### Graduate Colloquium

The Graduate Colloquium is a colloquium series presented primarily by graduate students. Any graduate student or faculty member is welcome to present research or other mathematical topics of interest. Talks are intended to be accessible to all graduate students, though everyone is welcome to attend. Talks are generally given on Thursday from 3:00pm to 4:00pm in Minard 212. Coffee and treats are served at each meeting.

##### Spring 2020 Talks

March 12, 2020

Speaker: Jimmy Thorne**Title:** Search for contra-continuous functions**Abstract:** A Contra-continuous function is a function where every preimage of a open set is closed. They provide an excellent playground to work with basic topological concepts, but can quickly lead to some challenging problems. We'll establish some basic properties of contra-continuous function and give some elementary examples. Then we'll state some open (or at least unknown to the speaker) questions.

March 5, 2020

Speaker: Brandon Allen**Title:** Antipodes: The Other Side(s) of the Element**Abstract:** We all have heard in our basic Topology course that the antipode of a point x on the sphere takes on the form f(x) = - x and we call f the antipodal map. But what is an antipode? Can we extend the notion of an antipode map algebraic objects? Is there a way to explicitly calculate an antipode? These are fundamental questions that this talk will answer.

We will look at a specific class of algebras that are known as Hopf algebras. A Hopf algebra is a bi-algebra with an antipode map. Hopf algebras were first used to study H-spaces in Algebraic Topology. Over the course of time, Hopf algebras have been used to study and verifying the enumeration of combinatorial objects. We will go over some combinatorial identities that can be proven through the notion of a Hopf Algebra, and special properties of the antipode map. At the end of the presentation we will go over some conjectures and results which is spearheading research today regarding the antipode map

February 27, 2020

Speaker: Dylan Heuer**Title:** Permutations and Alternating Sign Matrices in the Partial and Chained Settings**Abstract:** We define certain generalizations of permutations and alternating sign matrices - some of which were inspired by 3-person chess! We also explore various bijections analogous to those in the usual alternating sign matrix setting.

Note: This talk is in preparation for a "job talk" and one of its goals is to be understandable by a wide range of backgrounds, and even at an undergraduate level.

February 13, 2020

Speaker: Austin Uden**Title:** A glance at "Don't Break the Ice" through combinatorial game theory**Abstract:** The game “Don’t Break the Ice” is a classic children’s game that involves players taking turns hitting “ice” blocks out of a grid until a block containing a bear falls. Our research analyzed various versions of “Don’t Break the Ice” through Combinatorial Game Theory, specifically normal and misère play. Different winning strategies are presented, some applying to specific games and some generalized for all versions of the game.

February 6, 2020

Speaker: Faraad Armwood**Title:** Things about symplectic manifolds**Abstract:** The goal of this talk is to prove/sketch some of the first fundamental results about symplectic manifolds. No background knowledge is required.

March 19 - Spring Break

March 26 - Chelsey Morrow

April 2 - Lane Morrison

April 9 - Chase Reuter

April 16 -

April 23 - Nick Salmon

April 30 - Michael Marmorstein

##### Fall 2019 Talks

December 5, 2019

Speaker: Joey Forsman**Title:** Pattern Emergence**Abstract:** In the spirit of complex automata, we will see how certain systems behave when we impose strict rules and then perturb the initial positions. We will get a behind-the-scenes look at a modified version of Pascal's triangle, and we will also see how certain strategies applied in the game of Snake cause wild patterns. Given time, Sierpinski's triangle may make an appearance.

November 21, 2019

Speaker: Michael Marmorstein**Title:** My Sincerest Topologies**Abstract:** Two topologies occur naturally in Commutative Algebra related to a ring R: the Zariski topology on the space Spec(R), and the Krull topology with respect to an ideal I on a ring R. This presentation will introduce the two topologies, and explore basic properties of these two topologies and how they are related to algebraic properties and ideal structure of R.

