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The AMS Student Chapter Seminar is an informal, graduate student-run 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, 3:00 PM – 3:30 PM
* '''When:''' Wednesdays, 3:30 PM – 4:00 PM
* '''Where:''' Van Vleck, 9th floor lounge
* '''Where:''' Van Vleck, 9th floor lounge (unless otherwise announced)
* '''Organizers:''' Daniel Hast, Ryan Julian, Laura Cladek, Cullen McDonald, Zachary Charles
* '''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.


== Fall 2015 ==
The schedule of talks from past semesters can be found [[AMS Student Chapter Seminar, previous semesters|here]].


=== October 7, TBA ===
== Spring 2023 ==


=== October 14, TBA ===
=== January 25, Michael Jeserum ===


=== October 21, TBA ===


=== October 28, TBA ===


=== November 4, TBA ===
Title: Totally Realistic Supply Chains


=== November 11, TBA ===
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!


=== November 18, TBA ===
=== February 1, Summer al Hamdani ===


=== December 2, TBA ===


=== December 9, TBA ===


==Spring 2015==
Title: Monkeying Around: On the Infinite Monkey Theorem


===January 28, Moisés Herradón===
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: Winning games and taking names
=== February 8, Dionel Jaime ===


Abstract:  So let’s say we’re already amazing at playing one game (any game!) at a time and we now we need to play several games at once, to keep it challenging. We will see that doing this results in us being able to define an addition on the collection of all games, and that it actually turns this collection into a Group. I will talk about some of the wonders that lie within the group. Maybe lions? Maybe a field containing both the real numbers and the ordinals? For sure it has to be one of these two!


===February 11, Becky Eastham===


Title: A generalization of van der Waerden numbers: (a, b) triples and (a_1, a_2, ..., a_n) (n + 1)-tuples
Title: The weird world of polynomial curve fitting.


Abstract: Van der Waerden defined w(k; r) to be the least positive integer such that for every r-coloring of the integers from 1 to w(k; r), there is a monochromatic arithmetic progression of length k.  He proved that w(k; r) exists for all positive k, r.  I will discuss the case where r = 2.  These numbers are notoriously hard to calculate: the first 6 of these are 1, 3, 9, 35, 178, and 1132, but no others are known. I will discuss properties of a generalization of these numbers, (a_1, a_2, ..., a_n) (n + 1)-tuples, which are sets of the form {d, a_1x + d, a_2x + 2d, ..., a_nx + nd}, for d, x positive natural numbers.
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 18, Solly Parenti===
=== February 15, Sun Woo Park ===


Title: Chebyshev's Bias


Abstract: Euclid told us that there are infinitely many primes.  Dirichlet answered the question of how primes are distributed among residue classes.  This talk addresses the question of "Ya, but really, how are the primes distributed among residue classes?"  Chebyshev noted in 1853 that there seems to be more primes congruent to 3 mod 4 than their are primes congruent to 1 mod 4.  It turns out, he was right, wrong, and everything in between.  No analytic number theory is presumed for this talk, as none is known by the speaker.
Title: What I did in my military service (Universal covers and graph neural networks)


===February 25, David Bruce===
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!


Title: Mean, Median, and Mode - Well Actually Just Median
=== February 22: NO SEMINAR ===


Abstract: Given a finite set of numbers there are many different ways to measure the center of the set. Three of the more common measures, familiar to any middle school students, are: mean, median, mode. This talk will focus on the concept of the median, and why in many ways it's sweet. In particular, we will explore how we can extend the notion of a median to higher dimensions, and apply it to create more robust statistics. It will be awesome, and there will be donuts.
=== February 28, Owen Goff ===
Title: The RSK Correspondence


===March 4, Jing Hao===
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.


Title: Error Correction Codes
=== March 8, Pubo Huang  ===


Abstract: In the modern world, many communication channels are subject to noise, and thus errors happen. To help the codes auto-correct themselves, more bits are added to the codes to make them more different from each other and therefore easier to tell apart. The major object we study is linear codes. They have nice algebraic structure embedded, and we can apply well-known algebraic results to construct 'nice' codes. This talk will touch on the basics of coding theory, and introduce some famous codes in the coding world, including several prize problems yet to be solved!


===March 10 (Tuesday), Nathan Clement===


''Note: This week's seminar will be on Tuesday at 3:30 instead of the usual time.''
Title: 2-dimensional Dynamical Billiards


Title: Two Solutions, not too Technical, to a Problem to which the Answer is Two
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: A classical problem in Algebraic Geometry is this: Given four pairwise skew lines, how many other lines intersect all of them. I will present some (two) solutions to this problem. One is more classical and ad hoc and the other introduces the Grassmannian variety/manifold and a little intersection theory.
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 25, Eric Ramos===
=== March 15: NO SEMINAR (SPRING BREAK) ===


Title: Braids, Knots and Representations
=== March 22: Vicky Wen ===


Abstract: In the 1920's Artin defined the braid group, B_n, in an attempt to understand knots in a more algebraic setting. A braid is a certain arrangement of strings in three-dimensional space. It is a celebrated theorem of Alexander that every knot is obtainable from a braid by identifying the endpoints of each string. Because of this correspondence, the Jones and Alexander polynomials, two of the most important knot invariants, can be described completely using the braid group. In fact, Jones was able to show that knot invariants can often be realized as characters of special representations of the braid group.


