# Brandeis Graduate Student Seminar (GSS)

### Spring 2016

This semester the GSS will meet Fridays at 12:30pm in GS317. All math graduate students are welcome. Free pizza is provided, and beer is often available too.

• Jan 15 – Introduction and Lightning Talks
• We will call for volunteers to sign up for GSS spots for the semester and discuss other math graduate business. Afterwards, we will invite the audience to give five minute lightning talks on any subject. These should be comprehensible in 5 minutes and on any subject potentially of interest to someone else (including math). Possibilities include “how to speed read math papers”, “how to set up your website”, “how to improve your teaching”, “how to apply for jobs”, etc.
• Jan 22 – Langte Ma: Building the Symplectic Capping By Symplectic Configurations.
• Abstract: In this talk, I will introduce the basic notion of symplectic form and how symplectic manifolds look like. Then I will talk about how people study symplectic filling via symplectic capping and the construction of caps by some symplectic configurations.
• Jan 29 – Duncan Levear: Introduction to Data Structures: Hash Tables and Binary Search Trees
• Abstract: In this computer science talk, I will introduce basic data structures and describe more advanced ones. We will discuss the pros and cons of each, and look at some examples where the differences can be significant. The discussion will showcase some interesting applications for prime numbers and logarithms. Google has publicly stated “hash tables are the most important data structure known to mankind”; come find out why!
• Feb 5 – Job Rock: What is Haskell and why should I care?
• Abstract: I will demonstrate some fundamentals of Haskell. Along the way we’ll cover the differences between Haskell and object-oriented programming and why some of these differences benefit mathematicians. Bring your laptop and make sure to have Haskell installed before the talk so you can follow along. You should download the entire Haskell Platform from here: https://www.haskell.org/downloads.
• Feb 12 – Rose Morris-Wright: Folding Algorithms and Kempe’s Universality Theorem (slides)
• Abstract: The mathematics of folding is a fusion of geometry, discrete math, and computer science, with applications ranging from robotics to origami to protein folding. This talk will introduce some of the terminology used to model 1D folding apparatus and discuss some of the questions about these apparatus that are of interest to mathematicians. Lastly I will introduce Kempe’s Universality Theorem, a fundamental result of 1-D folding famous for showing the existence of a linkage that “signs your name”.
• Feb 26 – Frankie Petronella: Introduction to Iterative Methods
• Abstract: Solving or approximating solutions to large systems of linear equations using matrix-matrix operations can be painfully inefficient. In this talk I will be introducing the idea of Krylov Subspace Methods, these provide a nicer alternative. I will use a few examples to show why they are needed. I will then introduce the power iteration and use it to build a couple of the most common and popular krylov methods.
• Mar 4 – Yan Zhuang A Generalized Goulden–Jackson Cluster Method and Lattice Path Enumeration
• Abstract: How many length n words on the alphabet {a,b,c} contain exactly k occurrences of the subwords acb and bc? Questions like this can be answered using the Goulden–Jackson cluster method, a powerful theorem in enumerative combinatorics. We give an expository account of the cluster method beginning with an introduction to generating functions, and then present a new generalization of the cluster method for “monoid networks”, a combinatorial construction reminiscent of finite-state automata. We then show how this generalized cluster method can be used to solve problems in lattice path enumeration.
• Mar 11 (double feature) – Nick Wadleigh: An Inhomogeneous Dirichlet Theorem Using Shrinking Targets (slides)