November 14, 2019

Speaker: Jimmy Thorne**Title:** Understanding AlphaGo Zero**Abstract:** Deep learning and neural networks have been making a lot of news. In the Go community, one of the biggest events was when a computer program, called AlphaGo, final was able to beat a professional. It is has since changed how games are played, with classic play-outs being abandoned in favor of AI produced patterns. Since then, an even stronger program called AlphaGo Zero was formed that can beat the original AlphaGo. The peculiar fact about this new program is that it learned to play Go without any data sets, that is it taught itself. In this talk I'll speak about how computers attack games and how AlphaGo Zero works.

November 7, 2019

Speaker: Kevin Trowbridge**Title:** Herd Immunity: Protecting the Herd One Shot at a Time**Abstract:** About a month ago the Influenza vaccine came out for the season. With this it is hard to get away from the anti-vaxxers. In response, several people speak out about loved ones that can't get vaccines due to some immune deficiency and that they will always vaccinate to keep them safe. This notion is called herd immunity. Some people insist that herd immunity is not a real thing. These people are wrong, because herd immunity is mathematically sound. In this talk I will discuss what herd immunity is and how vaccines affect epidemiological models.

October 31, 2019

Speaker: Danny Luecke**Title:** An Introduction to RUME and TCUs**Abstract:** Research in Undergraduate Math Education (RUME) is a Special Interest Group of the MAA. This talk with will begin with a brief history of RUME and continuing into a brief history of tribal colleges and universities (TCUs). An overview of the current link between them specifically through the theoretical framework of Tribal Critical Race Theory will be given. Lastly, I will position my potential research topics within these two histories.

October 24, 2019

Speaker: Brandon Allen**Title:** Enumeration of Words of Length n in Coxeter Groups. **Abstract:** It is a natural question in an algebra course, "How many words of length n there are for a general group?" Even though this seems very benign as a question, it is actually been a question that has been daunting for the past hundred years starting with Max Dehn tackling the word problem in 1911 and 1912. In this talk we go over what is a Coxeter group, exciting and thrilling properties of Coxeter groups, and constructing digraphs from the properties of Coxeter groups which will give us the enumeration of words of length n for a Coxeter group. In this presentation there will be no assumptions of previous knowledge of Coxeter groups.

October 10, 2019

Speaker: Nicholas Salmon**Title:** Theorema Egregium**Abstract:** Have you ever come to prove a result so fantastic that people simply refer to it as “Theorema Egregium” (Remarkable Theorem)? Well, Gauss has! In this talk we go back to the early days of differential geometry and discuss a theorem which gave the study some legitimacy. For a quantity to have a good geometric interpretation, it ought to be independent of coordinates. Proving that the Gaussian Curvature is in fact independent of choice of coordinates is what Theorema Egregium accomplished. Only a background in multivariable calculus will be necessary!

October 3, 2019

Speaker: Eric Sarfo Amponsah**Title:** Applying methods of the theory of heterogeneous populations to the problem of pathogen co-existence**Abstract:** No two species can indefinitely occupy the same ecological niche according to the competitive exclusion principle. When competing strains of the same pathogen invade a homogeneous population, the strain with the largest basic reproductive ratio R0 will force the other strains to extinction. It is imperative to know the results for heterogeneous models since the host population may differ in susceptibility. Heterogeneous models tend to capture dynamics such as evolution, resistance to infection, etc., giving a more accurate results of the epidemics. The talk will focus on the behaviour of multi-pathogen heterogeneous models and will try to answer the question: What are the conditions that lead to pathogen coexistence?. The goal is to understand the mechanisms in heterogenous populations that mediates pathogen coexistence by studying (numerically and analytically) the existence and stability of coexistence equilibrium in heterogeneous models.

September 26, 2019

Speaker: Pratyush Mishra**Title:** The Arithmetics of Hyperbolic Place(H^2) and SL(2,Z) action on it**Abstract:** The talk with be an introduction to Hyperbolic geometry. During the first half, I will introduce some basics about the geodesics, hyperbolic isometries using the upper half space model. Then I will define an action of the modular group SL(2,Z) on H^2 to calculate the Fundamental domain and show that SL(2,Z)/(+-I) is generated by translations by one and the negative reciprocal map.

