# Difference between revisions of "AMS Student Chapter Seminar, previous semesters"

The AMS Student Chapter Seminar is an informal, graduate student-run seminar on a wide range of mathematical topics.

## Past Organizers

Fall 2018 - Spring 2021: Michel Alexis, David Wagner, Patrick Nicodemus, Son Tu, Carrie Chen

Here are all the talks given in previous semesters.

## Spring 2020

### February 5, Alex Mine

Title: Khinchin's Constant

### February 12, Xiao Shen

Title: Coalescence estimates for the corner growth model with exponential weights

Abstract: (Joint with Timo Seppalainen) I will talk about estimates for the coalescence time of semi-infinite directed geodesics in the planar corner growth model. Not much probability background is needed.

### February 19, Hyun Jong Kim

Title: Orbifolds for Music

Abstract: In the first-ever music theory article published by the journal Science, Dmitri Tymoczko uses orbifolds to describe a general framework for thinking about musical tonality. I am going to introduce the musical terms and ideas needed to describe how such orbifolds arise so that we can see an example of Tymoczko's geometric analysis of chord progressions.

### February 26, Solly Parenti

Title: Mathematical Measuring

Abstract: What's the best way to measure things? Come find out!

### March 11, Ivan Aidun

Title: The Notorious CRT

Abstract: You're walking up Bascomb hill when a troll suddenly appears and says he'll kill you unless you compute the determinant of

$\displaystyle{ \begin{bmatrix}0 & -7 & -17 & -5 & -13\\8 & -14 & 14 & 11 & 15\\-5 & -17 & 10 & 2 & 10\\17 & 3 & -16 & -13 & 7\\-1 & 2 & -13 & -11 & 10\end{bmatrix} }$

by hand. wdyd?

Title: TBD

Abstract: TBD

Title: TBD

Abstract: TBD

Title: TBD

Abstract: TBD

Title: TBD

Abstract: TBD

Title: TBD

Abstract: TBD

Title: TBD

Abstract: TBD

Title: TBD

Abstract: TBD

Title: TBD

Abstract: TBD

Title: TBD

Abstract: TBD

Title: TBD

Abstract: TBD

Title: TBD

Abstract: TBD

Title: TBD

Abstract: TBD

## Fall 2019

### October 9, Brandon Boggess

Title: An Application of Elliptic Curves to the Theory of Internet Memes

Abstract: Solve polynomial equations with this one weird trick! Math teachers hate him!!!

### October 16, Jiaxin Jin

Title: Persistence and global stability for biochemical reaction-diffusion systems

Abstract: The investigation of the dynamics of solutions of nonlinear reaction-diffusion PDE systems generated by biochemical networks is a great challenge; in general, even the existence of classical solutions is difficult to establish. On the other hand, these kinds of problems appear very often in biological applications, e.g., when trying to understand the role of spatial inhomogeneities in living cells. We discuss the persistence and global stability properties of special classes of such systems, under additional assumptions such as: low number of species, complex balance or weak reversibility.

### October 23, Erika Pirnes

(special edition: carrot seminar)

Title: Why do ice hockey players fall in love with mathematicians? (Behavior of certain number string sequences)

Abstract: Starting with some string of digits 0-9, add the adjacent numbers pairwise to obtain a new string. Whenever the sum is 10 or greater, separate its digits. For example, 26621 would become 81283 and then 931011. Repeating this process with different inputs gives varying behavior. In some cases the process terminates (becomes a single digit), or ends up in a loop, like 999, 1818, 999... The length of the strings can also start growing very fast. I'll discuss some data and conjectures about classifying the behavior.

### October 30, Yunbai Cao

Title: Kinetic theory in bounded domains

Abstract: In 1900, David Hilbert outlined 23 important problems in the International Congress of Mathematics. One of them is the Hilbert's sixth problem which asks the mathematical linkage between the mechanics from microscopic view and the macroscopic view. A relative new mesoscopic point of view at that time which is "kinetic theory" was highlighted by Hilbert as the bridge to link the two. In this talk, I will talk about the history and basic elements of kinetic theory and Boltzmann equation, and the role boundary plays for such a system, as well as briefly mention some recent progress.

