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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, 3:30 PM – 4:00 PM
* '''When:''' Wednesdays, 3:30 PM – 4:00 PM
* '''Where:''' Van Vleck, B115
* '''Where:''' Van Vleck, 9th floor lounge (unless otherwise announced)
* '''Organizers:''' Daniel Hast, Ryan Julian, 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 2016 ==
The schedule of talks from past semesters can be found [[AMS Student Chapter Seminar, previous semesters|here]].


=== October 12, Soumya Sankar ===
== Spring 2023 ==


Title: Primes of certain forms and covering systems
=== January 25, Michael Jeserum ===


Abstract: A lot of classical questions revolve around primes of the form 2^n + k, where k is an odd integer. I will talk about such primes, or the lack thereof, and use this to convert coffee into covering systems. Time permitting, I'll talk about a few cool results and conjectures related to the notion of covering systems.


=== October 19, Daniel Hast ===


Title: A combinatorial lemma in linear algebra
Title: Totally Realistic Supply Chains


Abstract: I'll talk about a fun little lemma in linear algebra and its combinatorial interpretation. (It might be "well-known" to someone, but I'd never heard of it before.) If there's time, I'll discuss some possible generalizations.
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!


=== October 26, Brandon Alberts ===
=== February 1, Summer al Hamdani ===


Title: An Introduction to Matroids


Abstract: What if you wanted to do linear algebra, but couldn't use addition or scalar multiplication? Can we still have a notion of independence and bases? The answer is yes, and these are called matroids. Not only will I introduce matroids, but I will give an example that shows not all matroids arise from vector spaces.


=== November 2, Vlad Matei ===
Title: Monkeying Around: On the Infinite Monkey Theorem


Title: Hadamard Matrices
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.


Abstract: A Hadamard matrix is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. The most important open question in the theory of Hadamard matrices is that of existence. The Hadamard conjecture proposes that a Hadamard matrix of order 4k exists for every positive integer k. The Hadamard conjecture has also been attributed to Paley, although it was considered implicitly by others prior to Paley's work.
=== February 8, Dionel Jaime ===


=== November 9, David Bruce ===


Title: TBA


Abstract: TBA
Title: The weird world of polynomial curve fitting.


=== November 16, TBA ===
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.


=== November 30, TBA ===
=== February 15, Sun Woo Park ===


=== December 7, TBA ===


=== December 14, TBA ===
Title: What I did in my military service (Universal covers and graph neural networks)


== Spring 2016 ==
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!


=== January 27, Wanlin Li ===
=== February 22: NO SEMINAR ===


Title: The Nottingham group
=== February 28, Owen Goff ===
Title: The RSK Correspondence


Abstract: It's the group of wild automorphisms of the local field F_q((t)). It's a finitely generated pro-p group. It's hereditarily just infinite. Every finite p-group can be embedded in it. It's a favorite test case for conjectures concerning pro-p groups.  It's the Nottingham group! I will introduce you to this nice pro-p group which is loved by group theorists and number theorists.
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.


=== February 3, Will Cocke ===
=== March 8, Pubo Huang  ===


Title: Who or What is the First Order & Why Should I Care?


Abstract: As noted in recent films, the First Order is very powerful. We will discuss automated theorem proving software, including what exactly that means. We will then demonstrate some theorems, including previously unknown results, whose proofs can be mined from your computer.


=== February 10, Jason Steinberg ===
Title: 2-dimensional Dynamical Billiards


Title: Mazur's Swindle
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: If we sum the series 1-1+1-1+1-1+... in two ways, we get the nonsensical result 0=1 as follows: 0=(1-1)+(1-1)+(1-1)+...=1+(-1+1)+(-1+1)+...=1. While the argument is invalid in the context of adding infinitely many numbers together, there are other contexts throughout mathematics when it makes sense to take arbitrary infinite "sums" of objects in a way that these sums satisfy an infinite form of associativity. In such contexts, the above argument is valid. Examples of such contexts are connected sums of manifolds, disjoint unions of sets, and direct sums of modules, and in each case we can use this kind of argument to achieve nontrivial results fairly easily. Almost too easily...
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.


=== February 17, Zachary Charles ===
=== March 15: NO SEMINAR (SPRING BREAK) ===


Title: #P and Me: A tale of permanent complexity
=== March 22: Vicky Wen ===


Abstract: The permanent is the neglected younger sibling of the determinant. We will discuss the permanent, its properties, and its applications in graph theory and commutative algebra. We will then talk about computational complexity classes and why the permanent lies at a very strange place in the complexity hierarchy. If time permits, we will discuss operations with even sillier names, such as the immanant.