September 19, 2019

Speaker: Faraad Armwood**Title:** Introduction to Complex Forms**Abstract: **Given any embedding of R^{2n} into C^n we get complex coordinates on R^{2n}. Therefore, any map f: R^{2n} -> R^{2n} gives rise to a map f': C^n -> C^n. Hence, an honest question is, if x_1,...,x_n,y_1,...,y_n are the coordinates on R^{2n}, how are the k-forms (embedded coordinates) expressed?

##### Spring 2019 Talks

March 21, 2019

Speaker: Dakota Ihli**Topic:** Use of Baire Category in Mathematics**Abstract:** Suppose you wish to prove "there exists an element of X with property P". Counterintuitively, it is sometimes the case that "most elements of X have property P" is actually easier to prove. All we need is a correct interpretation of the word "most". One such interpretation – Baire category – is very well-suited to situations where X admits a nice topology. In this talk I will describe this method and provide (hopefully lesser-known) examples of its use in mathematics.

February 21, 2019

Speaker: Pratyush Mishra**Topic:** From Braids to Mapping Class Groups**Abstract:** The main theme is to understand the connection between the Artin's braid group and the group of automorphisms of the closed unit disk (with a finite number of points removed). We will start with the construction of braids using covering space theory and then will see some tools to compute braid groups for some nice spaces (which is really difficult!! in general) using higher homotopies and see how braids arisesunexpectedly in other areas of mathematics.

Eventually we will see its connection to mapping class group of surfaces and some of its important results.

I shall be assuming a very basic understanding of algebraic topology from the audience (just fundamentals groups and covering space theory).

February 7, 2019

Michael Marmorstein**Title:** A mathematical model of jazz harmony**Abstract:** The presentation will present a technique for analyzing symmetries in musical harmony using group actions. In particular, the model will be used to analyze common idioms in jazz piano music.

##### Fall 2018 Talks

September 13th, 2018

Halley Fritze**Title:** What I did at Math Camp**Abstract:** In this talk I will go over my experiences at MSRI's Summer School in Symplectic Geometry and Chaos, and attempt to briefly summarize some of the research presented there. Such topics include the Lagrangian Tetragons, and maybe the 3 Body Problem and Kirkwood Gaps. I will try to give a construction of these Lagrangian Tetragons and may omit certain details due to lack time.

September 20th, 2018

Jimmy Thorne**Title:** Stone-Cech Compactification**Abstract:** Compactness is one of the most powerful properties we can ask of a topological space. Therefore it is natural to ask can a non-compact space be embedded as a dense set into a compact space. One common compactification is the Alexandroff one-point compactification, where the point ‘infinity’ is added to the space. In 1937, Marshall Stone and Eduard Cech formalized the work of Tychonoff to give a compactification of a topological space into a compact Hausdorff space. In this talk we will go through the Stone-Cech compactification of the natural numbers by recognizing this new space is equivalent to the space of ultrafilters on N. Then we will speak briefly on a few applications and areas of research involving the Stone-Cech compactification.

September 27th, 2018

Megan Jensen **Title:** Flexing Hexflexagon**Abstract:** Have you ever been so bored in class that you found yourself searching for something to do to keep you from going mad? With one strip of paper, a few folds and some tape, you can make yourself a mathematical toy that you can play with discreetly during class. A hexaflexagon is a manipulated piece of paper that can be flexed along its folds to hide and reveal its various “faces”. My presentation will unveil the mysteries of hexaflexagons and will demonstrate multiple ways of constructing various versions of them. I will also discuss how to keep track of and reach each face of a hexaflexagon efficiently by diagramming its transformations. By flexing hexaflexagons and adding some artistic flourishes, you can turn a smiley face into a frown, make a dinosaur chase people, watch a snake become decapitated or even make an edible hexaflexamexagon.

