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Mandy Smith, President
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Jonathan Alcaraz, Vice President
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Alec Martin, Treasurer
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Noble Williamson, Secretary
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| 22 May 2020 | Stumbling into Ergodic Theory via Birkhoff's Theorem |
| James K. Alcala | |
| In this talk, we will prove and discuss Birkhoff's pointwise ergodic theorem, a fundamental result central to ergodic theory and dynamical systems. The purpose of this talk will be to give listeners a flavor of the field via discussion and proof of some of its central results. Though the proofs we will uncover together are analysis-flavored, ergodic theory also has fruitful connections to other fields of mathematics such as number theory, geometry, and even Lie theory. |
| 15 May 2020 | Structural types for algebraic theories |
| Christian Williams | |
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Structural types are built from the same operations as terms, plus predicate logic. For example, "prime = not(1) and not(not(1)*not(1))" is the type of prime elements in a monoid. These types provide languages with an intrinsic form of higher reasoning. We give an algorithm which takes a language, given as a second-order algebraic theory (a structure with operations that can form and evaluate maps), and produces a structural type theory with polymorphism, subtyping, and recursion. We demonstrate with the ρ-calculus, a concurrent language with reflection, the logic for which provided original motivation. The draft is available at cbw124/stat. |
| 7 May 2020 | Lie Algebras and their Quantum Cousins |
| Maranda Smith | |
| In this talk, we will take a brief look into the world of Lie Algebras, their affine counterparts, and the work being done with modules over them. |
| 1 May 2020 | A Look at Surface Bundles |
| Jonathan Alcaraz | |
| Surface bundles are topological objects that we study using a whole lot of group theory. In this talk, I will go though some results regarding surface bundles and get in to some of the group theory surrounding these objects. |
| 17 April 2020 | Social Hour |
| 10 Apr 2020 | Homfly Satellite Invariants of Links |
| Alexander Pokorny | |
| In this talk, I will introduce the Homfly polynomial and see how far we can push the concept. In particular, we will define what I mean by a satellite invariant of a link. We will explore a natural parameter space of these invariants and show it has the form of an algebra. The goal of the talk is to try and learn as much about this algebra as we can. |
| 3 Apr 2020 | Local Fractal Zeta Functions |
| Xander Henderson | |
| In Fractal Zeta Functions and Fractal Drums, Lapidus et al. study the geometry of bounded subsets of Euclidean space by associating "fractal zeta functions" to these sets. These fractal zeta functions make it possible to rephrase many problems in geometry as problems in complex analysis. The theory can be generalized to a larger class of metric spaces, but the generalization relies on embeddings of a set into an ambient space of known dimension. In this talk, we discuss a theory of fractal zeta functions in homogeneous metric spaces. We first give a general theory of global zeta functions, then introduce a notion of "local fractal zeta functions." These local zeta functions have many properties analogous to those of the fractal zeta functions described by Lapidus et al., but which are defined without embeddings into an ambient space. |
| 6 Mar 2020 | "Should I Quit Mathematics?" and other scary questions |
| James K. Alcala | |
| When the going gets tough, how do we decide that our time spent as mathematicians is relevant and important? Is ‘I just like doing math’ good enough? Why does it sometimes feel like this answer isn’t sufficient? There are plenty of fields that are worthy of our time, attention, and career aspirations which don’t require us to spend all of these years locked away in an academic bubble. While listing all of the reasons why graduate school and academic life are difficult is redundant, it must be acknowledged that life can’t be put on hold, not even for a PhD. Given all of this, how do we convince ourselves that getting a PhD is worth it? Come to GSS on Friday, March 6th to hear how one mathematician found their place in the mathematical world, to explore how we overcome our struggles, and to honestly explore some pretty scary questions. Bring a pen and paper! |
| 28 Feb 2020 | An n-Dimensional Borg-Levinson Theorem |
| Robert Lopez | |
| The aim of this talk is to explore the inverse spectral problem pertaining to the elliptic operator $\delta + q$ acting on functions defined on a bounded domain in $\mathbb{R}^n$. The results of A. Nachman, J. Sylvester, and G. Uhlmann show that the coefficient function $q$ can be uniquely recovered by the spectral data of this operator. |
| 21 Feb 2020 | Theory of generalized complex structures |
| Matthew Burns | |
| The notion of a generalized complex structure on the generalized tangent bundle of a differential manifold M was introduced by Nigel Hitchin in 2002. This was introduced as a way of characterizing certain geometric structures by means of studying the functionals of differential forms on M. Some of these structures include complex and symplectic manifolds. I shall give a short survey of some key concepts related to these structures in the area of generalized geometry and do a few concrete examples of generalized complex structures. |
| 14 Feb 2020 | The combinatorics of linear programming problems |
| Ethan Kowalenko | |
| This talk will include some of the underlying combinatorial objects of my talk in Lie theory seminar this week, without any of the algebra or representation theory. Our goal is to describe oriented matroids as a generalization of hyperplane arrangements, tying in notions of face lattices, polytopes, and combinatorial spheres if and when convenient. Time willing, I will talk about the existence of the bane of my existence, which arises in dimensions higher than 3 in this purely combinatorial setting. |