=== February 24, Brandon Alberts ===
Title: On Mostow's Rigidity Theorem


Title: The Rado Graph
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: A graph so unique, that a countably infinite random graph is isomorphic to the Rado Graph with probability 1. This talk will define the Rado Graph and walk through a proof of this surprising property.
=== March 24: VISIT DAY SPECIAL SESSIONS  ===


=== March 2, Vlad Matei ===


Title: Pythagoras numbers of fields


Abstract: The Pythagoras number of a field describes the structure of the set of squares in the field. The Pythagoras number p(K) of a field K is the smallest positive integer p such that every sum of squares in K is a sum of p squares.
Title: Log concavity properties and combinatorial Hodge theory


A pythagorean field is one with Pythagoras number 1: that is, every sum of squares is already a square.
Speaker: Colin Crowley


These fields have been studied for over a century and it all started with David Hilbert and his famous 17th problem and whether or not positive polynomial function on '''R'''^n can be written as a finite sum of squares of polynomial functions.
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.


We explore the history and various results and some unanswered questions.
Title: Commutative algebra and geometry of systems of polynomials


=== March 9, Micky Steinberg ===
Speaker: Maya Banks


Title: The Parallel Postulate and Non-Euclidean Geometry.
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:
“Is Euclidean Geometry true? It has no meaning. We might as well ask if the metric system is true and if the old weights and measures are false; if Cartesian coordinates are true and polar coordinates false. One geometry cannot be more true than another: it can only be more convenient.” -Poincaré


Euclid’s Fifth Postulate is logically equivalent to the statement that there exists a unique line through a given point which is parallel to a given line. For 2000 years, mathematicians were sure that this was in fact a theorem which followed from his first four axioms. In attempts to prove the postulate by contradiction, three mathematicians accidentally invented a new geometry...


=== March 16, Keith Rush ===
Title: Markov chains and upper bounds on ranks of quadratic twists of an elliptic curve.


Title: Fourier series, random series and Brownian motion--the beginnings of modern analysis and probability
Speaker: Sun Woo Park


Abstract: A mostly historical and (trust me!) non-technical talk on the development of analysis and probability through the interplay between a few fundamental, well-known objects: namely Fourier, random and Taylor series, and the Brownian Motion. In my opinion this is a beautiful and interesting perspective that deserves to be better known. DISCLAIMER: I'll need to end at least 5 minutes early because I'm giving the grad analysis talk at 4.
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.


=== March 30, Iván Ongay Valverde ===


Title: Monstrosities out of measure


Abstract: It is a well known result that, using the Lebesgue measure, not all subsets of the real line are measurable. To get this result we use the property of invariance under translation and the axiom of choice. Is this still the case if we remove the invariance over translation? Depending how we answer this question the properties of the universe itself can change.
Title: Coherent Structures in Convection.  


=== April 6, Nathan Clement ===
Speaker: Varun Gudibanda


Title: Algebraic Doughnuts
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: A fun, elementary problem with a snappy solution from Algebraic Geometry. The only prerequisite for this talk is a basic knowledge of circles!


=== April 13, Adam Frees ===


Title: The proof is in the 'puting: the mathematics of quantum computing
Title: Morse Theory in Algebraic Topology (According to ChatGPT)


Abstract: First proposed in the 1980s, quantum computing has since been shown to have a wide variety of practical applications, from finding molecular energies to breaking encryption schemes. In this talk, I will give an introduction to quantum mechanics, describe the basic building blocks of a quantum computer, and (time permitting) demonstrate a quantum algorithm. No prior physics knowledge required!
Speaker: Alex Hof


=== April 20, Eva Elduque ===


Title: The Cayley-Hamilton Theorem


Abstract: The Cayley-Hamilton Theorem states that every square matrix with entries in a commutative ring is a root of its characteristic polynomial. We all have used this theorem many times but might have never seen a proof of it. In this talk I will give a slick proof of this result that uses density and continuity so as to prevent the non-algebraists in the room from rioting.
Title: Life in a Hyperbolic City


=== April 27, David Bruce ===
Speaker: Daniel Levitin


Title: A Crazy Way to Define Homology
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: This talk will be like a costume party!! However, instead of pretending to be an astronaut I will pretend to be a topologist, and try and say something about the Dold-Thom theorem, which gives a connected between the homotopy groups and homology groups of connected CW complexes. So I guess this talk will be nothing like a costume party, but feel free to wear a costume if you want.