### November 6, Tung Nguyen

Title: Introduction to Chemical Reaction Network

Abstract: Reaction network models are often used to investigate the dynamics of different species from various branches of chemistry, biology and ecology. The study of reaction network has grown significantly and involves a wide range of mathematics and applications. In this talk, I aim to show a big picture of what is happening in reaction network theory. I will first introduce the basic dynamical models for reaction network: the deterministic and stochastic models. Then, I will mention some big questions of interest, and the mathematical tools that are used by people in the field. Finally, I will make connection between reaction network and other branches of mathematics such as PDE, control theory, and random graph theory.

### November 13, Jane Davis

Title: Brownian Minions

Abstract: Having lots of small minions help you perform a task is often very effective. For example, if you need to grade a large stack of calculus problems, it is effective to have several TAs grade parts of the pile for you. We'll talk about how we can use random motions as minions to help us perform mathematical tasks. Typically, this mathematical task would be optimization, but we'll reframe a little bit and focus on art and beauty instead. We'll also try to talk about the so-called "storytelling metric," which is relevant here. There will be pictures and animations! 🎉

Sneak preview: some modern art generated with MATLAB.

### November 20, Colin Crowley

Title: Matroid Bingo

Abstract: Matroids are combinatorial objects that generalize graphs and matrices. The famous combinatorialist Gian Carlo Rota once said that "anyone who has worked with matroids has come away with the conviction that matroids are one of the richest and most useful ideas of our day." Although his day was in the 60s and 70s, matroids remain an active area of current research with connections to areas such as algebraic geometry, tropical geometry, and parts of computer science. Since this is a doughnut talk, I will introduce matroids in a cute way that involves playing bingo, and then I'll show you some cool examples.

### December 4, Xiaocheng Li

Title: The method of stationary phase and Duistermaat-Heckman formula

Abstract: The oscillatory integral $\int_X e^{itf(x)}\mu=:I(t), t\in \mathbb{R}$ is a fundamental object in analysis. In general, $I(t)$ seldom has an explicit expression as Fourier transform is usually inexplicit. In practice, we are interested in the asymptotic behavior of $I(t)$, that is, for $|t|$ very large. A classical tool of getting an approximation is the method of stationary phase which gives the leading term of $I(t)$. Furthermore, there are rare instances for which the approximation coincides with the exact value of $I(t)$. One example is the Duistermaat-Heckman formula in which the Hamiltonian action and the momentum map are addressed. In the talk, I will start with basic facts in Fourier analysis, then discuss the method of stationary phase and the Duistermaat-Heckman formula.

### December 11, Chaojie Yuan

Title: Coupling and its application in stochastic chemical reaction network

Abstract: Stochastic models for chemical reaction networks have become increasingly popular in the past few decades. When the molecules are present in low numbers, the chemical system always displays randomness in their dynamics, and the randomness cannot be ignored as it can have a significant effect on the overall properties of the dynamics. In this talk, I will introduce the stochastic models utilized in the context of biological interaction network. Then I will discuss coupling in this context, and illustrate through examples how coupling methods can be utilized for numerical simulations. Specifically, I will introduce two biological models, which attempts to address the behavior of interesting real-world phenomenon.

## Spring 2019

### February 6, Xiao Shen (in VV B139)

Title: Limit Shape in last passage percolation

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

### February 13, Michel Alexis (in VV B139)

Title: An instructive yet useless theorem about random Fourier Series

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

### February 20, Geoff Bentsen

Title: An Analyst Wanders into a Topology Conference

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

### February 27, James Hanson

Title: What is...a Topometric Space?

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

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

Title: Introducing GRAMS (Graduate Representative Association of Mathematics Students)

Abstract: Over the past couple months, a handful of us have been working to create the UW Graduate Representative Association of Mathematics Students (GRAMS). This group, about 5-8 students, is intended to be a liaison between the graduate students and faculty, especially in relation to departmental policies and decisions that affect graduate students. We will discuss what we believe GRAMS ought to look like and the steps needed to implement such a vision, then open up the floor to a Q&A. Check out our website for more information.

### March 13, Connor Simpson

Title: Counting faces of polytopes with algebra

Abstract: A natural question is: with a fixed dimension and number of vertices, what is the largest number of d-dimensional faces that a polytope can have? We will outline a proof of the answer to this question.

### March 26 (Prospective Student Visit Day), Multiple Speakers

#### Eva Elduque, 11-11:25

Title: Will it fold flat?

Abstract: Picture the traditional origami paper crane. It is a 3D object, but if you don’t make the wings stick out, it is flat. This is the case for many origami designs, ranging from very simple (like a paper hat), to complicated tessellations. Given a crease pattern on a piece of paper, one might wonder if it is possible to fold along the lines of the pattern and end up with a flat object. We’ll discuss necessary and sufficient conditions for a crease pattern with only one vertex to fold flat, and talk about what can be said about crease patterns with multiple vertices.

