Forthcoming events in this series
I will discuss a basic problem in analytic number theory which has appeared recently in my work. This will be a gentle introduction to the Gauss circle problem, hopefully with a discussion of some extensions and applications to understanding L-functions.
I will introduce the Hodge-de-Rham spectral sequence and formulate an algebraic Hodge decomposition theorem. Time permitting, I will sketch Deligne and Illusie’s proof of the Hodge decomposition using positive characteristic methods.
I will introduce various heights on number fields and outline how solving polynomial equations with heights conditions is related to Arakelov geometry and a continuous logic theory called GVF.
In 2019 Masser and Zannier proved that "most" abelian varieties over the algebraic numbers are not isogenous to the jacobian of any curve (where "most" refers to an ordering by some suitable height function). We will see how this result fits in the general Zilber-Pink Conjecture picture and we discuss some (rather concrete) analogous problems in a power of the modular curve Y(1).
The twin prime conjecture asserts that there are infinitely many primes p for which p+2 is also prime. This conjecture appears far out of reach of current mathematical techniques. However, in 2013 Zhang achieved a breakthrough, showing that there exists some positive integer h for which p and p+h are both prime infinitely often. Equidistribution estimates for primes in arithmetic progressions to smooth moduli were a key ingredient of his work. In this talk, I will sketch what role these estimates play in proofs of bounded gaps between primes. I will also show how a refinement of the q-van der Corput method can be used to improve on equidistribution estimates of the Polymath project for primes in APs to smooth moduli.
A central topic of study in analytic number theory is the behaviour of the Riemann zeta function. Many theorems and conjectures in this area are closely connected to concepts from probability theory. In this talk, we will discuss several results on the typical size of the zeta function on the critical line, over different scales. Along the way, we will see the role that is played by some probabilistic phenomena, such as the central limit theorem and multiplicative chaos.
It is known that there exists certain randomness in the values of the Moebius function. It is widely believed that this randomness predicts significant cancellations in the summation of the Moebius function times any 'reasonable' sequence. This rather vague principle is known as an instance of the 'Moebius randomness principle'. Sarnak made this principle precise by identifying the notion 'reasonable' as deterministic. More precisely, Sarnak's Moebius Disjointness Conjecture predicts the disjointness of the Moebius function from any arithmetic functions realized in any topological dynamical systems of zero topological entropy. In this talk, I will firstly introduce some background and progress on this conjecture. Secondly, I will talk about some of my work on this. Thirdly, I will talk some related problems to this conjecture.
We will first introduce the modular method for solving Diophantine Equations, famously used to
prove the Fermat Last Theorem. Then, we will see how to generalize it for a totally real number field $K$ and
a Fermat-type equation $Aa^p+Bb^q=Cc^r$ over $K$. We call the triple of exponents $(p,q,r)$ the
signature of the equation. We will see various results concerning the solutions to the Fermat equation with
signatures $(r,r,p)$ (fixed $r$). This will involve image of inertia comparison and the study of certain
$S$-unit equations over $K$. If time permits, we will discuss briefly how to attack the very similar family
of signatures $(p,p,2)$ and $(p,p,3)$.
Let $p$ be a prime, let $1 \le t < d < p$ be integers, and let $S$ be a non-empty subset of $\mathbb{F}_p$ (which may be thought of as being $\{0,1\}$). We will establish that if a polynomial $P:\mathbb{F}_p^n \to \mathbb{F}_p$ with degree $d$ is such that the image $P(S^n)$ does not contain the full image $A(\mathbb{F}_p)$ of any non-constant polynomial $A: \mathbb{F}_p \to \mathbb{F}_p$ with degree at most $t$, then $P$ coincides on $S^n$ with a polynomial $Q$ that in particular has bounded degree-$\lfloor d/(t+1) \rfloor$-rank in the sense of Green and Tao, and has degree at most $d$. Likewise, we will prove that if the assumption holds even for $t=d$ then $P$ coincides on $S^n$ with a polynomial determined by a bounded number of coordinates and with degree at most $d$.