=== May 4, Paul Tveite ===


Title: Kissing Numbers (not the fun kind)
Title: Logic: What is it good for?


Abstract: In sphere packing, the n-dimensional kissing number is the maximal number of non-intersecting radius 1 n-spheres that can all simultaneously be tangent to a central, radius 1 n-sphere. We'll talk a little bit about the known solutions and some of the interesting properties that this problem has.
Speaker: John Spoerl


=== May 11, Becky Eastham ===
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: Logic is Useful for Things, Such as Ramsey Theory


Abstract: Hindman’s Theorem, first proven in 1974, states that every finite coloring of the positive integers contains a monochromatic IP set (a set of positive integers which contains all finite sums of distinct elements of some infinite set).  The original proof was long, complicated, and combinatorial.  However, there’s a much simpler proof of the theorem using ultrafilters.  I’ll tell you what an ultrafilter is, and then I will, in just half an hour, prove Hindman’s Theorem by showing the existence of an idempotent ultrafilter.


== Fall 2015 ==
Title: Fourier restriction and Kakeya problems


=== October 7, Eric Ramos ===
Speaker: Mingfeng Chen


Title: Configuration Spaces of Graphs
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.


Abstract: A configuration of n points on a graph is just a choice of n distinct points. The set of all such configurations is a topological space, and so one can study its properties. Unsurprisingly, one can determine a lot of information about this configuration space from combinatorial data of the graph. In this talk, we consider some of the most basic properties of these spaces, and discuss how they can be applied to things like robotics. Note that most of the talk will amount to drawing pictures until everyone agrees a statement is true.
=== March 29, Ivan Aidun ===


=== October 14, Moisés Herradón ===


Title: The natural numbers form a field


Abstract: But of course, you already knew that they form a field: you just have to biject them into Q and then use the sum and product from the rational numbers. However, out of the many field structures the natural numbers can have, the one I’ll talk about is for sure the cutest. I will discuss how this field shows up in "nature" (i.e. in the games of some fellows of infinite jest) and what cute properties it has.
'''Title:''' Fractional Calculus


=== October 21, David Bruce ===
'''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?


Title: Coverings, Dynamics, and Kneading Sequences
=== April 5, Diego Rojas La Luz ===


Abstract: Given a continuous map f:X—>X of topological spaces and a point x in X one can consider the set {x, f(x), f(f(x)), f(f(f(x))),…} i.e, the orbit of x under the map f. The study of such things even in simple cases, for example when X is the complex numbers and f is a (quadratic) polynomial, turns out to be quite complex (pun sort of intended). (It also gives rise to main source of pretty pictures mathematicians put on posters.) In this talk I want to show how the study of such orbits is related to the following question: How can one tell if a (ramified) covering of S^2 comes from a rational function? No background will be assumed and there will be pretty pictures to stare at.


=== October 28, Paul Tveite ===


Title: Gödel Incompleteness, Goodstein's Theorem, and the Hydra Game
Title: Eating a poisoned chocolate bar


Abstract: Gödel incompleteness states, roughly, that there are statements about the natural numbers that are true, but cannot be proved using just Peano Arithmetic. I will give a couple concrete examples of such statements, and prove them in higher mathematics.
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?


=== November 4, Wanlin Li ===
=== April 12, Taylor Tan ===


Title: Expander Families, Ramanujan graphs, and Property tau


Abstract: Expander family is an interesting topic in graph theory. I will define it, give non-examples and talk about the ideal kind of it, i.e. Ramanujan graph. Also, I will talk about property tau of a group and how it is related to expander families. To make the talk not full of definitions, here are part of the things I'm not going to define: Graph, regular graph, Bipartite graph, Adjacency matrix of a graph and tea...


=== November 11, Daniel Hast ===
Title: A Proof From The Hall of Fame -- Topological Methods in Combinatorics


Title: Scissor groups of polyhedra and Hilbert's third problem
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.


Abstract: Given two polytopes of equal measure (area, volume, etc.), can the first be cut into finitely many polytopic pieces and reassembled into the second? To investigate this question, we will introduce the notion of a "scissor group" and compute the scissor group of polygons. We will also discuss the polyhedral case and how it relates to Dehn's solution to Hilbert's third problem. If there is time, we may mention some fancier examples of scissor groups.
=== April 19, NO SEMINAR  ===


=== November 18, James Waddington ===
=== April 26, Hyun Jong Kim  ===


''Note: This week's talk will be from 3:15 to 3:45 instead of the usual time.''


