Graduate Student Colloquium
Graduate Student Colloquium
2024-2025
2024-2025
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:
Davide Leonessi: dleonessi@gradcenter.cuny.edu
Carol Badre: cbadre@gradcenter.cuny.edu
Mac McCormick: dmccormick1@gradcenter.cuny.edu
Professor Christian Wolf: cwolf@gc.cuny.edu
Spring 2025
Date: Monday, March 3, 2025
Speaker: John Terilla
Title: Tokens, Characters, and the Mathematics of Tokenization in Language Models
Abstract: Modern large language models (LLMs) do not actually model text at the character level. Instead, they rely on tokens---subword units that come out of algorithms like byte-pair encoding (BPE) and Unigram. Importantly, tokenization typically sits outside the neural network pipeline: it is a preprocessing step that converts raw text into tokens before feeding them to the model. While tokenization has been highly practical, it can lead to odd and surprising behaviors in LLMs: adding or removing a single space in an input can sometimes drastically alter the model’s output.
In this talk, I present a rigorous mathematical framework for understanding tokenization. I will illustrate how standard tokenizers can be viewed as stochastic maps between string spaces. Then I’ll state and prove a striking lemma---one I personally found surprising---that clarifies the theoretical foundations and helps pinpoint where standard tokenization practices can break down. Along the way, I will highlight small but striking examples in real language models, illustrating why these mathematical subtleties really matter in practice.
Date: Monday, March 10, 2025
Speaker: Ivan Horozov
Title: Arithmetic Groups
Abstract: We will start with many examples of arithmetic groups. Most popular among them is the full modular group SL_2(Z) of 2×2-matrices with integer coefficients and determinant 1. This group is related to modular forms and elliptic curves. We will also define some congruence subgroups. However, we will be mostly interested in higher rank arithmetic groups such as GL_3(Z), which is the group of 3×3-matrices with integer coefficients and Sp_4(Z), (it will be defined at the talk.) Part of my research is to compute the cohomology of those and other arithmetic groups. It has applications to L-functions, Eisenstein series and automorphic forms. The techniques for such computations rely on Lie algebras, which also have applications in differential geometry and mathematical physics.
Date: Monday, March 24, 2025
Speaker: Renato Ghini Bettiol
Title: Minimal surfaces in elongated ellipsoids
Abstract: Imagine stretching an n-dimensional ellipsoid inside Euclidean space so that one of its semiaxes becomes very large. This causes a corresponding stretching of geometric objects that locally minimize length (geodesics) or area (minimal surfaces) inside that ellipsoid, making them less stable. In this talk, I will explain how this growing instability can be exploited to detect hard-to-find solutions to these variational problems that evade most cutting-edge methods of Geometric Analysis.
Date: Monday, April 7, 2025
Speaker: Vladimir Shpilrain
Title: Growth in products of matrices
Abstract: The problems that we consider in this talk are as follows. Let A and B be 2x2 matrices (over reals). Let w(A, B) be a word of length n. After evaluating w(A, B) as a product of matrices, we get a 2x2 matrix, call it W. What is the largest (by the absolute value) possible entry of W, over all w(A, B) of length n, as a function of n? What is the expected absolute value of the largest (by the absolute value) entry in a random product of n matrices, where each matrix is A or B with probability 0.5? What is the Lyapunov exponent for a random matrix product like that? We show how these questions cut across several areas of mathematics including group theory, number theory, and the theory of stochastic processes, with applications to information security.
Date: Monday, April 21, 2025
Speaker: David Aulicino
Title: The Wonderful World of Translation Surfaces
Abstract: This talk will be a survey of some recent results on the topic of translation surfaces by me and my collaborators. We will start with basic examples, which include the torus. We will show how translation surfaces were used to solve a problem concerning the Platonic solids, and focus on the dodecahedron in particular. Next we will present a generalization of the remarkable fact that the proportion of primitive lattice points in a disc in the plane is approximately 6/pi^2. All necessary background will be given, and directions for future investigations will be presented.
