# 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. Here are all the talks given in previous semesters.

## Contents

- 1 Spring 2016
- 1.1 January 27, Wanlin Li
- 1.2 February 3, Will Cocke
- 1.3 February 10, Jason Steinberg
- 1.4 February 17, Zachary Charles
- 1.5 February 24, Brandon Alberts
- 1.6 March 2, Vlad Matei
- 1.7 March 9, Micky Steinberg
- 1.8 March 16, Keith Rush
- 1.9 March 30, Iván Ongay Valverde
- 1.10 April 6, Nathan Clement
- 1.11 April 13, Adam Frees
- 1.12 April 20, Eva Elduque
- 1.13 April 27, David Bruce
- 1.14 May 4, Paul Tveite
- 1.15 May 11, Becky Eastham

- 2 Fall 2015
- 3 Spring 2015
- 4 Fall 2014

## 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

Title: The Rado Graph

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 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.

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!

### April 13, Adam Frees

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, David 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.

### 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.

### October 21, David 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.

### November 18, James Waddington

*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

### January 28, Moisés Herradón

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, David 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.

### April 8, James Waddington

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

### September 25, Vladimir Sotirov

Title: The compact open topology: what is it really?

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

### October 8, David Bruce

Title: 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!

### November 19, Adrian Tovar-Lopez

Title: Hodgkin and Huxley equations of a single neuron

### December 3, Zachary Charles

Title: Addition chains: To exponentiation and beyond

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