers with a title and abstract. Talks are 25 minutes long and should avoid assuming significant mathematical background beyond first-year graduate courses.


The schedule of talks from past semesters can be found [[AMS Student Chapter Seminar, previous semesters|here]].
The schedule of talks from past semesters can be found [[AMS Student Chapter Seminar, previous semesters|here]].


== Spring 2019 ==
== Spring 2023 ==


=== February 6, Xiao Shen (in VV B139)===
=== January 25, Michael Jeserum ===


Title: Limit Shape in last passage percolation


Abstract: Imagine the following situation, attached to each point on the integer lattice Z^2 there is an arbitrary amount of donuts.  Fix x and y in Z^2, if you get to eat all the donuts along an up-right path between these two points, what would be the maximum amount of donuts you can get? This model is often called last passage percolation, and I will discuss a classical result about its scaling limit: what happens if we zoom out and let the distance between x and y tend to infinity.


=== February 13, Michel Alexis (in VV B139)===
Title: Totally Realistic Supply Chains


Title: An instructive yet useless theorem about random Fourier Series
Abstract: Inspired by a group of fifth and sixth graders, we'll embark on a journey to discover how supply chains definitely work in real life. Along the way, we'll eat donuts, learn about graphs and the magical world of chip-firing, and maybe even make new friends!


Abstract: Consider a Fourier series with random, symmetric, independent coefficients. With what probability is this the Fourier series of a continuous function? An <math>L^{p}</math> function? A surprising result is the Billard theorem, which says such a series results almost surely from an <math>L^{\infty}</math> function if and only if it results almost surely from a continuous function. Although the theorem in of itself is kind of useless in of itself, its proof is instructive in that we will see how, via the principle of reduction, one can usually just pretend all symmetric random variables are just coin flips (Bernoulli trials with outcomes <math>\pm 1</math>).
=== February 1, Summer al Hamdani ===


=== February 20, Geoff Bentsen ===


Title: An Analyst Wanders into a Topology Conference


Abstract: Fourier Restriction is a big open problem in Harmonic Analysis; given a "small" subset <math>E</math> of <math>R^d</math>, can we restrict the Fourier transform of an <math>L^p</math> function to <math>E</math> and retain any information about our original function? This problem has a nice (somewhat) complete solution for smooth manifolds of co-dimension one. I will explore how to start generalizing this result to smooth manifolds of higher co-dimension, and how a topology paper from the 60s about the hairy ball problem came in handy along the way.
Title: Monkeying Around: On the Infinite Monkey Theorem


=== February 27, James Hanson ===
Abstract: Will monkeys keyboard bashing eventually type all of Hamlet? Yes, almost surely. We will discuss the history and proof of the infinite monkey theorem.


Title: What is...a Topometric Space?
=== February 8, Dionel Jaime ===


Abstract: Continuous first-order logic is a generalization of first-order logic that is well suited for the study of structures with a natural metric, such as Banach spaces and probability algebras. Topometric spaces are a simple generalization of topological and metric spaces that arise in the study of continuous first-order logic. I will discuss certain topological issues that show up in topometric spaces coming from continuous logic, as well as some partial solutions and open problems. No knowledge of logic will be required for or gained from attending the talk.


=== March 6, Working Group to establish an Association of Mathematics Graduate Students ===


Title: Introducing GRAMS (Graduate Representative Association of Mathematics Students)
Title: The weird world of polynomial curve fitting.


Abstract: Over the past couple months, a handful of us have been working to create the UW Graduate Representative Association of Mathematics Students (GRAMS). This group, about 5-8 students, is intended to be a liaison between the graduate students and faculty, especially in relation to departmental policies and decisions that affect graduate students. We will discuss what we believe GRAMS ought to look like and the steps needed to implement such a vision, then open up the floor to a Q&A. Check out our [http://sites.google.com/wisc.edu/grams/home website] for more information.
Abstract: You have some continuous function, and you decide you want to find a polynomial curve that looks a lot like your function. That is a very smart and easy thing to do. Nothing will go wrong.


=== March 13, Connor Simpson ===
=== February 15, Sun Woo Park ===


Title: Counting faces of polytopes with algebra


Abstract: A natural question is: with a fixed dimension and number of vertices, what is the largest number of d-dimensional faces that a polytope can have? We will outline a proof of the answer to this question.
Title: What I did in my military service (Universal covers and graph neural networks)


=== March 26 (Prospective Student Visit Day), Multiple Speakers ===
Abstract: I'll try to motivate the relations between universal covers of graphs and graph isomorphism classification tasks implemented from graph neural networks. This is a summary of what I did during my 3 years of leave of absence due to compulsory military service in South Korea. Don't worry, everything I'll present here is already made public and not confidential, so you don't need to worry about the South Korean government officials suddenly appearing during the seminar and accusing me of misconduct!