Date: Monday, April 28, 2025
Speaker: Ara Basmajian
Title: Counting problems on Hecke surfaces
Abstract: For an integer k ≥ 3, the (2, k, ∞)-Hecke surface is the punctured sphere with cone points of orders 2 and k; a special case being the modular surface (k = 3) which is the quotient of the upper half-plane by the modular group, PSL(2, Z). In this talk, after setting up the basics, we’ll focus on various classes of geodesics and their growth rates (with respect to word length and geometric length) leading to several counting problems. These counting problems are part of more general phenomena that intertwine the geometry and topology of curves on surfaces with number theoretic and combinatorial considerations. Finally, we’ll discuss solutions to some of these counting problems which are part of joint projects with Robert Suzzi Valli, Blanca Marmolejo and with Mingkun Liu.
Date: Monday, May 5, 2025
Speaker: Victor Pan
Title: A Nontrivial Mathematical Challenge in Numerical Linear Algebra
Abstract: NUMERICAL LINEAR ALGEBRA (NLA) is a workhorse of modern computations for Big Data represented with matrices of immense sizes that are too large to fit primary memory of computers. Some algorithms work efficiently in practice by accessing only a tiny fraction of matrix entries chosen randomly in a nontrivial way, and it is a challenge to identify the class of input matrices for which the algorithms succeed.
Date: Monday, May 12, 2025
TALK CANCELLED
Fall 2024
Date: Monday, September 16, 2024
Speaker: Christian Wolf
Title: Ergodic theory on coded shift spaces
Abstract: In this talk we present results about ergodic properties of coded shift spaces. Coded shift spaces are natural generalizations of sofic shifts which cover several well-known classes of shifts including S-gap shifts, generalized gap shifts and Beta shifts. We derive sufficient conditions for the uniqueness of measures of maximal entropy (mme) and equilibrium states for Hölder potentials based on the partition of the coded shift into its concatenation set (sequences that are concatenations of generating words) and its residual set (sequences added under the closure). We also obtain flexibility results for the entropy on the concatenation and residual set. Finally, we obtain a local structure theorem for coded shift spaces with unique mme’s.
Date: Monday, September 30, 2024
Speaker: Hans Schoutens
Title: Cohen-Macaulay modules, big and small.
Abstract: I will give a short overview of CM modules and their use in proving homological conjectures.
Date: Tuesday, October 15, 2024 (CUNY Monday)
Speaker: Melvyn Nathanson
Title: Finitely many implies infinitely many.
Date: Monday, October 21, 2024
Speaker: Alexey Ovchinnikov
Title: Finding parameter values in dynamical systems from measurements.
Abstract: We will discuss how to estimate parameter values in dynamical systems given by ordinary differential equations and by recurrence relations.
Date: Monday, October 28, 2024
Speaker: Adam Sheffer
Title: Distinct Distances Problems
Abstract: Erdős introduced a family of problems concerning distinct distances of point sets. Over the decades, a deep theory developed around these problems, connecting combinatorics, algebraic geometry, and additional fields. The current talk is a brief introduction to this topic, including open problems, known bounds, and a brief glance at the theory.
Date: Monday, November 4, 2024
Speaker: Krzysztof Klosin
Title: Introduction to modularity theorems in number theory.
Abstract: Modularity theorems are results that connect certain algebraic structures, for example elliptic curves, or more generally Galois representations to some analytic functions called modular forms. They are a central problem in modern number theory. We will discuss some examples and explain their meaning as well as give an idea of how far we are in the quest of proving them.
Date: Monday, November 18, 2024
Speaker: Sandra Kingan
Title: PageRank
Abstract: We will go over PageRank, the algorithm that started Google.
Date: Monday, December 9, 2024
Speaker: Rohit Parikh
Title: Beth definability, interpolation and language splitting
Abstract: Both the Beth definability theorem and Craig’s lemma (interpolation theorem from now on) deal with the issue of the entanglement of one language L1 with another language L2, that is to say, "how much can we say in language L1 which relates to some fact or otherwise in language L2?" If I get some bad news from the dentist, it will not affect my views in politics. But something I learn about climate change may well have that effect. This issue of "cross-talk" between different languages can be studied mathematically.
The notion of splitting we study looks into this issue. We briefly relate our own theorems in this area as well as the results of subsequent researchers like K. Georgatos, Kourousias and Makinson, and Peppas, Chopra and Foo. We also relate the work on splitting with the famous AGM axioms for belief revision.