Title: Euler Spoilers
Title: Machine Learning Tools for the Working Mathematician


Abstract: Leonhard Euler is often cited as one of the greatest mathematicians of the 18. Century. His solution to the Königsburg Bridge problem is an important result of early topology. Euler also did work in combinatorics and in number theory. Often his methods tended to be computational in nature (he was a computer in the traditional sense) and from these he proposed many conjectures, a few of which turned out to be wrong. Two failed conjectures of Euler will be presented.
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.


=== December 9, Brandon Alberts ===
=== May 3, Asvin G ===
Title: On the random graph on countably many vertices 


Title: The field with one element
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!


=== December 16, Micky Soule Steinberg ===
== Spring 2022 ==


Title: Intersective polynomials
=== February 9, Alex Mine ===
Title: Would you like to play a game?


==Spring 2015==
Abstract: We'll look at some fun things in combinatorial game theory.


===January 28, Moisés Herradón===
=== February 16, Michael Jeserum ===
Title: The Internet's Take on Number Bases


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


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!
<nowiki>*</nowiki>Disclaimer: audience may not actually learn what the best number base is.


===February 11, Becky Eastham===
=== February 23, Erika Pirnes ===
Title: Staying Balanced- studying the balanced algebra


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


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.
=== March 2, Jason Torchinsky ===
Title: Holmes and Watson and the case of the tropical climate


===February 18, Solly Parenti===
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.


Title: Chebyshev's Bias
=== March 7, Devanshi Merchant ===
Title: Mathematics of soap films


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.
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.  


===February 25, David Bruce===
=== March 30, Jacob Denson ===
Title: Proofs in 3 bits or less


Title: Mean, Median, and Mode - Well Actually Just Median
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.


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.
=== April 6, Aidan Howells ===
Title: Goodstein Sequences, Hercules, and the Hydra


===March 4, Jing Hao===
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>


Title: Error Correction Codes
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>


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!
=== April 13, Yu Fu ===
Title: How do generic properties spread?


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


''Note: This week's seminar will be on Tuesday at 3:30 instead of the usual time.''
=== April 20, Ivan Aidun ===
Title: The are no Orthogonal Latin Squares of Order 6


Title: Two Solutions, not too Technical, to a Problem to which the Answer is Two
Abstract: The title says it all.


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.
== Fall 2021 ==


===March 25, Eric Ramos===
=== September 29, John Cobb ===


Title: Braids, Knots and Representations
Title: Rooms on a Sphere


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.
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.


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.
=== October 6, Karan Srivastava ===


===April 8, James Waddington===
Title: An 'almost impossible' puzzle and group theory


Title: Goodstein's Theorem
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.


Abstract: One of the most important results in the development of mathematics are
=== October 13, John Yin ===
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
Title: TBA
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
Abstract: TBA
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===
=== October 20, Varun Gudibanda ===


Title: Finite Groups aren't too Square
Title: TBA


Abstract: We investigate how many non-p-th powers a group can have for a given prime p.
Abstract: TBA
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==
 
===September 25, Vladimir Sotirov===
 
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===
=== October 27, Andrew Krenz ===


Title: Hex on the Beach
Title: The 3-sphere via the Hopf fibration


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: 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  ===


===October 22, Eva Elduque===
Title: Probabilistic methods in math


Title: The fold and one cut problem
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.


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.
=== November 10, Ivan Aidun ===
[[File:Screen Shot 2021-11-15 at 3.25.38 PM.png|thumb]]
Title: Intersection Permutations


===November 5, Megan Maguire===
Abstract: During a boring meeting, your buddy slips you a Paris metro ticket with this cryptic diagram (see left).


Title: Train tracks on surfaces
What could it mean?  The only way to find out is to come to this Donut Talk!


Abstract: What is a train track, mathematically speaking? Are they interesting? Why are they interesting? Come find out!
=== December 1, Yuxi Han  ===


===November 19, Adrian Tovar-Lopez===
Title: Homocidal Chaffeur Problem


Title: Hodgkin and Huxley equations of a single neuron
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 3, Zachary Charles===
=== December 8, Owen Goff  ===


Title: Addition chains: To exponentiation and beyond
Title: The Mathematics of Cribbage


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?
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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