====Eva Elduque, 11-11:25====
=== February 22: NO SEMINAR ===


Title: Will it fold flat?
=== February 28, Owen Goff ===
Title: The RSK Correspondence


Abstract: Picture the traditional origami paper crane. It is a 3D object, but if you don’t make the wings stick out, it is flat. This is the case for many origami designs, ranging from very simple (like a paper hat), to complicated tessellations. Given a crease pattern on a piece of paper, one might wonder if it is possible to fold along the lines of the pattern and end up with a flat object. We’ll discuss necessary and sufficient conditions for a crease pattern with only one vertex to fold flat, and talk about what can be said about crease patterns with multiple vertices.
Abstract: In this talk I will show a brilliant 1-to-1 mapping between permutations on n elements and pairs of Standard Young Tableau of size n. This bijection, known as the Robinson-Schensted-Knuth correspondence, has many beautiful properties. It also tells you the best way for people to escape a series of rooms.


====Soumya Sankar, 11:30-11:55====
=== March 8, Pubo Huang  ===


Title: An algebro-geometric perspective on integration


Abstract: Integrals are among the most basic tools we learn in complex analysis and yet are extremely versatile. I will discuss one way in which integrals come up in algebraic geometry and the surprising amount of arithmetic and geometric information this gives us.


====Chun Gan, 12:00-12:25====
Title: 2-dimensional Dynamical Billiards


Title: Extension theorems in complex analysis
Abstract: We have all played, or watched, a game of pool, and you probably noticed that when a ball hits the cushion on the table, its angle of rebound is equal to its angle of incidence.


Abstract: Starting from Riemann's extension theorem in one complex variable, there have been many generalizations to different situations in several complex variables. I will talk about Fefferman's field's medal work on Fefferman extension and also the celebrated Ohsawa-Takegoshi L^2 extension theorem which is now a cornerstone for the application of pluripotential theory to complex analytic geometry.
Dynamical Billiards is an idealization and generalization of the popular game called pool (or billiards, or snooker), and it aims to understand the trajectory (as time goes to infinity) of a ball on a frictionless table that rebounds perfectly. During the talk, I will provide a lot of examples of dynamical billiards on an actual table and compare it with its mathematical counterpart. We will also see how we can relate billiards on a rectangular table to the classical example of circle rotation in dynamics.


====Jenny Yeon, 2:00-2:25====
=== March 15: NO SEMINAR (SPRING BREAK) ===


Title: Application of Slope Field
=== March 22: Vicky Wen ===


Abstract: Overview of historical problems in Dynamical Systems and what CRN(chemical reaction network) group at UW Madison does. In particular, what exactly is the butterfly effect? Why is this simple-to-state problem so hard even if it is only 2D (Hilbert's 16th problem)? What are some modern techniques availble? What do the members of CRN group do? Is the theory of CRN applicable? 


====Rajula Srivastava, 2:30-2:55====
Title: On Mostow's Rigidity Theorem


Title: The World of Wavelets
Abstract: Mostow rigidity is one of those famous theorems in hyperbolic geometry that links the topology and geometry of a hyperbolic space (aka a Riemannian manifold with constant curvature -1). It states that in higher dimension (n>2), the geometry of the space is completely determined by its fundamental group, which is a quiet strong and amazing result. In this talk I will try to explain the idea behind the proof and give some counterexamples in dimension 2.


Abstract: Why the fourier series might not be the best way to represent functions in all cases, and why wavelets might be a good alternative in some of these.
=== March 24: VISIT DAY SPECIAL SESSIONS  ===


====Shengyuan Huang, 3:00-3:25====


Title: Group objects in various categories


Abstract: I will introduce categories and talk about group objects in the category of sets and manifolds.  The latter leads to the theory of Lie group and Lie algebras.  We can then talk about group objects in some other category coming from algebraic geometry and obtain similar results as Lie groups and Lie algebras.
Title: Log concavity properties and combinatorial Hodge theory


====Ivan Ongay Valverde, 3:30-3:55====
Speaker: Colin Crowley


Title: Games and Topology
Abstract: Combinatorial Hodge theory is a newly created field (past decade) at the intersection of combinatorics and algebraic geometry. It has lead to proofs of long standing conjectures about matroids, which are objects that generalize finite graphs. I'll introduce some of the main objects, and tell a rough story of how this field came to be.


