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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.

Samuel Magill:
Ryan Utke:
Professor Alexander Gamburd:

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

Spring 2018 Schedule

February 26th, 2018

    Melvyn Nathanson  

    Title:  Matrix scaling and a problem in number theory

    Abstract:  Recently, there has been renewed interest in alternate minimization algorithms to generate doubly stochastic matrices, and their generalization to operator scaling.  This talk will introduce this subject, and                    describe a problem in diophantine approximation that these algorithms suggest.

March 5th, 2018

    Enrique Pujals

    Title: Back and Forth on the Lorenz Attractor: Hitchhiking on Geometrical Construction

    Abstract:    Almost five decades ago,  E. Lorenz published an article in the Journal of Atmospheric Sciences, which raised mathematical questions that played an essential role in the modern  development of the                         theory of dynamical systems. I will make an effort to describe  this  ride.

March 12th, 2018

    Blair Davey
    Title: Unique continuation of PDEs and Landis' conjecture

    Abstract:    In the late 1960s, E.M. Landis made the following conjecture: 
                    ‘’If $u$ and $V$ are bounded functions, and $u$ is a solution to the equation $\Delta u - V u = 0$ in $\mathbb{R}^n$ that decays like $|u(x)| \le c \exp(- C |x|^{1+})$, then $u$ must be identically zero.”

                    This conjecture is about the unique continuation of elliptic partial differential equations. I will discuss some of the broad questions in the study of unique continuation, explore a few of the techniques used,                     then present some recent progress that has been made towards resolving Landis' conjecture.

March 19th, 2018

   John Terilla

    Title:  Matrix product states

    Abstract:  Quantum many body problems involve vector spaces of very large dimension, 2^10^23, for example.  Matrix product states describe small subsets that are easier to work with.  No physical examples will be discussed.  Instead, I’ll give an example from representation theory related to human language.

March 26th, 2018

    Elena Kosygina 

Title: Self-interacting random walks

    Abstract: We shall discuss several models of random walks which interact with their own history (and possibly with a random environment). Unlike standard random walks, these walks are not Markovian (i.e. have    memory). One of the very attractive features of these models is that they are easy to describe and offer many interesting open problems. 

    In the second part of the talk I shall give an example of how in the case of excited random walks (ERW) on integers a mapping to a class of branching processes (which are Markovian) helped to obtain many results for ERWs. But this mapping works only in dimension 1, and new ideas are currently needed to make progress in higher dimensions.

April 9th, 2018

    Gunter Fuchs

    Title: The story of the continuum, continued.

    Abstract: The question what the size c of the real continuum is has been a driving force in set theory, from its very beginning. Georg Cantor, the founding father of set theory, dedicated much of his lifetime to trying to prove the continuum hypothesis, stating that c is the first uncountable cardinal, in vain. Hilbert's famous list of 23 problems that he presented in 1900 begins with this question, and it was not until 1963 that the question was "answered" by Paul Cohen in the sense that it was shown that one can neither prove nor refute the continuum hypothesis based on the axioms of set theory (which had been developed in the meantime). But this does not mean that the continuum hypothesis is settled. In a sense, the question was no longer whether the continuum hypothesis is true, but rather, whether it "should be" true. I want to look at this problem from the angle of forcing principles. The crucial technique used for Cohen's result is called forcing, a method that makes it possible to add objects, such as real numbers, to a given model of set theory. Forcing axioms say in a sense that the universe is saturated with respect to certain kinds of forcing notions, and the commonly applied forcing axioms imply in particular that there are many real numbers, so that the continuum hypothesis fails. I will talk a little bit about recent developments that show that there are natural forcing axioms that do not have this consequence, and in fact, some related forcing principles outright imply that the continuum hypothesis is true.

April 16th, 2018

    Shirshendu Chatterjee
    Title: Stochastic Modeling of Network Dynamics and Inference.

    Abstract: We will discuss about some popular stochastic models of epidemics, infection spreading and community detection. We will focus on the mathematical aspects and analysis of those models.

April 23rd, 2018

    Alexey Ovchinnikov 

    Differential algebra and modeling

    Many real-world processes and phenomena are modeled using systems of ordinary differential equations with parameters. Given such a system, we say that a parameter is globally identifiable if     it can be uniquely recovered from input and output data. We will discuss the basics of indentifiability and mathematics that leads to new algorithms that can tackle problems that could not be        tackled before.
April 30th, 2018

    Dragomir Saric

    Title: Quasiconformal and hyperbolic geometry of surfaces

    Abstract: A non-exceptional Riemann surface X has a unique hyperbolic metric in its conformal class. The complex analytic structure on Riemann surfaces allows us to define quasiconformal maps between them. We discuss some relations between the two different viewpoints of Riemann surfaces.