| 7 Feb 2020 | Adjunction Junction |
| Christian Williams | |
| Have you ever considered one thing and another thing? What about one thing or another thing? We use the words "and" / "or" all the time; at cocktail parties and even at work! But did you ever wonder what they mean... in math?? Of course, "and" / "or" are binary operations on predicates, satisfying equations; but with category theory we can understand exactly why they are so natural and fundamental - and this understanding leads to vast generalizations of the classical logical notions such as quantification. We will get to the bottom of what is so special about "and" / "or", and maybe even have fun along the way. |
| 31 Jan 2020 | Keeping up with the Bernoulli’s |
| Will Hoffer🌍 | |
| Did you know that Bernoulli numbers, Bernoulli’s identity, and Bernoulli’s principle are all due to three different Bernoulli’s? The Bernoulli family as it turns out is a mathematical dynasty, responsible for such famous results as discovering the constant e=2.718281828…, proving the law of large numbers, and likely L’Hopital’s rule as well. They proposed (and answered!) famous problems such as the Brachistochrone problem and the St. Petersburg paradox. But the family also comes with its share of sibling rivalry, jealously, and even disownment due to their spotlight as Switzerland’s preeminent mathematicians. Join us as we look at their mathematical contributions and make heads or tails out of the Johann’s and Jakob’s of the Bernoulli family. |
| 24 Jan 2020 | Correspondence between Dominated Splitting and Spectrum of the Jacobi Operator |
| Kateryna Alkorn | |
| We will talk about one of the basic results relating spectrum of Schrodinger operator and Uniform Hyperbolicity of the associated Schrodinger cocycle. After that we will proceed by extending the existing theory into a more general setting relating Dominated Splitting and Spectrum of Jacobi operator. We will talk about the problems we have encountered along the way and possible ways to resolve them. In the end we will see a proof of partial result from our hypothesis. |
| 17 Jan 2020 | Upper bounds of the cop number |
| Raymond Matson | |
| The game of cops and robbers is a type of graph searching problem where a team of cops try to capture a robber by moving onto the same vertex as the robber. The canonical question that arises is: what is the smallest number of cops needed to ensure that the cops will win for any graph of order $n$? Henri Meyniel conjectured that for any connected graph of order $n$, the number of cops needed is $O(\sqrt{n})$. We will explore the upper bound of some specific graphs as well as attempts to prove Meyniel's conjecture. |
| 10 Jan 2020 | An Introduction to Interval Bundles |
| Jonathan Alcaraz | |
| Interval bundles are a way of taking a space we know and making it more topologically complicated while still maintaining homotopy type and hence maintaining the fundamental group. In low-dimensional topology, we see these when studying 3-manifolds. It turns out that if a 3-manifold smells like a surface (ie, its fundamental group is a surface group), then it is an interval bundle over a surface. |
| 6 Dec 2019 | Department Potluck! |
| 29 Nov 2019 | No Meeting |
| 22 Nov 2019 | On the Stability of Self-Similar Blowup in Nonlinear Wave Equations |
| Michael McNulty | |
| When studying nonlinear wave equations, one concerns themselves with the well-posedness of the Cauchy problem. Does a solution exist for some amount of time? Does it exist for all time? Is it unique? Does the solution depend continuously on the initial data? Within the context of energy supercritical wave equations, a typical way for solutions to fail to exist for all time is through the phenomenon of self-similar blowup. After making this observation, one is left pondering the stability of this blowup. In other words, one wants to know if there is an entire open set of initial data leading to this blowup. Answering this question for particular wave equations is an active area of research with lots of techniques stemming from wave maps. In this talk, we will discuss current work in progress toward establishing the asymptotic nonlinear stability of self-similar blowup in the strong-field Skyrme model. |
| 15 Nov 2019 | Fractions, Continued: A Look at Continued Fractions |
| Nick Newsome | |
| Continued fractions have been the subject of study for many years. A continued fraction is an expression that iteratively describes any real number. Rational numbers have a finite continued fraction representation, while irrationals have an infinite continued fraction representation. Continued fractions have applications ranging from approximating real numbers to solving Diophantine equations to (as I discovered recently) the study of differential equations. Although they are not incredibly difficult to comprehend, continued fractions are not typically part of the standard undergraduate (or graduate) mathematics curriculum. To that end, this talk will endeavor to introduce the basics of continued fractions in the hopes of showing how they can be used to do some pretty cool stuff. For example, have you ever considered just how irrational an irrational number can be? Continued fractions give us a way to develop a sort of hierarchy of irrationality. We will also (hopefully) show how continued fractions can be used to solve a particular Diophantine equation. |
| 8 Nov 2019 | No Meeting |
| 1 Nov 2019 | The Variable, Free and Bound |
| Christian Williams | |
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In mathematics, computer science, and logic, one of the most useful ideas is also the most humble: the variable. But what exactly is a variable? When we write f(x)=x+3, how do we formalize the distinction between the "placeholder" x to be substituted, and the "real" value 3? Conventional algebra and logic do not answer this question; we simply take variables for granted. Though not widely known, the answer was given 25 years ago, in a paper called "Abstract Syntax with Variable Binding". This summer I got to visit the mathematician who realized this idea, and began to join in a grand project of laying the algebraic foundations of formal languages.