Abstract: Studying the topology of the real line leads to really interesting questions and facts. One of them is the relation between some kind of infinite games, called topological games, and specific properties of a subsets of reals. In this talk we will study the perfect set game.
Title: Commutative algebra and geometry of systems of polynomials


====Sun Woo Park, 4:00-4:25====
Speaker: Maya Banks


Title: Reconstruction of character tables of Sn
Abstract: When your favorite computer algebra system solves systems of polynomials, it does so by computing something called a Groebner Basis. Groebner bases are collections of polynomials that have many algebraic and geometric properties that make them especially well suited for solving both computational and theoretical problems in commutative algebra and algebraic geometry. I’ll talk about how we (and our computers) make use of these tools and what behind-the-scenes algebra and geometry makes them special.


Abstract: We will discuss how we can relate the columns of the character tables of Sn and the tensor product of irreducible representations over Sn. Using the relation, we will also indicate how we can recover some columns of character tables of Sn. 


====Max Bacharach, 4:30-4:55====


Title: Clothes, Lice, and Coalescence
Title: Markov chains and upper bounds on ranks of quadratic twists of an elliptic curve.


Abstract: A gentle introduction to coalescent theory, motivated by an application which uses lice genetics to estimate when human ancestors first began wearing clothing.
Speaker: Sun Woo Park


=== April 3, Yu Feng ===
Abstract: I will try to give a heuristic argument on how one can use Markov chains to understand the dimensions of some families of finite dimensional vector spaces over F2 (the finite field with 2 elements), which can be used to compute an upper bound on the rank of families of quadratic twists of an elliptic curve. The talk I will deliver will assume background in vector spaces / linear algebra over finite fields, and no prior knowledge about elliptic curves will be required.


Title: Suppression of phase separation by mixing


Abstract: The Cahn-Hilliard equation is a classical PDE that models phase separation of two components. We add an advection term so that the two components are stirred by a velocity. We show that there exists a class of fluid that can prevent phase separation and enforce the solution converges to its average exponentially.


=== April 17, Hyun-Jong Kim===
Title: Coherent Structures in Convection.


Title: Musical Harmony for the Mathematician
Speaker: Varun Gudibanda


Abstract: TBD
Abstract: Have you ever boiled water? If so, then that's really great I hope you made some tea. It also means that you are familiar with the concept of convection. In convective systems, there are fundamental structures which play an important role in dictating the heat transport and other properties of the system. Let's explore these structures and also learn about how a single number has divided a community of researchers for decades.


=== April 24, Carrie Chen ===


Title: TBD


Abstract: TBD
Title: Morse Theory in Algebraic Topology (According to ChatGPT)


== Fall 2019 ==
Speaker: Alex Hof


=== September 25, TBD===


Title: TBD


Abstract: TBD
Title: Life in a Hyperbolic City


=== October 2, TBD===
Speaker: Daniel Levitin


Title: TBD
Abstract: I will discuss the most important reason prospective students should come to UW Madison: the (almost) locally Euclidean geometry, and how much of a mess it would be to live in a hyperbolic city. I will then talk about some related concepts in geometric group theory. This should provide a soft introduction to the colloquium talk as well.


Abstract: TBD


=== October 9, TBD===


Title: TBD
Title: Logic: What is it good for?


Abstract: TBD
Speaker: John Spoerl


=== October 16, TBD===
Abstract: What are the logicians doing in the math department? Are they philosophers or computer scientists in disguise? (No.) How can I be as cool and mysterious as the logicians? We’ll see how the methods of logic are the most “effective” ways to do mathematics.


Title: TBD


Abstract: TBD


=== October 23, TBD===
Title: Fourier restriction and Kakeya problems


Title: TBD
Speaker: Mingfeng Chen


Abstract: TBD
Abstract: Fourier restriction problem was introduced by Elias Stein in the 1970s. It is a central problem in Harmonic analysis. Moreover, restriction problems have close connections with other important questions in Geometric Measure theory(Kakeya problem), Harmonic analysis, combinatorics, number theory and PDE. In this talk, I'm going to give a simple introduction to what it is and what we are going to do.