In 1993, Erdős, Sárközy and Sós posed the question of whether there exists a set $S$ of positive integers that is both a Sidon set and an asymptotic basis of order $3$. This means that the sums of two elements of $S$ are all distinct, while the sums of three elements of $S$ cover all sufficiently large integers. In this talk, I will present a construction of such a set, building on ideas of Ruzsa and Cilleruelo. The proof uses a powerful number-theoretic result of Sawin, which is established using cutting-edge algebraic geometry techniques.
Serre's conjecture (now a theorem) predicts that an irreducible 2-dimensional odd
Galois representation of $\mathbb Q$ with coefficients in $\bar{\mathbb F}_p$ comes from the mod p reduction of
a modular form. A key feature is that two modular forms of different weights can have the same
mod p reduction. Fixing a modular form $f$, the weight part of Serre's conjecture seeks to find all
the possible weights where one can find a modular form congruent to $f$ mod $p$. The recipe for these
weights was conjectured by Serre, and it depends only on the local Galois representation at $p$. I
will explain the ideas involved in Edixhoven's proof of the weight part, and if time allows, I
will briefly say something about what the generalizations beyond $\operatorname{GL}_2/\mathbb Q$ might look like.
The notion of a pseudocharacter was introduced by A.Wiles for GL_2 and generalized by R.Taylor to GL_n. It is a tool that allows us to deal with the
deformation theory of a residually reducible Galois representation when the usual techniques fail. G.Chenevier gave an alternative theory of "determinants" extending that of pseudocharacters to arbitrary rings. In this talk we will discuss some aspects of this theory and introduce a similar definition in the case of the symplectic group, which is the subject of a forthcoming work joint with J.Quast.
In this talk we will study the graph structure of supersingular isogeny graphs. These graphs are known to have very few loops and multi-edges. We formalize this idea by studying and finding bounds for their number of loops and multi-edges. We also find conditions under which these graphs are simple. To do so, we introduce a method of counting the total number of collisions (which are special endomorphisms) based on a trace formula of Gross and a known formula of Kronecker, Gierster and Hurwitz.
The method presented in this talk can be used to study many kinds of collisions in supersingular isogeny graphs. As an application, we will see how this method was used to estimate a certain number of collisions and then show that isogeny graphs do not satisfy a certain cryptographic property that was falsely believed (and proven!) to hold.
In 2018, Keating, Rodgers, Roditty-Gershon and Rudnick conjectured that the variance of sums of the divisor
function in short intervals is described by a certain piecewise polynomial coming from a unitary matrix integral. That is
to say, this conjecture ties a straightforward arithmetic problem to random matrix theory. They supported their
conjecture by analogous results in the setting of polynomials over a finite field rather than in the integer setting. In this
talk, we'll discuss arithmetic problems over F_q[T] and their connections to matrix integrals, focusing on variations on
the divisor function problem with symplectic and orthogonal distributions. Joint work with Matilde Lalín.
The Dedekind zeta function generalises the Riemann zeta
function to other number fields than the rationals. The analytic class number
formula says that the leading term of the Dedekind zeta function is a
product of invariants of the number field. I will say some things
about the class number formula, about L-functions, and about Stark's
conjecture which generalises the class number formula.
A pair of elliptic curves is said to be $N$-congruent if their mod $N$ Galois representations are isomorphic. We will discuss a construction of the moduli spaces of $N$-congruent elliptic curves, due to Kani--Schanz, and describe how this can be exploited to compute explicit equations. Finally we will outline a proof that there exist infinitely many pairs of elliptic curves with isomorphic mod $12$ Galois representations, building on previous work of Chen and Fisher (in the case where the underlying isomorphism of torsion subgroups respects the Weil pairing).
Isogeny-based cryptography is a candidate for post-quantum cryptography. The underlying hardness of isogeny-based protocols is the problem of computing endomorphism rings of supersingular elliptic curves, which is equivalent to the path-finding problem on the supersingular isogeny graph. Can path-finding be reduced to knowing just one endomorphism? An endomorphism gives an explicit orientation of a supersingular elliptic curve. In this talk, we use the volcano structure of the oriented supersingular isogeny graph to take ascending/descending/horizontal steps on the graph and deduce path-finding algorithms to an initial curve. This is joint work with Sarah Arpin, Kristin E. Lauter, Renate Scheidler, Katherine E. Stange and Ha T. N. Tran.
