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 Fall 2018 ScheduleSeptember 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 closedOctober 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 ColloquiumOctober 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 MathematicianSpring 2018 ScheduleFebruary 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, 2018Elena 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 Title: Differential algebra and modeling Abstract: 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 ScheduleSeptember 18th, 2017 Russell Miller qcpages.qc.cuny.edu/~rmiller/ Title: Classification of Algebraic Fields Abstract: 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 Tradlerwebsupport1.citytech.cuny.edu/faculty/ttradler/ Title: Operads, Associahedra, and Infinity-AlgebrasAbstract: 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 fsw01.bcc.cuny.edu/luis.fernandez01/ Title: Almost complex maps from the 2-dimensional sphere to the 6-dimensional sphere Abstract: 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 SeminarOctober 16th, 2017 Christian Wolfmath.sci.ccny.cuny.edu/people?name=Wolf
Title: Zero-temperature measures in dynamical systems Abstract: 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 Douglashttps://www.gc.cuny.edu/Page-Elements/...Centers.../Mathematics/.../Andrew-Douglas Title: Lie algebras and representation theory Abstract: 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 SullivanNovember 6th, 2017 Olga KharlampovichTitle: 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 |

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