=== October 30, TBD===
=== March 29, Ivan Aidun ===


Title: TBD


Abstract: TBD


=== November 6, TBD===
'''Title:''' Fractional Calculus


Title: TBD
'''Abstract:''' We teach our calculus students about 1<sup>st</sup> and 2<sup>nd</sup> derivatives, but what about 1/2th derivatives?  What about πth derivatives?  Can we make sense of these derivatives?  Can we use them for anything?


Abstract: TBD
=== April 5, Diego Rojas La Luz ===


=== November 13, TBD===


Title: TBD


Abstract: TBD
Title: Eating a poisoned chocolate bar


=== November 20, TBD===
Abstract: Today we are going to talk about Chomp, a game where you take turns eating chocolate and you try not to die from poisoning. This is one of those very easy-to-state combinatoric games which happens to be very hard to fully analyze. We'll see that we can say some surprising things regarding winning strategies, so stay tuned for that. Who wants to play?


Title: TBD
=== April 12, Taylor Tan ===


Abstract: TBD


=== December 4, TBD===


Title: TBD
Title: A Proof From The Hall of Fame -- Topological Methods in Combinatorics


Abstract: TBD
Abstract: Consider the collection of all n-sets from a 2n+k element ground set. This collection can be partitioned into k+2 partite classes such that there are no intersections between n-sets in the same partite class. In 1955, Kneser conjectured that this bound was sharp, but the problem remained open for two decades until László Lovász gave a proof through topological methods in 1978, thereby inventing the field of topological combinatorics. Another few decades later, a greatly simplified proof (it fits in one paragraph!) was discovered by Joshua Greene and his beautiful proof will be presented in all its glory.


=== December 12, TBD===
=== April 19, NO SEMINAR  ===


Title: TBD
=== April 26, Hyun Jong Kim  ===


Abstract: TBD
 
Title: Machine Learning Tools for the Working Mathematician
 
Abstract: Mathematicians often have to learn new concepts. I will briefly present <code>trouver</code>​, a Python librarythat I have been developing that uses machine learning models to help this process. In particular, <code>trouver</code>​ can categorize types of mathematical text, identify where notations are introduced in such mathematical text, and attempt to summarize what these notations denote. I will also talk about some high-level ideas go into training such machine learning models in the modern day without huge amounts of data and computational resources.
 
=== May 3, Asvin G ===
Title: On the random graph on countably many vertices 
 
Abstract: I will tell you about "the" graph on countably many vertices. It has many remarkable properties - for instance, any "property" true of it is true for almost all finite graphs!
 
== Spring 2022 ==
 
=== February 9, Alex Mine ===
Title: Would you like to play a game?
 
Abstract: We'll look at some fun things in combinatorial game theory.
 
=== February 16, Michael Jeserum ===
Title: The Internet's Take on Number Bases
 
Abstract: Inspired by a TikTok video, we'll embark on a journey to find the best number base to work in*.
 
<nowiki>*</nowiki>Disclaimer: audience may not actually learn what the best number base is.
 
=== February 23, Erika Pirnes ===
Title: Staying Balanced- studying the balanced algebra
 
Abstract: The balanced algebra has two generators, R and L, and its defining relations are that any pair of balanced words commutes. For example, RL and LR are balanced (contain the same number of both generators), so in the balanced algebra, (RL)(LR)=(LR)(RL). The goal is to find out which pairs are required to commute in order to make any pair of balanced words commute. This talk includes beautiful mountain landscapes and requires very minimal background knowledge.
 
=== March 2, Jason Torchinsky ===
Title: Holmes and Watson and the case of the tropical climate
 
Abstract: With a case as complex as the tropical climate, who else could you call? In this talk, we will discuss a strategy for getting models to team up to create a faithful simulation through an analogy of the original sleuthing dynamic duo, Sherlock Holmes and Dr. James Watson.
 
=== March 7, Devanshi Merchant ===
Title: Mathematics of soap films
 
Abstract: Nature is a miser when it comes to energy. This tendency, in case of soap films motivates mathematicians to study minimal surfaces. This study leads to some beautiful geometry that we will explore.
 
=== March 30, Jacob Denson ===
Title: Proofs in 3 bits or less
 
Abstract: What can you prove with a string of bits? Is there a proof of Fermat's Last Theorem of the form: "101"? Let's eat donuts, and then talk about it.
 