• Abstract: Dirichlet’s theorem states that for a real $m\times n$ matrix,  $\|Aq+p\|^m < \frac{1}{t}, \|q\|^n\leq t$ has nontrivial integer solutions for all $t\geq 1$.  Davenport and Schmidt have observed that if $\frac{1}{t}$ is replaced with $\frac{c}{t}, c<1$, almost no $A$ has the property that there exist solutions for sufficiently large $t$.  Replacing $\frac{c}{t}$ with an arbitrary function, it’s natural to ask when precisely does the set of such $A$ drop to a null set.  We do not answer this question, but the analogous inhomogeneous question seems to be amenable to the tools of dynamics on the space of affine lattices. Namely, we give a necessary and sufficient condition on a function $\psi$ such that for almost all pairs $(A,b)$ (where $A$ is an $m\times n$ matrix and $b$ an $m$-tuple), the system $\|Aq+b+p\|^m < \psi(t), \|q\|^n\leq t$ has  integer solutions for all large enough $t$.
• Mar 11, 1pm (double feature) – Devin Murray: Residually finite Groups
• Abstract: Finite group theory is great, but a lot of groups aren’t finite. It would be awesome if we could use techniques from finite group theory to study infinite groups. We will define residually finite groups and discuss some of the interesting results and entirely reasonable theorems about residually finite groups that very unreasonably turn out to be false for groups in general. Time permitting we may also talk about other generalizations of “finiteness” into the world of infinite groups.
• Mar 18 – Josh EikePatterson-Sullivan measures on boundaries of metric spaces
• Abstract: If we have a group acting on a proper metric space X, we can construct a Patterson-Sullivan measure on the boundary of X. I’ll show how this construction works and briefly discuss its applications. Time permitting, I’ll also explain how it relates to my current research on Ruth and Harold’s contracting boundary for CAT(0) metric spaces.
• Mar 18, 2pm (encore)- Rose Morris-Wright: Folding Algorithms and Kempe’s Universality Theorem (slides)
• Abstract: The mathematics of folding is a fusion of geometry, discrete math, and computer science, with applications ranging from robotics to origami to protein folding. This talk will introduce some of the terminology used to model 1D folding apparatus and discuss some of the questions about these apparatus that are of interest to mathematicians. Lastly I will introduce Kempe’s Universality Theorem, a fundamental result of 1-D folding famous for showing the existence of a linkage that “signs your name”.
• Apr 1, 2pm – Cristobal LemusLagrange Inversion and Lattice Path counting.
• Abstract: Lagrange Inversion is a very useful tool for finding the coefficients of a generating function. In this talk we first introduce the method and give a few classical examples. Then we discuss multivariable Lagrange and apply it to some systems of generating functions. Lastly we talk about counting a few families of Lattice paths with added statistics.
• Apr 8, 2pm- Matt Garcia: Calculus without Limits: the Kock-Lawvere axiom and an alternative approach to differential geometry
• Abstract: Sophus Lie famously remarked on his work in differential geometry and Lie theory, “I found these theories originally by synthetic considerations. But soon I realized, as expedient the synthetic method is for discovery, as difficult it is to give a clear exposition on synthetic investigations…” In this talk, we will discuss some key axiomatic features of the models (read: categories) that allow for valid synthetic reasoning to be concisely expressed, such as the Kock-Lawvere axiom and the necessity to reason constructively. Then, we will attempt to illustrate the flavor of a synthetic proof of one of the classical results of differential geometry: the Frobenius theorem. Time permitting, we might outline how to construct such a categorical model.
• Apr 15, 2pm – Jordan Awan:  An Extension of the Tutte Polynomial for Digraphs and Oriented Matroids
• The Tutte Polynomial is a bivariate generating function defined for graphs, which contains structure of the graph such as the number of forests and spanning trees, the Chromatic Polynomial, and the number of acyclic and totally cyclic orientations. We define a generating function called the A-polynomial for digraphs which, when restricted to graphs, is equivalent to the Tutte Polynomial. We will show that the A-polynomial has many characteristics in common with the Tutte, and present results on what additional structure is encoded in the A-polynomial.
• Apr 21 (Th) – Angelica Deibel: Random Artin Groups
• Abstract: What is a random Artin group? What does it mean to say a random Artin group has some property? What does it mean for an Artin group to be of FC-type? When are random Artin groups of FC-type? I will at least partially answer all of these questions!

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