2017-2018

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.

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.

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 Tradler

websupport1.citytech.cuny.edu/faculty/ttradler/

Title: Operads, Associahedra, and Infinity-Algebras

Abstract:

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 Seminar

October 16th, 2017

Christian Wolf

math.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 Douglas

https://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 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