=== April 6, Aidan Howells ===
Title: Goodstein Sequences, Hercules, and the Hydra
 
Abstract: We'll discuss Goodstein sequences, Goodstein's theorem, and the Kirby–Paris theorem. We'll relate this to the hydra game of Kirby and Paris. The next time you are  supposed to be working, instead check out the hydra game here: <nowiki>http://www.madore.org/~david/math/hydra0.xhtml</nowiki>
 
Can you beat the hydra? Can you devise a winning strategy, and prove that it always wins? If that's too easy, a harder Hydra game is here: <nowiki>http://www.madore.org/~david/math/hydra.xhtml</nowiki>
 
=== April 13, Yu Fu ===
Title: How do generic properties spread?
 
Abstract: Given a family of algebraic varieties, a natural question to ask is what type of properties of the generic fiber, and how those properties extend to other fibers. Let's explore this topic from an arithmetic point of view by looking at an example: given a 1-dim family of pairs of elliptic curves with the generic fiber be a pair of isogenous elliptic curves, how the property of 'being isogenous' extend to other fibers?
 
=== April 20, Ivan Aidun ===
Title: The are no Orthogonal Latin Squares of Order 6
 
Abstract: The title says it all.
 
== Fall 2021 ==
 
=== September 29, John Cobb ===
 
Title: Rooms on a Sphere
 
Abstract: A classic combinatorial lemma becomes very simple to state and prove when on the surface of a sphere, leading to easy constructive proofs of some other well known theorems.
 
=== October 6, Karan Srivastava ===
 
Title: An 'almost impossible' puzzle and group theory
 
Abstract: You're given a chessboard with a randomly oriented coin on every square and a key hidden under one of them; player one knows where the key is and flips a single coin; player 2, using only the information of the new coin arrangement must determine where the key is. Is there a winning strategy? In this talk, we will explore this classic puzzle in a more generalized context, with n squares and d sided dice on every square. We'll see when the game is solvable and in doing so, see how the answer relies on group theory and the existence of certain groups.
 
=== October 13, John Yin ===
 
Title: TBA
 
Abstract: TBA
 
=== October 20, Varun Gudibanda ===
 
Title: TBA
 
Abstract: TBA
 
=== October 27, Andrew Krenz ===
 
Title: The 3-sphere via the Hopf fibration
 
Abstract: The Hopf fibration is a map from $S^3$ to $S^2$.  The preimage (or fiber) of every point under this map is a copy of $S^1$.  In this talk I will explain exactly how these circles “fit together” inside the 3-sphere.  Along the way we’ll discover some other interesting facts in some hands-on demonstrations using paper and scissors.  If there is time I hope to also relate our new understanding of $S^3$ to some other familiar models.
=== November 3, Asvin G  ===
 
Title: Probabilistic methods in math
 
Abstract: I'll explain how you can provr that something has to be true because it's probably true in a couple of examples. One of the proofs is by Erdos on the "sum set problem" and it is a proof that "only an alien could have come up with" according to a friend.
 
=== November 10, Ivan Aidun ===
[[File:Screen Shot 2021-11-15 at 3.25.38 PM.png|thumb]]
Title: Intersection Permutations
 
Abstract: During a boring meeting, your buddy slips you a Paris metro ticket with this cryptic diagram (see left).
 
What could it mean?  The only way to find out is to come to this Donut Talk!
 
=== December 1, Yuxi Han  ===
 
Title: Homocidal Chaffeur Problem
 
Abstract: I will briefly introduce the canonical example of differential games, called the homicidal chauffeur problem and how to use PDE to run down pedestrians optimally.
 
=== December 8, Owen Goff  ===
 
Title: The Mathematics of Cribbage
 
Abstract: Cribbage is a card game that I have played many times over the years, that involves, among other things, finding subsets of set of numbers that equal a specific value (in the game that value is 15). In this donut talk I will attempt to use the power of combinatorics to find the optimal strategy for this game, particularly to solve one problem -- is there a way you can guarantee yourself at least one extra point by adding an additional card to your set?

Revision as of 01:03, 1 July 2023

The AMS Student Chapter Seminar (aka Donut Seminar) is an informal, graduate student seminar on a wide range of mathematical topics. The goal of the seminar is to promote community building and give graduate students an opportunity to communicate fun, accessible math to their peers in a stress-free (but not sugar-free) environment. Pastries (usually donuts) will be provided.