October 4th, 2018

Joey Forsman**Title:** Interesting Properties of Bizarre Chess Pieces**Abstract:** In this talk we'll look at various properties of particular chess pieces on an infinite chess board. We'll consider the knight foremost. Besides this, we'll extend generally to any *new* piece with an arbitrary moveset and consider analogous properties here. Such include but hopefully not limited to: number of squares accessible after n moves, emergent patterns when considering the question: "How many squares accessible in a number of moves divisible by n?", and so-called "modular magic squares" found when considering the former question.

October 18th, 2018

Morgan O’Brien**Title:** Normal Numbers and Ergodic Theory**Abstract:** Normal numbers are numbers in which every finite sequence of digits between 0 and b-1 occur infinitely often in its base b expansion with a certain frequency. For example, it is conjectured that pi is a normal number, which is the origin of the commonly stated "every story ever written is contained somewhere within the decimal expansion of pi". In this talk, I will prove that almost every real number is normal (in a measure theoretic sense), and will present most of the prerequisite facts in Ergodic Theory needed to be able to understand why.

October 25, 2018

Jessica Striker**Title:** Poset and polytope perspectives on alternating sign matrices (or how to write a thesis and graduate)**Abstract:** In this talk, I’ll tell you about my thesis work in narrative form, interspersed with helpful tips on researching mathematics and surviving graduate school.

November 1st, 2018

Eric Sarfo Amponsah**Topic: **Applying the theory of heterogeneity to the problem of pathogen coexistence**Abstract:** In this talk, I will give a general overview of my current research topic in math biology (epidemiology). I will discuss the importance of heterogeneity, some heterogeneous models and some general results of pathogen coexistence.

November 8th, 2018

Cody Martin**Title:** HNN Extensions of Left-Orderable Groups**Abstract:** For certain classes of groups (such as solvable, nilpotent, Abelian) is left-orderability preserved under an HNN extension? In this talk, I'll first give a brief overview of left-orderable groups and HNN extensions. Then, I'll state the answer to the above question from a recent, joint work with Azer Akhmedov.

November 15th, 2018

Nick Salmon**Title:** Applying Stiefel-Whitney Classes to Division Algebras**Abstract:** The Real and Complex numbers are familiar to every mathematician. Both are examples of a general construction known as a Division Algebra over the Reals. A natural question to ask is what types of division algebras are there? We investigate some necessary conditions that involves the topology of spheres and projective spaces, particularly some characteristic classes of their tangent bundles. A basic understanding of homology/cohomology will be useful for this talk.

November 29th, 2018

Atif Ishan**Title:** Financial Modelling**Abstract:** The talk will be abbout Itô Calculus and Lévy processes. We will see the key role of Itô Lemma in financial modeling.

##### Spring 2018 Talks

Date: February 15

Speaker: Faraad Armwood**Title:** How to Effectively Communicate Mathematics**Abstract:** The goal of this talk is to provide ideas and techniques that can be employed to better communicate mathematics. The emphasis will be on colloquium talks, lecturing and general writing.

Date: March 1

Speaker: Michael Marmorstein**Title:** The Long Line**Abstract:** The main goal of this talk is to construct the Long Line, a pathological topological space, and to establish several of its main properties. In particular, we will prove that the long line is a path-connected space, locally homeomorphic to the real number line, but that it is not possible to embed it in Euclidean space.

Date: March 8

Speaker: Morgan O'Brien**Title:** An Introduction to p-adic Numbers**Abstract:** One way to construct the real numbers it to consider the completion of the rational numbers with respect to the metric defined by the absolute value. For a prime number p, we can define a different metric on the rational numbers related to p and talk about the completion of the rationals with respect to that metric. This completion is usually called the field of p-adic numbers. In this talk, we will discuss what this metric is as well as some interesting properties of both this metric and the p-adic numbers.

Date: March 29

Speaker: Jimmy Thorne

Date: April 5

Speaker: Artem Novozhilov

Date: April 12

Speaker: Shantanu Awasthi

_{}