The Riemann hypothesis (RH) is one of the great open problems in
mathematics. It arose from the study of prime numbers in an analytic
context, and—as often occurs in mathematics—developed analogies in an
algebraic setting, leading to the influential Weil conjectures. RH for
curves over finite fields was proven in the 1940’s by Weil using
algebraic-geometric methods, and later reproven by Stepanov and
Bombieri by elementary means. In this talk, we use RH for curves to
prove the Weil bound for certain (Kloosterman) exponential sums, which
in turn is a fundamental tool in the study of prime numbers.
The Chabauty-Kim method is an effective algorithm for finding the $S$-integral points of hyperbolic curves by directly using the hyperbolicity in group-cohomological arguments. Central objects in the theory are affine spaces known as a Selmer schemes. We'll introduce the CK method and Selmer schemes, and demonstrate some additional structures possessed by Selmer schemes which can aid in implementing the CK method.
The Balog-Szemerédi-Gowers theorem is a powerful tool in additive combinatorics, that allows one to roughly convert any “large energy” estimate into a “small sumset” estimate. This has found applications in a lot of results in additive combinatorics and other areas. In this talk, we will provide a friendly introduction and overview of this result, and then discuss some proof ideas. No hardcore additive combinatorics pre-requisites will be assumed.
Modular forms are holomorphic functions on the upper half plane satisfying a transformation property under the action of Mobius transformations. While they are a priori complex-analytic objects, they have applications to number theory thanks to their connection with Galois representations. Weight one modular forms are special because their Galois representations factor through a finite quotient. In this talk, we will explain a different degeneracy: they contribute to the cohomology of a line bundle over the modular curve in degrees 0 and 1. We propose an arithmetic explanation for this: an action of a unit group associated to the Galois representation of the modular form. This extends the conjectures of Venkatesh, Prasanna, and Harris. Time permitting, we will discuss a generalization to Hilbert modular forms.
I'll introduce the classical arithmetic topology dictionary of Mumford-Manin-Mazur-Morishita-etc. I'll then present an interesting instance of parallel phenomena related to symplectic structures on moduli spaces of certain bundles. The arithmetic side turns out to be an application of Poitou-Tate duality. Depending on time, I'll delve into the delicate details which make the analogy useful for Diophantine geometers.
In this talk, I would like to discuss Deligne’s version of Geometric Class Field theory, with special emphasis on the correspondence between rigidified 1-dimensional l-adic local systems on a curve and 1-dimensional l-adic local systems on Pic with certain compatibilities. We should like to give a sense of how this relates to the OG class field theory, and how Deligne demonstrates this correspondence via the geometry of the Abel-Jacobi Map. If time permits, we would also like to discuss the correspondence between continuous 1-dimensional l-adic representations of the etale fundamental group of a curve and local systems.
Many classical arithmetic problems ranging from the elementary ones such as the density of square-free numbers to more difficult such as the density of primes, can be extended to integer matrices. Arithmetic problems over higher dimensions are typically much more difficult. Indeed, the Bateman-Horn conjecture predicting the density of numbers giving prime values of multivariate polynomials is very much open. In this talk I give an overview of the unfortunately brief history of integer random matrices.
The distribution of primes in arithmetic progressions (AP) s a central question of analytic number theory. It is closely connected to the additive behaviour of primes (for example in the Goldbach problem) and application of sieves (for example in the Twin Prime problem). In this talk I will outline the basic results without going into technical details. The central questions I will consider are: What are the different tools used to study primes in AP? In what ranges of moduli are they useful? What error terms can be achieved? How do recent developments fit into the bigger picture?
The moments of Hardy’s function have been of interest to number theorists since the early 20th century, and to random matrix theorists especially since the seminal work of Keating and Snaith, who were able to conjecture the leading order behaviour of all moments. Studying joint moments offers a unified approach to both moments and derivative moments. In his 2006 thesis, Hughes made a version of the Keating-Snaith conjecture for joint moments of Hardy’s function. Since then, people have been calculating the joint moments on the random matrix side. I will outline some recent progress in these calculations. This is joint work with Theo Assiotis, Benjamin Bedert, and Mustafa Alper Gunes.