  • When: Wednesdays, 3:30 PM – 4:00 PM
  • Where: Van Vleck, 9th floor lounge (unless otherwise announced)
  • Organizers: Ivan Aidun, Kaiyi Huang, Ethan Schondorf

Everyone is welcome to give a talk. To sign up, please contact one of the organizers with a title and abstract. Talks are 25 minutes long and should avoid assuming significant mathematical background beyond first-year graduate courses.

The schedule of talks from past semesters can be found here.

Spring 2023

January 25, Michael Jeserum

Title: Totally Realistic Supply Chains

Abstract: Inspired by a group of fifth and sixth graders, we'll embark on a journey to discover how supply chains definitely work in real life. Along the way, we'll eat donuts, learn about graphs and the magical world of chip-firing, and maybe even make new friends!

February 1, Summer al Hamdani

Title: Monkeying Around: On the Infinite Monkey Theorem

Abstract: Will monkeys keyboard bashing eventually type all of Hamlet? Yes, almost surely. We will discuss the history and proof of the infinite monkey theorem.

February 8, Dionel Jaime

Title: The weird world of polynomial curve fitting.

Abstract: You have some continuous function, and you decide you want to find a polynomial curve that looks a lot like your function. That is a very smart and easy thing to do. Nothing will go wrong.

February 15, Sun Woo Park

Title: What I did in my military service (Universal covers and graph neural networks)

Abstract: I'll try to motivate the relations between universal covers of graphs and graph isomorphism classification tasks implemented from graph neural networks. This is a summary of what I did during my 3 years of leave of absence due to compulsory military service in South Korea. Don't worry, everything I'll present here is already made public and not confidential, so you don't need to worry about the South Korean government officials suddenly appearing during the seminar and accusing me of misconduct!

February 22: NO SEMINAR

February 28, Owen Goff

Title: The RSK Correspondence

Abstract: In this talk I will show a brilliant 1-to-1 mapping between permutations on n elements and pairs of Standard Young Tableau of size n. This bijection, known as the Robinson-Schensted-Knuth correspondence, has many beautiful properties. It also tells you the best way for people to escape a series of rooms.

March 8, Pubo Huang

Title: 2-dimensional Dynamical Billiards

Abstract: We have all played, or watched, a game of pool, and you probably noticed that when a ball hits the cushion on the table, its angle of rebound is equal to its angle of incidence.

Dynamical Billiards is an idealization and generalization of the popular game called pool (or billiards, or snooker), and it aims to understand the trajectory (as time goes to infinity) of a ball on a frictionless table that rebounds perfectly. During the talk, I will provide a lot of examples of dynamical billiards on an actual table and compare it with its mathematical counterpart. We will also see how we can relate billiards on a rectangular table to the classical example of circle rotation in dynamics.

March 15: NO SEMINAR (SPRING BREAK)

March 22: Vicky Wen

Title: On Mostow's Rigidity Theorem

Abstract: Mostow rigidity is one of those famous theorems in hyperbolic geometry that links the topology and geometry of a hyperbolic space (aka a Riemannian manifold with constant curvature -1). It states that in higher dimension (n>2), the geometry of the space is completely determined by its fundamental group, which is a quiet strong and amazing result. In this talk I will try to explain the idea behind the proof and give some counterexamples in dimension 2.

March 24: VISIT DAY SPECIAL SESSIONS

Title: Log concavity properties and combinatorial Hodge theory

Speaker: Colin Crowley

Abstract: Combinatorial Hodge theory is a newly created field (past decade) at the intersection of combinatorics and algebraic geometry. It has lead to proofs of long standing conjectures about matroids, which are objects that generalize finite graphs. I'll introduce some of the main objects, and tell a rough story of how this field came to be.

Title: Commutative algebra and geometry of systems of polynomials

Speaker: Maya Banks

Abstract: When your favorite computer algebra system solves systems of polynomials, it does so by computing something called a Groebner Basis. Groebner bases are collections of polynomials that have many algebraic and geometric properties that make them especially well suited for solving both computational and theoretical problems in commutative algebra and algebraic geometry. I’ll talk about how we (and our computers) make use of these tools and what behind-the-scenes algebra and geometry makes them special.


Title: Markov chains and upper bounds on ranks of quadratic twists of an elliptic curve.

Speaker: Sun Woo Park

Abstract: I will try to give a heuristic argument on how one can use Markov chains to understand the dimensions of some families of finite dimensional vector spaces over F2 (the finite field with 2 elements), which can be used to compute an upper bound on the rank of families of quadratic twists of an elliptic curve. The talk I will deliver will assume background in vector spaces / linear algebra over finite fields, and no prior knowledge about elliptic curves will be required.


