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Graduate Student Colloquium

The Graduate Student Colloquium is geared towards introducing graduate students to different areas of mathematics.  Each week a different member of the Graduate Center Mathematics Department faculty will discuss a topic that is accessible to all graduate students.  All graduate students without an advisor are required to attend, but even those with an advisor are welcome!  Faculty members are strongly discouraged from attending.

All meetings of the Graduate Student Colloquium will be held on Mondays from 4:15-5:00 PM in the  Room 5417, unless otherwise stated.

Organizers:
Samuel Magill: smagill@gradcenter.cuny.edu
Ryan Utke: rutke@gradcenter.cuny.edu
Professor Alexander Gamburd: agamburd@gmail.com



Spring 2019 Schedule

February 11th, 2019

Scott Wilson

Title: The ubiquitous de Rham complex

Abstract: I'll give a tour of some history and personal close encounters with the de Rham complex, including some present-day research.

February 18th, 2019

College closed

February 25th, 2019 (4:45pm)

Roman Kossak

Title: Resplendent Structures

Abstract: In model theory a structure is a set --- the domain of the structure --- with a set of functions and relations on it.

I will talk about resplendent structures. Examples include: arbitrary sets with no functions and relations, the set of rational  numbers with

the ordering relation, the countable random graph, and the field of complex numbers. The ring of integers, and the field

of real numbers are not resplendent. I will give a general definition of resplendency and I will discuss some applications. 

If you want to read something before the talk, here is a short introductory article: https://www.ams.org/notices/201106/rtx110600812p.pdf


March 4th, 2019

Cancelled (College closed due to weather)

March 11th, 2019

Vladimir Shpilrain

Title: Applications of number theory, algebra, and probability theory to information security

Abstract. I'll give a survey of how the above named areas of mathematics can be applied to information security.

March 18th, 2019

Cancelled

March 25th, 2019

Ilya Kofman

Title: Growth and Geometry

Abstract: We discuss knots, graphs and related topological objects whose complexity has an exponential growth rate given by hyperbolic volume.

April 1st

Tobias Johnson

Title: The frog model and other processes in discrete probability

Abstract: Imagine that every vertex of a graph contains a sleeping frog. At time 0, the frog at one vertex wakes up and begins a random walk. When it moves to a new vertex, the sleeping frog there wakes up and begins its own random walk, which in turn wakes up any sleeping frogs it lands on, and so on. This process is called the frog model, and despite the cutesy name, it's a serious object of study for which many basic questions remain open.

I'll talk about the frog model on trees, where the model displays some interesting phase transitions. In particular, I'll (mostly) answer a question posed by Serguei Popov in 2003 by showing that on a binary tree, all frogs wake up with probability one, while on a 5-ary or higher tree, some frogs remain asleep forever with probability one. I'll also introduce a few seminal results in discrete probability and statistical physics to put my work in context. This is joint work with Christopher Hoffman and Matthew Junge.

April 8th

Martin Bendersky

Title: An Intruduction to Homotopy Theory

Abstract: Some remarks about the homotopy groups of the spheres and applications to algebraic and geometric problems.

April 15th

Pat Hooper

Title: Infinite Rational IETs and Periodic Polyhedra

Abstract:

I will briefly discuss two ongoing research projects involving two undergraduate students: Pavel Javornik and Anna Tao.


Pavel Javornik and I are thinking about the geodesic flow on an periodic polyhedral surface built out of infinitely many squares. We are interested in a number of geometric and dynamical aspects of geodesics on the surface. I will concentrate on discussing properties of closed geodesics. By taking rescaled limits of sequences of closed geodesics, we find some interesting fractal snowflakes (among other phenomena).


Anna Tao and I have been thinking about questions related to infinite rational interval exchange maps. It is well know that Euler's constant e is the sum of 1/n!. This means you can cut the interval (0,e) into subinitervals of length 1/n!. We then permute the intervals (in a particular way) and flip the interval upside down obtaining a map from $(0,e)$ to itself. The vast majority of points are periodic under this map, but there is an aperiodic set of Hausdorff dimension zero. In a sense, we completely understand the dynamics of this map.

April 22nd

Cancelled (Spring Break)

April 29th

Jesenko Vukadinovic



Fall 2018 Schedule

September 17th, 2018

    Louis-Pierre Arguin

   Title:    Large values of the Riemann zeta function on short intervals

    Abstract: The moments of the zeta function play a fundamental role in analytic number theory. In this talk, we will review the basics of the zeta function. We        will look at the probabilistic interpretation of the moments. We will see see how modern techniques inspired by extreme value theory can say something about     the moments in short intervals.

September 24th, 2018

    Mahmoud Zeinalian

    Title:    Poisson geometry and closed string interactions

    Abstract: A function on a symplectic manifold generates a flow. The flows associated to two functions rarely commute. The extent to which such flows don't        commute is measured by a Lie bracket on the space of functions. The relationship this Lie bracket has with the addition and multiplication of functions is            encapsulated in the definition of a Poisson algebra. 

    Some of the most interesting objects in mathematics have natural symplectic structures. The space of all Hyperbolic structures on a closed and oriented           surface, for instance, has a natural symplectic structure. A closed curve on the surface can be thought of as a function that assigns to a hyperbolic structure     the length of the unique geodesic in its free homotopy class.

    The flow associated to a curve that has self intersections is complicated. There are, however, illuminating results about their associated Poisson brackets.        This geometric problem has produced a wealth of new mathematics, most of which was in fact created at GC CUNY. The original Chas-Sullivan work was            informed by the geometric results of Scott Wolpert and Bill Goldman. Their work, in turn, has informed a lot of the subsequent progress. In this talk, I will            describe the relevant Poisson geometry. Our Wednesday seminars, to which you are cordially invited to attend, revolve around related phenomena.

October 1st

    Khalid Bou-Rabee

    Title: The Classification of infinite groups

    Abstract: This sounds like a very daunting problem. And it is. Why tackle such a thing? Is there any hope? Will these contemplations make pizza taste            better? I hope to answer some of these questions with intoxicating buzz words: zeta functions, subgroup growth, commensurability growth, residual finiteness     growth, arithmetic lattices, linear algebraic groups, etc.

October 8th

    College closed

October 15th

    Benjamin Steinberg

    Title: Character Theory and Rationality of Zeta Functions of Languages & Symbolic Dynamical Systems 

     Abstract:    We talk about  zeta functions of symbolic dynamical systems and languages. We sketch a proof using character theory of a theorem of Berstel and Reutenauer on the rationality of zeta functions of         cyclic regular languages. As a consequence we obtain Manning's theorem on the rationality of zeta functions of sofic shifts. Basic knowledge of calculus and linear algebra is assumed and at the very end a small     bit of group theory.

October 22nd

    Math Department Colloquium

October 29th
  
    Jozef Dodziuk

    Title: Ubiquitous Laplacian

    Abstract: I will describe how certain differential and difference operators that are commonly called Laplacian appear in different contexts: graphs, random walks, Riemannian geometry, heat         conduction, propagation of waves, ...

November 5th

    Jason Behrstock

    Title: Introduction to Geometric Group Theory

    Abstract: A central aspect of geometric group theory is that finitely generated 
    groups can be treated as geometric objects and that this view point allows 
    one to prove results that are otherwise difficult or inaccessible through 
    purely algebraic means. In this talk, I'll give a quick introduction to the basic objects of study and a taste of the types of theorems one can prove from this            approach.

November 12th

    Hans Schoutens

    Title: Ultraproducts for the Lay Mathematician