#### Soumya Sankar, 11:30-11:55

Title: An algebro-geometric perspective on integration

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

#### Chun Gan, 12:00-12:25

Title: Extension theorems in complex analysis

Abstract: Starting from Riemann's extension theorem in one complex variable, there have been many generalizations to different situations in several complex variables. I will talk about Fefferman's field's medal work on Fefferman extension and also the celebrated Ohsawa-Takegoshi L^2 extension theorem which is now a cornerstone for the application of pluripotential theory to complex analytic geometry.

#### Jenny Yeon, 2:00-2:25

Title: Application of Slope Field

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

#### Rajula Srivastava, 2:30-2:55

Title: The World of Wavelets

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

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

Title: Group objects in various categories

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

#### Ivan Ongay Valverde, 3:30-3:55

Title: Games and Topology

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

#### Sun Woo Park, 4:00-4:25

Title: Reconstruction of character tables of Sn

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

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

Title: Clothes, Lice, and Coalescence

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

### April 3, Yu Feng

Title: Suppression of phase separation by mixing

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

### April 17, Hyun Jong Kim

Title: Musical Harmony for the Mathematician

Abstract: Harmony can refer to the way in which multiple notes that are played simultaneously come together in music. I will talk about some aspects of harmony in musical analysis and composition and a few ways to interpret harmonic phenomena mathematically. The mathematical interpretations will mostly revolve around symmetry and integer arithmetic modulo 12.

### April 24, Carrie Chen

Title: Pedestrian model

Abstract: When there are lots of people in a supermarket, and for some reason they have to get out as soon as possible, how do you expect the crowd to behave? Suppose each person is a rational individual and assume that each person has all knowledge to other people’s position at every time and further the number of people is huge, we can model it using mean field game model and get the macroscopic behaviour.

## Fall 2018

Title: Geometric Algebra

Abstract: Geometric algebra, developed at the end of the 19th century by Grassman, Clifford, and Lipschitz, is the forgotten progenitor of the linear algebra we use to this day developed by Gibbs and Heaviside. In this short introduction, I will use geometric algebra to do two things. First, I will construct the field of complex numbers and the division algebra of the quaternions in a coordinate-free way. Second, I will derive the geometric interpretation of complex numbers and quaternions as representations of rotations in 2- and 3-dimensional space.

### October 3, Juliette Bruce

Title: Kissing Conics

Abstract: Have you every wondered how you can easily tell when two plane conics kiss (i.e. are tangent to each other at a point)? If so this talk is for you, if not, well there will be donuts.

### October 10, Kurt Ehlert

Title: How to bet when gambling

Abstract: When gambling, typically casinos have the edge. But sometimes we can gain an edge by counting cards or other means. And sometimes we have an edge in the biggest casino of all: the financial markets. When we do have an advantage, then we still need to decide how much to bet. Bet too little, and we leave money on the table. Bet too much, and we risk financial ruin. We will discuss the "Kelly criterion", which is a betting strategy that is optimal in many senses.

### October 17, Bryan Oakley

Title: Mixing rates

Abstract: Mixing is a necessary step in many areas from biology and atmospheric sciences to smoothies. Because we are impatient, the goal is usually to improve the rate at which a substance homogenizes. In this talk we define and quantify mixing and rates of mixing. We present some history of the field as well as current research and open questions.

### October 24, Micky Soule Steinberg

Title: What does a group look like?

Abstract: In geometric group theory, we often try to understand groups by understanding the metric spaces on which the groups act geometrically. For example, Z^2 acts on R^2 in a nice way, so we can think of the group Z^2 instead as the metric space R^2.

We will try to find (and draw) such a metric space for the solvable Baumslag-Solitar groups BS(1,n). Then we will briefly discuss what this geometric picture tells us about the groups.