The purpose of this talk is to give a very light introduction to braid and knot theory. The majority of the talk will be comprised of drawing pictures, and nothing will be treated rigorously.
Title: On Mostow's Rigidity Theorem


===April 8, James Waddington===
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.


Title: Goodstein's Theorem
=== March 24: VISIT DAY SPECIAL SESSIONS  ===


Abstract: One of the most important results in the development of mathematics are
Gödel's Incompleteness theorems. The first incompleteness theorem shows that no
list of axioms one could provide could extend number theory to a complete and
consistent theory. The second showed that one such statement was no
axiomatization of number theory could be used to prove its own consistency.
Needless to say this was not viewed as a very natural independent statement
from arithmetic.


Examples of non-metamathematical results that were independent of PA, but true
of second order number theory, were not discovered until much later. Within a
short time of each three such statements that were more "natural" were
discovered. The Paris–Harrington Theorem, which was about a statement in Ramsey
theory, the Kirby–Paris theorem, which showed the independence of Goodstein's
theorem from Peano Arithmetic and the Kruskal's tree theorem, a statement about
finite trees.


In this talk I shall discuss Goodstein's theorem which discusses the end
Title: Log concavity properties and combinatorial Hodge theory
behavior of a certain "Zero player" game about k-nary expansions of numbers.
I will also give some elements of the proof of the Kirby–Paris theorem.


===April 22, William Cocke===
Speaker: Colin Crowley


Title: Finite Groups aren't too Square
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: We investigate how many non-p-th powers a group can have for a given prime p.
Title: Commutative algebra and geometry of systems of polynomials
We will show using some elementary group theory, that if np(G) is the number of non-p-th powers
in a group G, then G has order bounded by np(G)(np(G)+1). Time permitting we will show this bound
is strict and that mentioned results involving more than finite groups.


==Fall 2014==
Speaker: Maya Banks


===September 25, Vladimir Sotirov===
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: [[Media:Compact-openTalk.pdf|The compact open topology: what is it really?]]


Abstract:  The compact-open topology on the space C(X,Y) of continuous functions from X to Y is mysteriously generated by declaring that for each compact subset K of X and each open subset V of Y, the continous functions f: X->Y conducting K inside V constitute an open set. In this talk, I will explain the universal property that uniquely determines the compact-open topology, and sketch a pretty constellation of little-known but elementary facts from domain theory that dispell the mystery of the compact-open topology's definition.


===October 8, David Bruce===
Title: Markov chains and upper bounds on ranks of quadratic twists of an elliptic curve.


Title: Hex on the Beach
Speaker: Sun Woo Park


Abstract: The game of Hex is a two player game played on a hexagonal grid attributed in part to John Nash. (This is the game he is playing in /A Beautiful Mind./) Despite being relatively easy to pick up, and pretty hard to master, this game has surprising connections to some interesting mathematics. This talk will introduce the game of Hex, and then explore some of these connections. *As it is a lot more fun once you've actually played Hex feel free to join me at 3:00pm on the 9th floor to actually play a few games of Hex!*
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.


===October 22, Eva Elduque===


Title: The fold and one cut problem


Abstract: What shapes can we get by folding flat a piece of paper and making (only) one complete straight cut? The answer is surprising: We can cut out any shape drawn with straight line segments. In the talk, we will discuss the two methods of approaching this problem, focusing on the straight skeleton method, the most intuitive of the two.
Title: Coherent Structures in Convection.  


===November 5, Megan Maguire===
Speaker: Varun Gudibanda


Title: Train tracks on surfaces
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.


Abstract: What is a train track, mathematically speaking? Are they interesting? Why are they interesting? Come find out!


===November 19, Adrian Tovar-Lopez===


Title: Hodgkin and Huxley equations of a single neuron
Title: Morse Theory in Algebraic Topology (According to ChatGPT)


===December 3, Zachary Charles===
Speaker: Alex Hof


Title:  Addition chains: To exponentiation and beyond


Abstract: An addition chain is a sequence of numbers starting at one, such that every number is the sum of two previous numbers. What is the shortest chain ending at a number n? While this is already difficult, we will talk about how addition chains answer life's difficult questions, including: How do we compute 2^4? What can the Ancient Egyptians teach us about elliptic curve cryptography? What about subtraction?
 
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 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?
 
=== 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 <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?

Latest 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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