# Brandeis Graduate Student Seminar (GSS) – Fall 2015

### Fall 2015

The GSS meets Fridays at 2pm in GS300. All math graduate students are welcome. Free pizza is provided, and beer is often available too.

• Aug 28 – Introduction and Panel Discussion
• Featured panelists: Aru Ray, Yan Zhuang, Devin Murray & Shaunak Deo
• Moderator: Jordan Tirrell
• We will introduce the Graduate Student Seminar and organize speakers and our weekly teas. This will be followed by a Q&A panel for graduate students. Our goal is to encourage discussion and disseminate graduate school wisdom (topics may include courses, requirements, advisors, thesis writing, job applications, socializing, sanity etc). Beer will be free.
• ​​Sept 4 – Yan ZhuangThrough the Probabilistic Lens
• Abstract: In this expository talk, we present probabilistic proofs of three non-probabilistic results: Erdős’s theorem on sum-free sets (number theory), Sperner’s theorem (extremal combinatorics), and the Weierstrass approximation theorem (analysis). No background knowledge is assumed beyond the very basics of probability theory.
• Sept 11 – Tue Ly​​: Geometric and probabilistic ideas in Diophantine approximation.
• Abstract: In this talk, I will briefly introduce some classical results of Diophantine approximation on the real numbers, and discuss about some tools in geometry and probability to prove their higher dimensional analogues.
• Sept 18 – Duncan Levear: Linear Programming and Applications
• Abstract: Many problems in Operations Research can be framed as linear programs: optimizing a linear function subject to linear constraints. By proving some results about these problems, chiefly the Fundamental Theorem of Linear Programming, we will discover shortcuts to finding an optimal solution. This will shed light on ubiquitous problems from our daily lives: such as scheduling, dieting/mixing, and cooking Thanksgiving dinner.
• Sept 25 – McKee Krumpak: Regular polytopes and the reflection groups associated to them
• Abstract: Approximately covered topics will be the enumeration of the regular polytopes and some geometric intuition for their symmetry groups. Possibly something about the associated Coxeter diagram of a reflection group, depending on how things go.
• Oct 2 – Job​​ Rachowicz: Applications of Category Theory
• Abstract: In this talk we’ll address the concern that category theory is just `abstract nonsense.’ We’ll (very) briefly remind ourselves of the basics and some well-known categories. Then we’ll talk about some applications of category theory in and out of mathematics.
• ​​Oct 9 – Frankie Petronella: Why you should care about PDE’s
• Abstract: In this talk I will be discussing PDE’s. I will provide a short introduction into the subject area. I will discuss some of the methods used to prove the existence and uniqueness of solutions; including one that I have used. I will provide an example that illustrates why existence of solutions is not guaranteed. Finally, I will discuss some of the more fun applications of PDE’s.
• Oct 16 – Devin​​ Murray: Using non-standard analysis to study groups and metric spaces
• Abstract: There are two normal formal foundations for analysis, the standard one uses the delta-epsilon definition of limits and goes forward from there. The non-standard approach (which is perhaps more in line with Newtons original idea) is to enrich the real numbers with ‘infinitesimals’ and to define everything with these infinitesimals instead of limits. From my exposure to non-standard analysis in the past, I got the impression that it was mostly for overly-enthusiastic undergraduates, model theorists, and quacks. I will attempt to correct this misconception and show that non-standard analysis contains a powerful set of tools which aid in discussing and solving many delicate and interesting problems in groups theory and metric geometry.
• Oct 30 – Josh​​ Eike: Counter-Intuitive Games and Their Strategies
• Abstract: Mathematicians, economists, and others like to come up with clever games that challenge or expose people’s intuitions. The Monty Hall Problem and the Prisoner’s Dilemma are classics, but there are many more. We will see how the Brandeis math department fares against such challenges. There will be tricks and there will be treats. Keep your wits about you and you might leave with a prize!
• Nov 6 – Tarakaram​ Gollamudi:​ 4 Riemann – Roch’s I know
• Abstract: It has long been understood that there are deep connections between algebraic number theory and algebraic geometry. In this talk, I introduce Arakelov theory which expose some of these deep connections. In particular, I will talk about how we can extend definition of “genus” to Number Fields, and I will introduce Reimann-Roch for Number Fields, which is an analogue of RR for Curves. If time permits I will also introduce Alexander Grothendieck RR for Number Fields. I will try to make the talk self contained. But little exposure to RR for algebraic curves, Minksowski Theory and extensions of valuations would be helpful. (Thanks to Carl Wang for his help over the Summer).
• ​​Nov 13 – Ying Zhou: A formula for the permutation of maximal green sequences in $A_n$ straight orientation and beyond.
• Abstract: In this talk I will define quivers, quiver mutations, maximal green sequences, the associated permutation of a maximal green sequence and then give a formula for the permutation of maximal green sequences in $A_n$ straight orientation. This is a joint work with Professor Kiyoshi Igusa. Recent research such as the work in one-sink orientations of $A_n$ and the technique of quiver cutting will also be discussed.
• Nov 20 – Special Guest Jessica Sorrell: Lattice-based Cryptography from the Learning with Errors Problem
• Abstract: The most ubiquitous cryptographic algorithms today rely on the hardness of factoring or the discrete log problem, but these can be efficiently solved by a quantum computer implementing Shor’s algorithm. The search for a secure and efficient cryptographic successor has motivated renewed interest in the study of hard problems on lattices. We will talk about one such problem, the learning with errors problem, and show how one may use the hardness of this problem and its ring variant to construct public key encryption systems.
• ​​Dec 4 – Jordan Awan: Graph Coloring
• Abstract: We will introduce the problem of coloring the faces (or vertices) of a planar undirected graph. After developing some machinery, including Euler’s formula, we will prove the 5-color theorem.
• Dec 4 – Angelica​​ Deibel: The generic free basis property in Right-Angled Coxeter Groups
• Abstract: I will prove that every Right-Angled Coxeter Group either is virtually abelian or satisfies the generic free basis property.