### October 31, Sun Woo Park

Title: Induction-Restriction Operators

Abstract: Given a "nice enough" finite descending sequence of groups $\displaystyle{ G_n \supsetneq G_{n-1} \supsetneq \cdots \supsetneq G_1 \supsetneq \{e\} }$, we can play around with the relations between induced and restricted representations. We will construct a formal $\displaystyle{ \mathbb{Z} }$-module of induction-restriction operators on a finite descending sequence of groups $\displaystyle{ \{G_i\} }$, written as $\displaystyle{ IR_{\{G_i\}} }$. The goal of the talk is to show that the formal ring $\displaystyle{ IR_{\{G_i\}} }$ is a commutative polynomial ring over $\displaystyle{ \mathbb{Z} }$. We will also compute the formal ring $\displaystyle{ IR_{\{S_n\}} }$ for a finite descending sequence of symmetric groups $\displaystyle{ S_n \supset S_{n-1} \supset \cdots \supset S_1 }$. (Apart from the talk, I'll also prepare some treats in celebration of Halloween.)

### November 7, Polly Yu

Title: Positive solutions to polynomial systems using a (mostly linear) algorithm

Abstract: "Wait, did I read the title correctly? Solving non-linear systems using linear methods?” Yes you did. I will present a linear feasibility problem for your favourite polynomial system; if the algorithm returns an answer, you’ve gotten yourself a positive solution to your system, and more than that, the solution set admits a monomial parametrization.

### November 14, Soumya Sankar

Title: The worlds of math and dance

Abstract: Are math and dance related? Can we use one to motivate problems in the other? Should we all learn how to dance? I will answer these questions and then we will have some fun with counting problems motivated by dance.

### November 28, Niudun Wang

Abstract: When using fractions to approximate a real number, continued fraction is known to be one of the fastest ways. For instance, 3 is close to pi (somehow), 22/7 was the best estimate for centuries, 333/106 is better than 3.1415 and so on. Beyond this, I am going to show how continued fraction can also help us with finding the unit group of some real quadratic fields. In particular, how to solve the notorious Pell's equation.

### December 5, Patrick Nicodemus

Title: Applications of Algorithmic Randomness and Complexity

Abstract: I will introduce the fascinating field of Kolmogorov Complexity and point out its applications in such varied areas as combinatorics, statistical inference and mathematical logic. In fact the Prime Number theorem, machine learning and Godel's Incompleteness theorem can all be investigated fruitfully through a wonderful common lens.

### December 12, Wanlin Li

Title: Torsors

Abstract: I will talk about the notion of torsor based on John Baez's article 'Torsors made easy' and I will give a lot of examples. This will be a short and light talk to end the semester.

## Spring 2017

### January 25, Brandon Alberts

Title: Ultraproducts - they aren't just for logicians

Abstract: If any of you have attended a logic talk (or one of Ivan's donut seminar talks) you may have learned about ultraproducts as a weird way to mash sets together to get bigger sets in a nice way. Something particularly useful to set theorists, but maybe not so obviously useful to the rest of us. I will give an accessible introduction to ultraproducts and motivate their use in other areas of mathematics.

### February 1, Megan Maguire

Title: Hyperbolic crochet workshop

Abstract: TBA

### February 15, Paul Tveite

Title: Fun with Hamel Bases!

Abstract: If we view the real numbers as a vector field over the rationals, then of course they have a basis (assuming the AOC). This is called a Hamel basis and allows us to do some cool things. Among other things, we will define two periodic functions that sum to the identity function.

### February 22, Wil Cocke

Title: Practical Graph Isomorphism

Abstract: Some graphs are different and some graphs are the same. Sometimes graphs differ only in name. When you give me a graph, you've picked an order. But, is it the same graph across every border?

### March 1, Megan Maguire

Title: I stole this talk from Jordan.

Abstract: Stability is cool! And sometimes things we think don't have stability secretly do. This is an abridged version of a very cool talk I've seen Jordan give a couple times. All credit goes to him. Man, I should have stolen his abstract too.

### March 7, Liban Mohamed

Title: Strichartz Estimates from Qualitative to Quantitative

Abstract: Strichartz estimates are inequalities that give one way understand the decay of solutions to dispersive PDEs. This talk is an attempt to reconcile the formal statements with physical intuition.

### March 15, Zachary Charles

Title: Netflix Problem and Chill

Abstract: How are machine learning, matrix analysis, and Napoleon Dynamite related? Come find out!

### April 12, Micky Steinberg

Title: Groups as metric spaces

Abstract: Given a group as a set of generators and relations, we can define the “word metric” on the group as the length of the shortest word “between” two elements. This isn’t well-defined, since different generating sets give different metrics, but it is well-defined up to “quasi-isometry”. Come find out what we can do with this! There will lots of pictures and hand-waving!

### April 19, Solly Parenti

Title: Elementary Integration

Abstract: Are you like me? Have you also told your calculus students that finding the antiderivative of e^(-x^2) is impossible? Do you also only have a slight idea about how to prove it? Come find out more about the proof and free yourself of that guilt.