Fall 2017 Schedule

September 18th, 2017

    Russell Miller

                                                                         Title:  Classification of Algebraic Fields

    Every field has a smallest subfield:  either a copy of the rational numbers, or a copy of the p-element field, depending on whether the characteristic is 0 or a positive prime p.  A field is algebraic if it is an algebraic extension of this subfield, with every element of the field satisfying some polynomial over the subfield.  We will focus here on characteristic 0.

    The first question to be discussed is what counts as a classification.  Normally one would like to be able to give a list of all the        isomorphism types of such fields, in such a way that, for each field, we can find its place on the list, and for each place on the list, we can determine the corresponding field.  Saying "we can find" and "we can determine" suggest questions of effectiveness:  we would like to be able to compute the bijection between fields and places on the list.  Therefore, computability theory will enter into the talk.  Surprisingly, so will topology, along with (less surprisingly) a bit of algebra and field theory, and we will explain how all these areas intersect and yield our classification.

September 25th, 2017 

    Thomas Tradler
Thomas                   Tradler

                                              Title:  Operads, Associahedra, and Infinity-Algebras

   I will introduce operads, which are gadgets that describe the underlying algebraic structure of some algebraic notion, such as, for example, the notion of an "associative algebra." Operads are very well suited to study problems up to homotopy. I will show this for the case of homotopy associative algebras and show its connection to the combinatorics of associahedra. Although these ideas were studied and well-understood in the 1960s, 1970s and 1990s, similar techniques can be used to analyze other algebraic notions, that do not precisely fit into this language. I will show this for the notion of an "associative algebra with a co-inner product," which comes up, for example, in the study of string topology.

October 2nd, 2017

    Luis Fernandez

               Title:  Almost complex maps from the 2-dimensional sphere to the 6-dimensional sphere

   I will explain how to do complex analysis without complex numbers, thus introducing the concept of an almost complex structure, and what it means for a map to be almost complex (or pseudoholomorphic). Among the spheres, only the 2-dimensional sphere and the 6-dimensional sphere admit an almost complex structure. In fact, the 2-sphere is just the Riemann sphere. To define an almost complex structure in the 6-dimensional sphere I will introduce the octonions (or Cayley numbers) as a generalization of the quaternions. Once we have almost complex structures in the 2- and the 6- sphere, we can talk about the space of almost complex maps between them. I will discuss why these maps are,  in particular, harmonic, and state some results about the dimension of the space of such maps. NOTE: no technical knowledge will be necessary to understand most of the talk.
October 9th, 2017

    No Seminar

October 16th, 2017 

    Christian Wolf


                                                       Title: Zero-temperature measures in dynamical systems


   We give an introduction to the mathematical theory of zero-temperature measures which play an important role in statistical physics. In particular, we discuss various forms of complexity of a given dynamical systems  including entropy, free energy and topological pressure. The talk will be accessible to first year graduate students.

October 23rd, 2017

    Andrew Douglas

                                                       Title: Lie algebras and representation theory


    I'll give an introduction to Lie algebras and their representations. I'll say a word about applications of Lie algebras, particularly 

in physics. Then, I'll briefly describe some of my recent and current research projects on Lie algebras.

October 30th, 2017 

    Math Department Colloquium: Dennis Sullivan
November 6th, 2017

    Olga Kharlampovich

Title: Model theory and algebraic geometry in groups and group algebras.

Abstract:  We discuss the modern theory of equations in groups,  algebraic geometry and model theory in free and hyperbolic groups.  The development of algebraic geometry comes together with advances in the theory of  fully residually free (limit groups) and fully residually hyperbolic groups, which are coordinate groups of irreducible algebraic varieties.  We describe finitely generated  groups elementarily equivalent to a free non-abelian group, and definable sets in a free group. 

The Diophantine problem in free and hyperbolic groups is decidable. We also discuss the contrasting results on the undecidability of equations in group algebras of  free and limit groups over a field and  results on definability of the geometry of a group in the theory of its group algebra. No any special knowledge is required. This is a combination of the ICM (International Congress of Mathematicians) talk of 2014 with new results about group algebras.

November 13th, 2017 
    Joseph Maher

    Title:  Random walks on groups

    Abstract: We'll give a gentle introduction to some recent work on random
walks on groups.  We'll concentrate on the case of groups which act on 
hyperbolic spaces, which includes hyperbolic groups and mapping class 
groups of surfaces.  We'll discuss what's known (and not known) about 
typical elements of groups arising from random walks, and indicate how you 
can use random methods to show some non-random results.

November 20th, 2017
    Bianca Santoro