Essentially, it is a new take on the ideas I presented last year: rather than thinking of algebraic theories as categories, one can think of them as functors T:C->Set, from a category of contexts to the category of sets -- one interprets T(c) as the set of terms which can be derived from a context c (for simple languages, C=N, and a context "n" simply represents having n free variables). From this perspective, one can reformulate all of ordinary algebra; but this "category of presheaves" has a richer structure, in which we can say and do much more: in particular, we can formalize variables in a natural way. This is the key to having a universal language in which to express all of those we use in math and programming. Join me as we explore the beautiful world of "functors as languages", where we will connect such different concepts as the lambda calculus, species and simplicial sets, algebra and logic -- there will even be integrals. Hope to see you there. |
| 25 Oct 2019 | A Look at Microlocal Analysis |
| Michael McNulty | |
| In the study of differential equations, one is always interested in studying equations of the form $Pu=f$ where $u$ is some unknown function, $f$ is some known function, and $P$ is some differential operator. Typically, one hopes to solve for the unknown $u$ or to extract necessary properties of it. In other words, one wants to make sense of the right-hand side of $u=P^{-1}f$ or to know what spaces $u$ could live in. So, what does it mean to invert a differential operator? Where could the solution possibly live? Microlocal analysis provides a framework in which one can answer these questions. A good place to start the descent into microlocal analysis is with the study of pseudodifferential operators and how they propagate singularities. We will see how singularities of a distribution propagate along particular directions in the cotangent bundle which are determined uniquely by the pseudodifferential operator acting on them. |
| 18 Oct 2019 | Finitely Additive Invariant Set Functions and Paradoxical Decompositions, or: How I Learned to Stop Worrying and Love the Axiom of Choice |
| Adam D. Richardson | |
| This talk introduces the historic $\sigma$-additive measure problem in $\mathbb{R}^n$ and describes how the existence of nonmeasurable sets provided an answer to this problem that led mathematicians to explore the consequent finitely additive measure problem in $\mathbb{R}^n$. The Axiom of Choice plays an inextricable role in these problems. The existence of a finitely additive measure on $S^1$ is developed carefully using results from functional analysis before the problem is explored in general. The application of the Axiom of Choice in these problems can yield paradoxical decompositions of subsets of $\mathbb{R}^n$ (and by extension $\mathbb{R}^n$ itself) such as the seminal Hausdorff half-third paradox as well as the eponymous Banach-Tarski paradox. The development of these paradoxes is group theoretic in nature, and some of the group properties which yield such decompositions are discussed. This talk seeks to tell the mathematical origin story of such paradoxes, including detailing the Hausdorff half-third paradox, while highlighting how the controversial Axiom of Choice led to these wholly counterintuitive yet absolutely fascinating measure-theoretic results. |
| 11 Oct 2019 | Introduction to Operads |
| Joe Moeller | |
| I'll give an intuitive introduction to the notion of "operad", a categorical tool for describing algebraic structures. Then we'll look at a few of the main examples that people use in homotopy theory and combinatorics. Time permitting, I'll also talk about the recognition principle and Kozsul duality. |
| 4 Oct 2019 | A Crash Course on $q$-Calculus |
| Michael Pierce | |
| So $q$-calculus, also called quantum calculus, by itself is just a "generalization" of arithmetic and calculus. However some arithmetic factoids in $q$-calculus come up in representation theory and mathematical physics research. The goal of this talk though is to introduce $q$-calculus as a stand-alone topic, and to familiarize the audience with some of those factoids so that it won't be too jarring to them if those factoids pop up in research. If time permits I'll also talk about current research being done into pure $q$-calculus, and explain a bit about why that word generalization above is in quotes. |
| 27 Sep 2019 | Interpreting Mathematics Rigorously |
| Alexander Martin | |
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Rigor is the language of mathematics, but currently our languages are inherently not rigorous. It is possible for a student, using a pencil on paper with the notations and terms presented in the class, to make an error in a mathematical statement. I have been working on a project to develop a language which accounts for mathematical rigor. One major design goal is that assumptions, claims, and implication relations come from making a statement as opposed to from reading it, making it impossible to state something incorrect. The language is explicitly compiled by a computer so it makes sense to distinguish between valid statements and nonsense, nonsense being something which throws an error upon compilation. Any statement which can be stated (i.e. successfully compiled) is then tautologically true by design, so we will see how this works and what it means.
Another goal is to be able to write new definitions, make new claims, and prove them in a way which a computer can understand and without modifying the language itself. This language involves interpreting mathematical statements as directed acyclic graphs, so we will see how that works. Statements are saved in XML files (like SVG and XHTML if you have ever seen those) and I have written a compiler in javascript (web browser). This talk is an overview of what I have developed so far, both the language itself and the tools to interact with it, and what I hope the future holds for the project. |
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