September 12

• Speaker: N/A

• Title: First Meeting

• Meeting agenda: This will be a short meeting.  Ajmain will speak a little bit about the purpose of this seminar. We will draft a schedule of talks for the Fall 2024 semester.  Please attend this meeting if you are curious about attending/participating in the seminar.  

September 19

• Speaker: Ajmain Yamin

• Title: Introduction to Local Class Field Theory

• Meeting agenda: Ajmain introduced the statements of Local Class Field Theory and outlined an application to proving the Local Kronecker Weber Theorem.

September 26

• Speaker: Nathaniel Kingsbury

• Title: Renewal Theorems, Symbolic Dynamics, and the Markoff-Hurwitz Equation

• Abstract: In 2016, CUNY's own Alex Gamburd, together with two of his students Michael Magee and Ryan Ronan, gave an asymptotic count of the number of solutions of the Markoff-Hurwitz equation. Their techniques were based on a method of counting lattice points in the hyperbolic plane (and closed geodesics on hyperbolic manifolds) developed by Steven Lalley in 1989, inspired by renewal theorems in probability. This talk will be a high-level overview of the techniques involved in proving their results.

October 10

• Speaker: Josiah Sugarman

• Title: Kesten's Theory of symmetric random walks and the spectra of elements in group rings.

• Abstract: I will discuss work of Kesten on the spectra of operators associated with symmetric random walks on discrete groups. The spectrum of this operator is related to how quickly the walk mixes. I will relate the spectra of these operators to other ones appearing in the work of Lubotsky, Phillips, and Sarnak as well as my own work associated with discrete groups embedded in continuous ones. I will give a few examples and applications.

October 17

• Speaker: Josiah Sugarman

• Title: Some results and applications regarding the spectra for elements in matrix rings over the group ring.

• Abstract: Building on the discussion from last week, I will discuss a slight generalization where we replace elements of the group ring by matrices with entries in the group ring. I will discuss some theorems, conjectures and graph theoretic analogies in this setting. Each of the applications mentioned in the previous talk have versions in this setting and I will discuss those as well. This is ongoing work with Kieran O'Reilly.

October 24

• Speaker: James Myer

• Title: (Toward) an Algorithm to Explicitly Produce a Regular Model of a Hyperelliptic Curve in “Bad” Mixed Characteristic (0, 2): A Criterion to Verify Regularity, and Requisite Blowups via Inductive Valuations

• Abstract: We discuss progress toward an algorithm to explicitly produce a regular model of a hyperelliptic curve in “bad” mixed characteristic (0, 2) via normalization (in the function field of the hyperelliptic curve — always meant in the sequel) of a candidate “Obus-Srinivasan” model of the projective line described explicitly by Andrew Obus & Padmavathi Srinivasan via inductive ((Saunders) Mac Lane) valuations, and a criterion to verify its regularity. The normalization of a point on an Obus-Srinivasan model whose tube avoids a certain “bad” locus — the (union of the) monodromy disc(s) — and so “doesn’t contain any genus” — is conjectured to enjoy a hyperlocal proxy model of the projective line whose corresponding valuations are inductive, and thus, afford us an explicit description of the blowups to which they correspond.

October 31

• Speaker: Wuxuan Tan

• Title: Modular Representation of p-adic Groups

• Abstract: I will discuss mod-p representation theory of p-adic general linear groups of degree 2 and classify all smooth irreducible representations for Qp. The classification was first obtained by Barthel-Livné, and completed by Breuil. Furthermore, I will briefly introduce the mod-p Langlands correspondence.

November 7

• Speaker: Valeriy Sergeev

• Title: Elliptic Curves and F-singularities

• Abstract: In this talk I will introduce some basic facts about elliptic curves, focusing on what happens in prime characteristic. I will then introduce some tools from commutative algebra that were created to study singularities in prime characteristic and discuss their relationship with elliptic curves.

