Talk

Acquiring geometric information about algebraic varieties via counting $F_q$-points and employing various $p$-adic techniques
has always been a challenge, classically and neoclassically (..., perfectoid spaces). Surprisingly, this can be done for
$S^3 \setminus K$ for certain(!) hyperbolic knots, including the famous K12n242, which approach requires
instanton slices. These are Quot-stacks of torsion-free modules over isolated singularities with prescribed conductors,
in general.

Together with Galvez-Carilllo and Tonks we established that every simplicial set yields a bi-algebra, whose cubical structure can be understood via the category of simplicial strings, aka. necklaces, using Joyal duality. The reason for the existence is the fact that simplices form an operad and after dualization give a co-operad with multiplication which is the correct input for the GCKT machine.

Let D ≡ 1 mod 4 be a fundamental discriminant of a real quadratic field. We will construct an analogue of the classical Dedekind eta function for Hecke groups, giving rise to a family of holomorphic modular functions that vanish at the cusp at infinity. If time permits, we will discuss the asymptotic growth and sign patterns of the Fourier coefficients of these modular functions.
 

Commutative algebras can be viewed as certain function algebras on geometric spaces. This perspective motivates the philosophy of noncommutative geometry, which studies noncommutative algebras as if they are functions on noncommutative spaces.

Smooth numbers, integers whose prime factors are all small, play a central role in many areas of analytic and computational number theory. The Gauss circle problem, on the other hand, concerns counting lattice points inside a large circle and understanding the error term in this count. In this talk, we will give an introduction to both of these topics, describe a new problem arising from their connection, and discuss some of the results towards these problems. The last part of this talk will be ongoing joint work with Stelios Sachpazis.

We discuss a classical problem in global Riemannian geometry:
Loosely speaking, it asks: "how round can one make a given manifold"?

More concretely: if one avoids the obvious scaling trick: given a metric g with non-negative scalar curvature on a smooth manifold M,
can one find g' such that distances are not decreased when measured with g'
instaed of g, but such that the scalar curvature increases?

A classical result by Llarull states that this is not the case if g is the round metric on the sphere S^n (n>1);

Class field theory describes the structure of all abelian field extensions of a number field, and promises the existence of the so-called Hilbert class field.
However, its explicit construction has been a long standing open problem, and doing this with the special values of arithmetic functions is known as Hilbert's 12th problem.
We discuss some cases in which this problem has been solved, including imaginary quadratic fields through CM theory and singular moduli.