Talk

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.
We aim to understand and to construct strict higher categorical versions of this. The first ingredients are Street’s orientals and a monoidal product explained by Ozornova and Rovelli for higher categories defined by polytopes with an initial and a final vertex. Note that even the classical path space argument is actually a transition from a 2 to a 1 category. Following the Kapranov-Voevodsky dream the next steps are cubes and permutahedra. For the latter there is is the Manin-Schechtman theory of higher Bruhat orders and the relatively new operad with multiplication structure on them defined by Koshevoy and Schechtman. This project is joint with Ozornova, Koshevoy and Schechtman.



 

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.

In the 1980s, Jean Bellissard developed a surprisingly elegant and powerful framework, allowing for modelling quantum solid-state systems as noncommutative spaces. Their algebraic topology --- that is, topological K-theory --- encodes rich information that can be exactly measured in a physical experiment.

Over the past decade, new ideas and methods have emerged in this area, surrounding the following two questions:
1. How to effectively model a solid-state system, in a way that captures only the physically relevant topological information?
2. How to efficiently compute the numerical invariants of a model system, particularly when only partial information is available?

The goal of my talk is to provide an overview of the K-theoretic approach to solid-state physics, and discuss some ideas and results related to these questions.
 

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);
Goette and Semmelmann generalize this to further classes of manifolds, in particular symmetric spaces of compact type with
non-vanishing Euler characteristic.

We discuss the possible approaches to study this question and a number of important generalizations/variations:

a) the condition is purely metric: therefore also the conclusion should hold under low assumptions on the regularity (this is joint work with Simone Cecchini, Bernhard Hanke, Lukas Schoenlinner
b) certain instances where the manifold has Euler characteristic zero (but is not a sphere): this is joint work with Georg Frenck, Lukas Schoenlinner, Thomas Tony.

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.
Finally, we introduce novel p-adic constructions and outline a method to compute these special values using the p-adic infinitesimal deformation theory of Eisenstein series and theta series.
 

We prove that if X, Y are derived equivalent complex K3 surfaces, then the integral Chow motives are typically not isomorphic, but are always stably isomorphic. This unfies the result of Huybrechts for isomorphism of rational Chow motives with the result of Mukai for the stable equivalence of integral Hodge structures. Under some mild assumptions, the same result holds for K3 surfaces over nonclosed fields of characteristic zero. For the proofs we rely on known results about zero-cycles on K3 surfaces and set up the machinery of Tate-stable equivalence for integral Chow motives. This is a joint work in progress with Hsueh-Yung Lin and Pavel Sechin.

Let s(n) denote the sum of proper divisors of an integer n. In honour of the Number Theory Boat Seminar, we coin the term "unsinkable numbers" for integers n whose sequence of iterates under s never eventually "sinks" to 1. We will give an overview of what is known about unsinkable numbers as well as some open conjectures.

We will then turn to a different aspect of the sum-of-proper-divisors function: the distribution of the digits of s(n). We will see that s(n) obeys Benford's law and that, asymptotically, almost every value of s(n) contains every decimal digit among both its leading and trailing k(x) digits, where k(x)→∞. We will also discuss recent work showing that values of s(n) missing a decimal digit are surprisingly rare, with prime inputs providing the main contribution, confirming a special case of a conjecture of Erdős, Granville, Pomerance, and Spiro.

This talk is based on joint work with various subsets of the following co-authors: Kübra Benli, Giulia Cesana, Cécile Dartyge, Charlotte Dombrowsky, and Paul Pollack.