Graduate Student Colloquium
Graduate Student Colloquium
2023-2024
2023-2024
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!
All meetings of the Graduate Student Colloquium will be held on Mondays from 4:00-6:00 PM in person at the Graduate Center Room 5417, unless otherwise stated.
Organizers:
Megha Bhat: mbhat@gradcenter.cuny.edu
Davide Leonessi: dleonessi@gradcenter.cuny.edu
Carol Badre: cbadre@gradcenter.cuny.edu
Professor Christian Wolf: cwolf@gc.cuny.edu
Speaker: Ivan Horozov
Title: Arithmetic groups and modular forms
Abstract: Langlands program is a central part in number theory. It relates (representation of) the Galois group to modular forms. This talk will be about the modular side of the program. I will give examples of arithmetic groups similar to the modular group SL(2,Z) of 2x2 matrices with determinant 1. I will also introduce (holomorphic) modular forms for SL(2,Z). Spaces of modular forms have interpretation in terms of cohomology. Computing the cohomology gives dimensions of modular forms of certain types. Part of my research is to compute the cohomology of SL(n,Z) and GL(n,Z) or n by n matrices for n=3,4,5 in certain coefficient systems. They have interpretation of modular (automorphic) forms of certain types.
Speaker: Sergei Artemov
Title: Serial Properties, Selector Proofs, and the Provability of Consistency
Abstract: Hilbert treated the consistency of a formal theory as a series of statements "D is free of contradictions" for each derivation D. For him, a consistency proof is a two-stage process that produces both
(i) an operation that, given D, yields a proof that D is free of contradictions,
(ii) a proof that (i) works for all inputs D.
This two-stage approach to proving consistency naturally extends to a general notion of proof of an infinite series of sentences in a given theory (we call it "selector proof"). We provide a body of examples that demonstrate that selector proofs have already been tacitly employed by mathematicians and play a prominent role in metamathematics. Within this framework, we provide a proof of consistency of Peano Arithmetic PA and formalize this proof in PA. This undermines the well-known unprovability of consistency paradigm "there exists no consistency proof of a system that can be formalized in the system itself" (Encyclopedia Britannica).
Speaker: Russell Miller
Title: Paths Through Trees vs. Equations in a Galois Group
Abstract: We will present known results from computability theory about computing paths through finite-branching trees. Then we will address the problem of computing solutions to simple equations involving elements of Galois groups, such as the absolute Galois group of the rational numbers, and explain how the results for paths through trees affect our ability to solve these equations.
Speaker: Robert Thompson
Title: Homotopy theory and an application to robotics
Abstract: In this talk I will give a brief exposition of some of the basic concepts of homotopy theory. The idea is to attach algebraic structures to topological spaces in a way that enables one to classify spaces and maps between them. I will then illustrate how this can be used to analyze a mechanical system.
Speaker: John Terilla
Title: The Structure of Meaning in Language: Parallel Narratives in Linear Algebra and Category Theory
Abstract: I will discuss the mathematics involved in the following article from the February 2024 issue of the AMS Notices Article https://www.ams.org/journals/notices/202402/rnoti-p174.pdf. I’ll focus on what I think are the most interesting parts, and try to go a bit beyond what’s in the paper.
Speaker: Luis Fernandez
Title: Complex geometry without complex numbers?
Abstract: One can view complex analysis as the study of maps from the real plane to itself whose derivative commutes with some operator, "multiplication by i". Likewise, one can view complex manifolds as just real manifolds with an extra operator J that behaves like multiplication by i, in the sense that the square of J is -Id. I will make these ideas precise and explain the concepts of complex and almost complex manifolds. Then I will set up the main open problem: when is an almost complex manifolds actually complex?
Speaker: Olympia Hadjiliadis
Title: A Speed-based Estimator of Signal-to-Noise Ratios
Abstract: We present an innovative method to measure the signal-to-noise ratio (SNR) in a B rownian motion model. That is, the ratio of the mean to the standard deviation of the Brownian motion. Our method is based on the method of moments estimation of the drawdown and drawup speeds in a Brownian motion model, where the drawdown process is defined as the current drop of the process from its running maximum and the drawup process is the current rise of the process above its running minimum. The speed of a drawdown of K units (or a drawup of K units) is then the time between the last maximum (or minimum) of the process and the time the drawdown (or drawup) process hits the threshold K. We target the SNR directly rather than estimating the Uniformly Minimum Variance Unbiased Estimator (UMVUE) of the mean and dividing that by the square root of the UMVUE of the variance. Numerical results show that our estimator consistently outperforms the traditional estimator when the noise (i.e., the standard deviation) is significantly stronger than the signal (i.e., the mean). Finally, we explicitly derive the asymptotic distribution of our estimator.