Title: Coherent Structures in Convection.

Speaker: Varun Gudibanda

Abstract: Have you ever boiled water? If so, then that's really great I hope you made some tea. It also means that you are familiar with the concept of convection. In convective systems, there are fundamental structures which play an important role in dictating the heat transport and other properties of the system. Let's explore these structures and also learn about how a single number has divided a community of researchers for decades.


Title: Morse Theory in Algebraic Topology (According to ChatGPT)

Speaker: Alex Hof


Title: Life in a Hyperbolic City

Speaker: Daniel Levitin

Abstract: I will discuss the most important reason prospective students should come to UW Madison: the (almost) locally Euclidean geometry, and how much of a mess it would be to live in a hyperbolic city. I will then talk about some related concepts in geometric group theory. This should provide a soft introduction to the colloquium talk as well.


Title: Logic: What is it good for?

Speaker: John Spoerl

Abstract: What are the logicians doing in the math department? Are they philosophers or computer scientists in disguise? (No.) How can I be as cool and mysterious as the logicians? We’ll see how the methods of logic are the most “effective” ways to do mathematics.


Title: Fourier restriction and Kakeya problems

Speaker: Mingfeng Chen

Abstract: Fourier restriction problem was introduced by Elias Stein in the 1970s. It is a central problem in Harmonic analysis. Moreover, restriction problems have close connections with other important questions in Geometric Measure theory(Kakeya problem), Harmonic analysis, combinatorics, number theory and PDE. In this talk, I'm going to give a simple introduction to what it is and what we are going to do.

March 29, Ivan Aidun

Title: Fractional Calculus

Abstract: We teach our calculus students about 1st and 2nd derivatives, but what about 1/2th derivatives?  What about πth derivatives?  Can we make sense of these derivatives?  Can we use them for anything?

April 5, Diego Rojas La Luz

Title: Eating a poisoned chocolate bar

Abstract: Today we are going to talk about Chomp, a game where you take turns eating chocolate and you try not to die from poisoning. This is one of those very easy-to-state combinatoric games which happens to be very hard to fully analyze. We'll see that we can say some surprising things regarding winning strategies, so stay tuned for that. Who wants to play?

April 12, Taylor Tan

Title: A Proof From The Hall of Fame -- Topological Methods in Combinatorics

Abstract: Consider the collection of all n-sets from a 2n+k element ground set. This collection can be partitioned into k+2 partite classes such that there are no intersections between n-sets in the same partite class. In 1955, Kneser conjectured that this bound was sharp, but the problem remained open for two decades until László Lovász gave a proof through topological methods in 1978, thereby inventing the field of topological combinatorics. Another few decades later, a greatly simplified proof (it fits in one paragraph!) was discovered by Joshua Greene and his beautiful proof will be presented in all its glory.

April 19, NO SEMINAR

April 26, Hyun Jong Kim

Title: Machine Learning Tools for the Working Mathematician

Abstract: Mathematicians often have to learn new concepts. I will briefly present trouver​, a Python librarythat I have been developing that uses machine learning models to help this process. In particular, trouver​ can categorize types of mathematical text, identify where notations are introduced in such mathematical text, and attempt to summarize what these notations denote. I will also talk about some high-level ideas go into training such machine learning models in the modern day without huge amounts of data and computational resources.

May 3, Asvin G

Title: On the random graph on countably many vertices

Abstract: I will tell you about "the" graph on countably many vertices. It has many remarkable properties - for instance, any "property" true of it is true for almost all finite graphs!

Spring 2022

February 9, Alex Mine

Title: Would you like to play a game?

Abstract: We'll look at some fun things in combinatorial game theory.

February 16, Michael Jeserum

Title: The Internet's Take on Number Bases

Abstract: Inspired by a TikTok video, we'll embark on a journey to find the best number base to work in*.

*Disclaimer: audience may not actually learn what the best number base is.

February 23, Erika Pirnes

Title: Staying Balanced- studying the balanced algebra

Abstract: The balanced algebra has two generators, R and L, and its defining relations are that any pair of balanced words commutes. For example, RL and LR are balanced (contain the same number of both generators), so in the balanced algebra, (RL)(LR)=(LR)(RL). The goal is to find out which pairs are required to commute in order to make any pair of balanced words commute. This talk includes beautiful mountain landscapes and requires very minimal background knowledge.