In this talk, we present our study of Brown’s definition of the probabilistic zeta function of a finite lattice, and propose a natural alternative that may be better-suited for non-atomistic lattices. The probabilistic zeta function admits a general Dirichlet series expression, which need not be ordinary. We investigate properties of the function and compute it on several examples of finite lattices, establishing connections with well-known identities. Furthermore, we investigate when the series is an ordinary Dirichlet series. Since this is the case for coset lattices, we call such lattices coset-like. In this regard, we focus on partition lattices and d-divisible partition lattices and show that they typically fail to be coset-like. We do this by using the prime number theorem, establishing a connection with number theory.
We'll sketch how the $K$-rational solutions of a system $X$ of multivariate polynomials can be viewed as the solutions of a "classical mechanics" problem on an associated affine space.
When $X$ has a suitable topology, e.g. if its $\mathbb{C}$-solutions form a Riemann surface of genus $>1$, we'll observe some of the advantages of this new point of view such as a relatively computable algorithm for effective finiteness (with some stipulations). This is joint work with Minhyong Kim.
It is a widely accepted philosophy in additive number theory that convex sets ought not to exhibit much additive structure. We could measure this by estimating the sizes of their sumsets. In this talk, we will hopefully move from the philosophical to the concrete, by giving ways to see that convex sets and functions have poor additive structure. We will also discuss some recent developments in the area.
The speakers will be giving short presentations introducing topics in algebraic number theory, arithmetic topology, random matrix theory, and analytic number theory.
Undergrads and Master's students are encouraged to come and sample a taste of research in these areas.
In a world of polynomial Pell’s equations, where the integers are replaced by polynomials with complex coefficients, and its smallest solution is used to generate all other solutions $(u_{n},v_{n})$, $n\in\mathbb{Z}$. One junior number theory group will embark on a journey in search of the properties of the factors of $v_{n}(t)$. There will be Galois extensions, there will be estimations and of course there will be loglogs.
The Jacquet-Langlands correspondence gives a relationship between automorphic representations on $GL_2$ and its twisted forms, which are the unit groups of quaternion algebras. Writing this out in more classical language gives a combinatorial way of producing the eigenvalues of Hecke operators acting on modular forms. In this talk, we will first go over notions of modular forms and quaternion algebras, and then dive into an explicit example by computing some eigenvalues of the lowest level quaternionic modular form of weight $2$ over $\mathbb{Q}$.
A nice collection of problems in additive combinatorics focus on analysing solutions to additive equations over sequences that exhibit some flavour of convexity. This, for instance, includes genuine convex sequences as well as images of arbitrary sets under convex functions. In this talk, I will survey some of the literature surrounding these type of questions, along with some motivation from analytic number theory as well as the current best known results towards these problems.
We can define a continued fraction for formal series $f(t)=\sum_{i=-\infty}^d c_it^i$ by repeatedly removing the polynomial part, $\sum_{i=0}^d c_it^i$, (the equivalent of the integer part) and inverting the remaining part, as in the real case. This way, the partial quotients are polynomials. Both the usual continued fractions and the polynomial continued fractions carry properties of best approximation. However, while for square roots of rationals the real continued fraction is eventually periodic, such periodicity does not always occur for $\sqrt{D(t)}$. The correct analogy was found by Abel in 1826: the continued fraction of $\sqrt{D(t)}$ is eventually periodic if and only if there exist nontrivial polynomials $x(t)$, $y(t)$ such that $x(t)^2-D(t)y(t)^2=1$ (the polynomial Pell equation). Notice that the same holds also for square root of integers in the real case. In 2014 Zannier found that some periodicity survives for all the $\sqrt{D(t)}$: the degrees of their partial quotients are eventually periodic. His proof is strongly geometric and it is based on the study of the Jacobian of the curve $u^2=D(t)$. We give a brief survey of the theory of polynomial continued fractions, Jacobians and an account of the proof of the result of Zannier.