Speaker: Jun Hu
Title: Dynamics of cubic rational maps
Abstract: The Julia sets of quadratic rational maps have a dichotomy classification: either connected or a Cantor set. In general, this dichotomy doesn't hold for rational maps of degree greater than 2. We will show how to extend this dichotomy to or find a classification for certain sub spaces of cubic rational maps. If time is allowed, I will discuss other problems on the study of dynamics of cubic rational maps.
Speaker: Elena Kosygina
Title: Self-interacting random walks
Abstract: TSelf-interacting random walks (SIRW) are random walks which can change their jump distributions depending on their past trajectory. I shall describe several models of SIRW, discuss some of the known results, work in progress, and open problems. I shall also suggest 1-2 open questions which might be starting points of further discussions.
Fall 2023
Date: Monday, September 11, 2023
Speaker: Christian Wolf
Title: Symbolic Dynamics, Entropy and Computability
Abstract: In this talk I will provide an overview about a collection of problems in the area of computability in dynamical systems. This area has recently drawn quit some attention in both, the dynamical systems and computability communities. To keep it accessible I will only discuss symbolic dynamical systems over finite alphabets and will emphasize computability properties of natural invariant measures, entropy, pressure, etc. The talk will be self-contained and and I will mostly focus on ideas rather than technical statements.
Date: Monday, September 18, 2023
Speaker: Sergiy Merenkov
Title: Fractals in geometry and dynamics
Abstract: I will discuss how fractal (self-similar or rugged) spaces arise in various dynamical settings, give an overview of some known results on quasisymmetric uniformization and rigidity of fractals, and state some open problems and conjectures.
Date: Monday, October 2, 2023
Speaker: Alina Vdovina
Title: Ramanujan cube complexes and non-residually finite CAT(0) groups in any dimension
Abstract: We will show the interplay between Geometry, Number Theory and Analysis which leads to unexpected results. A number of possible research projects will be discussed as well.
Date: Monday, October 16, 2023
Speaker: Michael Shub
Title: Structure and behavior in dynamical systems
Abstract: I will survey a few structures perhaps ; Morse-Smale systems, Uniformly hyperbolic and non-unifriomly hyperbolic systems, horseshoes and blenders and homology theory and some behaviors as stability, chaos and randomness.
Date: Monday, October 23, 2023
Speaker: Jason Behrstock
Title: Mapping class groups
Abstract: The mapping class group of a surface is a fundamental object in geometric group theory, low-dimensional topology, dynamics, algebraic geometry, and elsewhere. In this talk, I'll give a brief introduction to mapping class groups and discuss how they can be studied via curves on surfaces.
Date: Monday, October 30, 2023
Speaker: Alexey Ovchinnikov
Title: Parameter estimation in ODE systems: data interpolation, differential algebra, and polynomial system solving
Abstract: Frequently, ODE systems depend on unknown parameters, which could be important quantities on their own. We will discuss an approach to estimate these parameter values that does not rely on optimization. It uses differential algebra, output data interpolation for derivative estimation, and on polynomial system solving. This approach provides a framework for a robust implementation.
Date: Monday, November 13, 2023
Speaker: Victor Pan
Title: New Algorithmic Support for the Fundamental Theorem of Algebra
Abstract: Herman Weyl in his constructive proof of the Fundamental Theorem of Algebra in 1924 rationally approximated all complex roots of a polynomial equation within any fixed positive error tolerance even though there is no explicit rational formulas for roots of x² = 2 (≈ 500BC) and even if radicals are allowed for x⁵ −x−1 = 0 (Galois 1830,32). Hundreds of efficient root-finders have been proposed and keep appearing. Smale in 1981 and Schönhage in 1982 proposed to study solution in terms of its complexity (arithmetic ops+ computational precision or bit-operation complexity). Our modification of Weyl’s root-finder is nearly optimal (up to polylogarithmic factors in degree and precision) and promises to be user’s choice algorithm. In words, it runs nearly as fast as one accesses the coefficients with the precision required for the task.
Date: Monday, November 20, 2023
Speaker: Guy Moshkovitz
Title: Recent progress in the study of tensor ranks
Abstract: While for matrices there is one notion of rank with many equivalent definitions, for tensors (a.k.a. higher-dimensional matrices) there are many inequivalent notions. Understanding the interplay between these notions is closely related to important questions in additive combinatorics, number theory, algebraic geometry, and computer science. In this talk I will discuss recent (exciting!) progress in the study of tensor ranks from the last few years.
Date: Monday, November 27, 2023
Speaker: Adam Sheffer
Title: A structural Szemerédi–Trotter theorem
Abstract: The Szemerédi–Trotter theorem is the fundamental theorem of geometric incidences. This combinatorial theorem has an unusually wide variety of applications, and is used in combinatorics, theoretical computer science, harmonic analysis, number theory, model theory, and more.
Surprisingly, hardly anything is known about the structural question - characterizing the cases where the theorem is tight. This is a basic survey talk and does not require previous knowledge of the field. It will also include recent results which are a joint work with Olivine Silier.