### April 26, Ben Bruce

Title: Permutation models

Abstract: Permutation models belong to a version of axiomatic set theory known as "set theory with atoms." I will give some examples of permutation models and highlight their connection to the axiom of choice and notions of infinity. There will be concrete examples, and no prior knowledge of set theory is required.

### May 3, Iván Ongay-Valverde

Title: Living with countably many reals?

Abstract: Can I make you believe that a countable set of reals are all the reals? If we just have countably many reals, what happens with the others? Do they have any special properties? Let's play a little with our notion of 'reality' and allow to ourselves to find crazy reals doing weird things. Hopefully, no-one's headache will last forever.

## Fall 2016

### October 12, Soumya Sankar

Title: Primes of certain forms and covering systems

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

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.

### October 26, Brandon Alberts

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.

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.

### November 9, Juliette Bruce

Title: Some Numbers Are Sometimes Bigger Than Others (Sometimes...)

Abstract: I will write down two numbers and show that one of them is larger than the other.

### November 16, Solly Parenti

Title: The Congruent Number Problem

Abstract: To add to the over-romanticization of number theory, I will talk about a simple to state problem about triangles that quickly leads into very difficult open problems in modern number theory.

### November 30, Iván Ongay Valverde

Title: Games for fun, games to change the world, games, games, games

Abstract: We will talk about infinite perfect information games. We will discuss different uses for these games, and we will see that some of them have interesting information for us that helps determine some properties of subsets of reals. Can games change the world? Can we use them in a non-intrusive way? Join to have fun with games, since they are games!

### December 7, Will Mitchell

Title: An unsolved isomorphism problem from plane geometry

Abstract: A geometric n-configuration is a collection of points and lines in the Euclidean plane such that each point lies on exactly n lines and each line passes through n points. While the study of 3-configurations dates to the nineteenth century, the first example of a 4-configuration appeared only in 1990. I will say a few things about 4-configurations and state an unsolved problem, and I hope that someone in the audience will decide to work on it. There will be nice pictures and a shout-out to the singular value decomposition.

### December 14, Paul Tveite

Title: Infinite Chess - Mate in Infinity

Abstract: There's a long history of stating puzzles using chess boards and chess pieces. Particularly endgame puzzles, like so-called "mate in n" problems. When we extend these questions to chess on an infinite board, we get some surprisingly mathematically deep answers.

## Spring 2016

### January 27, Wanlin Li

Title: The Nottingham group

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.

### February 3, Will Cocke

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: Mazur's Swindle

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...

### February 17, Zachary Charles

Title: #P and Me: A tale of permanent complexity

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

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.

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.

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

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.

We explore the history and various results and some unanswered questions.

### March 9, Micky Steinberg

Title: The Parallel Postulate and Non-Euclidean Geometry.

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: Fourier series, random series and Brownian motion--the beginnings of modern analysis and probability

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.

### 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.

### April 6, Nathan Clement

Title: Algebraic Doughnuts

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

Title: The proof is in the 'puting: the mathematics of quantum computing

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!

### 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.

### April 27, Juliette Bruce

Title: A Crazy Way to Define Homology

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)

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.

### May 11, Becky Eastham

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

### October 7, Eric Ramos

Title: Configuration Spaces of Graphs

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.

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.

### October 21, Juliette Bruce

Title: Coverings, Dynamics, and Kneading Sequences

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

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.

### November 4, Wanlin Li

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: Scissor groups of polyhedra and Hilbert's third problem

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.

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

Title: Euler Spoilers

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.

### December 9, Brandon Alberts

Title: The field with one element

### December 16, Micky Soule Steinberg

Title: Intersective polynomials

## Spring 2015

Title: Winning games and taking names

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

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.

### February 18, Solly Parenti

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.

### February 25, Juliette Bruce

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

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.

### March 4, Jing Hao

Title: Error Correction Codes

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: Two Solutions, not too Technical, to a Problem to which the Answer is Two

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.

### March 25, Eric Ramos

Title: Braids, Knots and Representations

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: Goodstein's Theorem

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 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

Title: Finite Groups aren't too Square

Abstract: We investigate how many non-p-th powers a group can have for a given prime p. 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

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, Juliette Bruce

Title: Hex on the Beach

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!*

### 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.

### November 5, Megan Maguire

Title: Train tracks on surfaces

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