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 4:00-4:45 PM in the Science Center, Room 4102, unless otherwise stated.Organizers: Professor Alexander Gamburd: agamburd@gmail.com Richard Gustavson: rgustavson@gradcenter.cuny.edu Jacob Russell: jrussellmadonia@gradcenter.cuny.edu Spring 2017 ScheduleFebruary 27th, 2017 No seminarMarch 6th, 2017, Room 4214 (Math Lounge) Alice Medvedev What good is model theory, a branch of mathematical logic? Does an injective polynomial function with the same number of inputs and outputs have to be surjective? If you have n polynomials P_{1}, P_{2},...,P in _{n}n variables x, and the function _{1}, x_{2},...,x_{n}f(x) := (_{1},x_{2},...,x_{n}P(_{1}x),...,_{1}, x_{2},...,x_{n}P(_{n}x_{1},x)) is injective, does it follow that this function _{2},...,x_{n}f is surjective? Of course, the answer depends on where the coefficients of the polynomials and the inputs of the function come from; and on the value of n. Some instances of this question are obviously true, others are obviously false, and yet others are not so obvious. Ax used mathematical logic, together with some obvious cases, to resolve some natural non-obvious cases, such as complex numbers and n>1. I will present Ax's proof, identifying key logic ingredients. If time permits, I will also talk about model-theoretic versions of Galois theory and algebraic geometry - without fields, rings, or any kind of algebra. March 13th, 2017 Ara Basmajian Geometric Invariants in Hyperbolic GeometryMarch 20th, 2017, Room 4214 (Math Lounge) Perry Susskind Connecticut College Applying for mathematics department positions at liberal arts institutions: a conversation March 27th, 2017, Room 4214 (Math Lounge) Yunping JiangAn introduction to chaotic and complex dynamical systems I will give a brief introduction about history and recent developments in chaotic and complex dynamical systems, including period three implies chaos, Sharkovskii’s order, period doublings and the universality, Smale’s horseshoe, Julia and Fatou sets, and the Mandelbrot set. I will also mention some work we did in the past. Depending on times, I would like also to talk about the Mobius randomness and Sarnak’s conjecture in number theory which relates closely to chaotic dynamical systems. April 3rd, 2017Dennis Sullivan Some basic math things I like, where they lead and potential usages I like the idea of a cycle representing a homology class and intersection properties of cycles and homologies. This leads to a subject called string topology. It also leads to generalizing the notion of a commutative and associative ring structure in the presence of a boundary operator. I like locally euclidean coordinates for manifolds from rough to smooth. In various dimensions the possibilities with varying amounts of calculus have a nice pattern with a big spike in dimension four. There is a close relationship to mathematical physics models. I like fluid motion and turbulence in dimension three. The nonlinear PDE model is currently intractable but scientists make progress by discrete approximation and simulation. I have been trying to use various math ideas to do something useful here especially using the first two sets of ideas in response to the intractability. April 10rd, 2017 Spring Break, no seminar April 17th, 2017 Spring Break, no seminarApril 24th, 2017 Benjamin SteinbergWe give an overview of the representation theory of finite monoids and discuss some applications including finite Markov chains. No prior knowledge of these topics are assumed. May 1st, 2017 Mahmoud Zeinalian On the geometry and algebra of the space of all Riemann surfaces. Making sense of the space of all geometric objects of a specific type, up to an appropriate notion of equivalence, has been a fruitful endeavor in mathematics. For example, the fundamental theorem of algebra implies the space of all n un-ordered, not necessarily distinct, points on the 2-sphere is the n-dimensional complex projective space CP^n. Or, the space all ellipses in the plane, or any abstract 2-dimensional real vector space, that are centered at the origin, up to magnification, is a new 2-dimensional space with a natural notion of distance. This notion of distance is god-given — i.e. it doesn’t come from a choice of a metric on the plane — and makes the space of such ellipses into a model of non-Euclidean geometry. Among more sophisticated examples are the space of all Riemann surfaces of a given genus. These spaces have interesting topology, geometry, and dynamics. In several ways, CUNY has been a center for studying these spaces. In fact, if you consider them for all genera at once, they also become a fundamental tool in the algebraists' theory of operations with applications to the quantum theory of fields. In this talk, I would like to explore with you some of the geometric and algebraic features of these spaces. I will discuss some actual math content and end with a description of some of the current research branches stemming from these spaces. |

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