November 14 & 21 @ 11:15AM - 12:15PM over Zoom

• Speaker: Ajith Nair

• Title: Representation theory of SL(2,R) and Selberg trace formula

• Abstract: The Selberg trace formula is a generalization of the classical Poisson summation formula. It expresses a relation between the eigenvalues of Laplacian acting on a compact quotient of the upper half plane (spectral side) and the lengths of certain closed geodesics on the compact quotient (geometric side). Moreover, the eigenvalues of the Laplacian have an interpretation in terms of the representation theory of SL(2,R). In this expository talk, after a brief overview of the representation theory of SL(2,R), we will discuss the trace formula and some applications.

• Recording: part 1 link (Passcode: *nz?k&q7 ), part 2 link (Passcode: 9dcSW18^ )

December 5

• Speaker: Joseph Dominick DiCapua

• Title: Explicit Interpolation Theorem for Norm Compatible Sequences

• Abstract: We give a new explicit proof of Coleman's interpolation theorem for norm compatible sequences. The proof uses isomorphisms of formal group laws, eigenspaces of principal units over a Lubin-Tate tower of field extensions, and Hilbert's Theorem 90. This is joint work with me and my advisor, Victor Kolyvagin.

December 12

• Speaker: Jeffrey Kornhauser

• Title: The Riemann-Roch Theorem for Compact Riemann Surfaces

• Abstract: In this talk, I will start by giving a crash course on cohomology groups of sheaves of abelian groups on topological spaces. Then we will specialize this to our object of interest: sheaves of meromorphic functions on compact Reimann surfaces. We can then define genus, and state and give a proof sketch of the classical Riemann-Roch theorem. Some applications will be given, and time permitting, we can briefly discuss the Serre-Duality theorem, and the Riemann-Hurwitz Formula.

January 30

• Speaker: N/A

• Title: First Meeting

• Meeting agenda: This will be a short meeting.  Ajmain will speak a little bit about the purpose of this seminar. We will draft a schedule of talks for the Fall 2024 semester.  Please attend this meeting if you are curious about attending/participating in the seminar.  

February 6

• Speaker: Nathaniel Kingsbury

• Title: Counting Primes: an Intro to Analytic Number Theory

• Meeting agenda: Let π(x) denote the number of prime numbers less than x. The Prime Number Theorem states that π(x) is asymptotic to x/log(x). In this talk, I will discuss some of the key techniques and ideas going into the prime number theorem. Along the way, we will introduce some of the key classical arithmetical functions, such as the Mobius function and the von Mangoldt function, Chebyshev's elementary bounds for π(x) and Bertrand's postulate, and (time permitting) the very basics of Sieve methods.

February 13

• Speaker: Ajmain Yamin

• Title: The Volume of the Moduli Space of Lattices

• Abstract: I will discuss Siegel's volume formula for the modulo space of lattices SL(n,Z)\SL(n,R)/SO(n,R).  This volume is expressed with a product of values of the Riemann zeta function at 2, 3, ... , n.  A key ingredient in the proof is the Poisson summation formula for lattices in Euclidean space.  If time permits, I'll discuss how this volume formula fits in the general framework of Tamagawa numbers. 

February 20

• Speaker: Jeffrey Kornhauser

• Title: The Hasse-Minkowski Theorem

• Abstract: The Hasse-Minkowski theorem, which is a local to global principal, is one of the most important results in all of number theory. It states that a quadratic form in n variables over a number field K represents 0 if and only if it represents 0 in every completion of K. In this talk, following Borevich Shafarevich, I will discuss a proof of the theorem over the rationals. First, I will provide some background material on quadratic forms, p-adic numbers, and the Hilbert symbol. Then the focus will be on degree 3 case. Time permitting, we will sketch the remaining cases.

February 27

• Speaker: Nathaniel Kingsbury

• Title: Title: Two Proofs of Quadratic Reciprocity

• Abstract: I will present two classical proofs of quadratic reciprocity: one based on Fourier analysis in the form of Gauss sums, and the other based on Hilbert’s theory of the splitting of primes in a Galois extension, which we will state precisely but not prove. We will then discuss the interrelations between these two proofs: in particular, I aim to convince you that these are the same proof, and that the technology of Gauss sums is a way to do Galois theory “more concretely.” Time permitting, I will also discuss the problem of the “sign of the quadratic Gauss sum,” discuss why we must go beyond algebra to solve it, and determine the sign using the technique of Poisson summation.