March 2, Jason Torchinsky

Title: Holmes and Watson and the case of the tropical climate

Abstract: With a case as complex as the tropical climate, who else could you call? In this talk, we will discuss a strategy for getting models to team up to create a faithful simulation through an analogy of the original sleuthing dynamic duo, Sherlock Holmes and Dr. James Watson.

March 7, Devanshi Merchant

Title: Mathematics of soap films

Abstract: Nature is a miser when it comes to energy. This tendency, in case of soap films motivates mathematicians to study minimal surfaces. This study leads to some beautiful geometry that we will explore.

March 30, Jacob Denson

Title: Proofs in 3 bits or less

Abstract: What can you prove with a string of bits? Is there a proof of Fermat's Last Theorem of the form: "101"? Let's eat donuts, and then talk about it.

April 6, Aidan Howells

Title: Goodstein Sequences, Hercules, and the Hydra

Abstract: We'll discuss Goodstein sequences, Goodstein's theorem, and the Kirby–Paris theorem. We'll relate this to the hydra game of Kirby and Paris. The next time you are  supposed to be working, instead check out the hydra game here: http://www.madore.org/~david/math/hydra0.xhtml

Can you beat the hydra? Can you devise a winning strategy, and prove that it always wins? If that's too easy, a harder Hydra game is here: http://www.madore.org/~david/math/hydra.xhtml

April 13, Yu Fu

Title: How do generic properties spread?

Abstract: Given a family of algebraic varieties, a natural question to ask is what type of properties of the generic fiber, and how those properties extend to other fibers. Let's explore this topic from an arithmetic point of view by looking at an example: given a 1-dim family of pairs of elliptic curves with the generic fiber be a pair of isogenous elliptic curves, how the property of 'being isogenous' extend to other fibers?

April 20, Ivan Aidun

Title: The are no Orthogonal Latin Squares of Order 6

Abstract: The title says it all.

Fall 2021

September 29, John Cobb

Title: Rooms on a Sphere

Abstract: A classic combinatorial lemma becomes very simple to state and prove when on the surface of a sphere, leading to easy constructive proofs of some other well known theorems.

October 6, Karan Srivastava

Title: An 'almost impossible' puzzle and group theory

Abstract: You're given a chessboard with a randomly oriented coin on every square and a key hidden under one of them; player one knows where the key is and flips a single coin; player 2, using only the information of the new coin arrangement must determine where the key is. Is there a winning strategy? In this talk, we will explore this classic puzzle in a more generalized context, with n squares and d sided dice on every square. We'll see when the game is solvable and in doing so, see how the answer relies on group theory and the existence of certain groups.

October 13, John Yin

Title: TBA

Abstract: TBA

October 20, Varun Gudibanda

Title: TBA

Abstract: TBA

October 27, Andrew Krenz

Title: The 3-sphere via the Hopf fibration

Abstract: The Hopf fibration is a map from $S^3$ to $S^2$. The preimage (or fiber) of every point under this map is a copy of $S^1$. In this talk I will explain exactly how these circles “fit together” inside the 3-sphere. Along the way we’ll discover some other interesting facts in some hands-on demonstrations using paper and scissors. If there is time I hope to also relate our new understanding of $S^3$ to some other familiar models.

November 3, Asvin G

Title: Probabilistic methods in math

Abstract: I'll explain how you can provr that something has to be true because it's probably true in a couple of examples. One of the proofs is by Erdos on the "sum set problem" and it is a proof that "only an alien could have come up with" according to a friend.

November 10, Ivan Aidun

Screen Shot 2021-11-15 at 3.25.38 PM.png

Title: Intersection Permutations

Abstract: During a boring meeting, your buddy slips you a Paris metro ticket with this cryptic diagram (see left).

What could it mean?  The only way to find out is to come to this Donut Talk!

December 1, Yuxi Han

Title: Homocidal Chaffeur Problem

Abstract: I will briefly introduce the canonical example of differential games, called the homicidal chauffeur problem and how to use PDE to run down pedestrians optimally.

December 8, Owen Goff

Title: The Mathematics of Cribbage

Abstract: Cribbage is a card game that I have played many times over the years, that involves, among other things, finding subsets of set of numbers that equal a specific value (in the game that value is 15). In this donut talk I will attempt to use the power of combinatorics to find the optimal strategy for this game, particularly to solve one problem -- is there a way you can guarantee yourself at least one extra point by adding an additional card to your set?

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