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A seminar talk by Morgan Brown, University of Miami.

I will present a form of quantum circuit complexity that extends to open systems. To illustrate the methodology, I  focus on a basic model where the Hilbert space of states is represented by the Euclidean plane. Specifically, the  investigation is about the dynamics of mixed quantum states as they undergo interactions with a sequence of gates.

The mirror symmetry phenomenon was discovered by string theorists more than thirty years ago as an equivalence of two physical theories associated with very different geometries. The categorical point of view on this remarkable conjecture was famously introduced by M. Kontsevich in his 1994 ICM talk. As it became clear over time, homological mirror symmetry provides new approaches to many topics in symplectic and algebraic geometry. In these two talks, I will present a brief overview of the conjecture as well as some results inspired by mirror symmetry and obtained in joint work with L. Katzarkov about classical problems on discriminants and singularity theory.

A talk by Carlos Varea, (Universidade Tecnológica Federal do Paraná - Campus Cornélio Procópio, Brazil) on January 16, 14:00 Sofia time.

We study various correspondences between finite-dimensional nilpotent algebras and (quasi)groups similar to those given by the circle product in the case of associative algebras or by the Baker-Campbell-Hausdorff formulas in the case of Lie algebras or their generalizations. In the particular case of Malcev's correspondence, we obtain some new results about groups using Lie algebras and vice versa.

In this talk I will derive an algebraic formula for the Milnor number of a smoothable complex analytic curve singularity X by relating it to the Euler characteristic of its smoothing, which in turn I will relate to the multiplicity of the Jacobian ideal of X and and the multiplicity of X at its singular point. If time permits I will discuss generalizations to higher dimensions. This is a report on a joint work with Gaffney and Bengus-Lasnier.

In this talk I will use stratified Morse theory to relate the number of critical points of a generic linear functional on a complex analytic manifold M to the Euler characteristics of M and a generic hyperplane slice of M.

The Følner function of a group is defined on positive integers n as the smallest size of a Følner set, the boundary of which is at most 1/n times the size of the set. Its values are then finite if and only if the group is amenable. It can be thought of as encoding "how amenable a group is". We will give an overview of how our understanding of Følner functions has progressed. We will mostly talk about two major types of development. The first one concerns proving, for a given type of function, the existence of a group that has a Følner function of that type. The other one is connections between the asymptotics of Følner functions and those of the growth function.

This presentation introduces a novel holographic correspondence in d-dimensional de Sitter (dS_d) spacetime, connecting bulk dS_d scalar unitary irreducible representations (UIRs) with their counterparts at the dS_d boundary, all while preserving reflection positivity. The proposed approach, with potential applicability to diverse dS_d UIRs, is rooted in the geometry of the complex dS_d spacetime and leverages the inherent properties of the (global) dS_d plane waves, as defined within their designated tube domains.