March 13

• Speaker: Josiah Sugarman

• Title: The Kesten measure for certain generating sets of arithmetic groups

• Meeting agenda: I will introduce Bass-Serre theory and apply it to certain arithmetic groups, G. This will allow us to express them as free products with amalgamation. I will then discuss free probability which will help us compute the Kesten measure for certain generating sets.

March 20

• Speaker: Valeriy Sergeev

• Title: Commutative Algebra with a view towards Fermat's Last Theorem

• Abstract: We will introduce some tools from commutative algebra that were used in the proof of Fermat's Last theorem. This will include a discussion of complete intersection rings, Fitting ideals and criteria for isomorphisms of rings that played a role in the proof.

March 27

• Speaker: Connor Stewart

• Title: Scratching the Surface of the Grothendieck-Teichmuller Group

• Abstract: The absolute Galois group of ℚ is notoriously poorly understood – the problem of identifying its finite quotients, or the Galois groups of finite extensions of ℚ, is the Inverse Galois Problem, a major open problem in number theory. In his Esquisse d’un Programme, Grothendieck outlined a vast approach to studying the absolute Galois group via its actions on objects that are not obviously arithmetic, such as maps on topological surfaces or moduli spaces of marked varieties. In this talk, we will explore the idea of looking at actions of the absolute Galois group on moduli spaces – in the case of genus 0 curves, as studied by Drinfeld and others. This will lead us to look at a larger group containing the absolute Galois group, called the Grothendieck-Teichmuller group.

• Recording: link (Passcode:  B$Hx1uxM )

April 3

• Speaker: Ajmain Yamin

• Title: Sums of Binary Hamiltonian Forms

• Abstract: We investigate sums of binary Hamiltonian forms, analogous to Zagier's sums of binary quadratic forms and Karabulut's sums of binary Hermitian forms. We prove that these sums live in finite dimensional vector spaces and compute explicit bases for some of these spaces.  This results in new identities satisfied by certain elementary arithmetic functions.  This talk presents work done in collaboration with Gautam Chinta (CCNY). 

April 10

• Speaker: Wuxuan Tan

• Title: Introduction to Diophantine Approximation

• Abstract: The fundamental problem in the subject of Diophantine approximation is the question of how closely an irrational number can be approximated by a rational number. In this talk, I will discuss the results in diophantine approximation to algebraic numbers and Thue's method. I will then discuss their application to diophantine equations and generalizations.

April 24

• Speaker: Jeffrey Kornhauser

• Title: Burnside's Theorem

• Abstract: One of the major applications of representation theory is a proof that all finite groups of order divisible by at most two primes is solvable. William Burnside originally proved this in 1904 and it took nearly seventy years for a proof that avoids representation theory! In this talk, I will break down the essential ingredients needed in the theory, prove the dimension theorem, and use tools from number theory to ultimately deduce Burnside's celebrated result.

May 1

• Speaker: Josiah Sugarman

• Title: Explicit lower bound on the Spectral Gap of the Quaquaversal Operator

• Abstract: The spectral gap of the Quaquaversal Operator controls the rate of equdistribution for the Quaquaversal tiling. A certain tiling, due to Conway and Radin, that has the property that the orientations of its tiles equidistribute faster than what is possible for analogous tilings in two dimensions. I will outline a proof of an explicit lower bound for the spectral gap of this operator. This resolves a 2006 conjecture of Draco, Sadun and van Wieren and gives a nearly exact equidistribution rate for this operator. The strategy involves expressing the fourier coefficients of a certain family of cusp forms in terms of this operator, bounding them via the Ramanujan Conjectures in order to extract spectral information. This idea is essentially due to Lubotzky, Phillips, and Sarnak. 

May 7

• Speaker: Nathaniel Kingsbury

• Title: Three Examples of the Polynomial Method

• Abstract: The Polynomial Method is a powerful and relatively new tool in additive and geometric combinatorics for bounding the sizes of certain kinds of sets by realizing them as zero sets of polynomials. I will present three examples of the Polynomial Method applied over finite fields: a polynomial proof of the Erdős-Heilbronn conjecture, Dvir’s proof of the finite field Kakeya conjecture, and a power-saving upper bound on the size of a cap set.