I will talk about valuations not necessarily of rank 1 and give the algebraic basics of valuation theory. Such valuations can present some supplementary pathological behaviour compared to the rank 1 situation, however they can still be seen as generalizations of multiplicities or orders of vanishing. Indeed one can embed any valuation into a spherically complete ring which will be a Hahn series ring.
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 is the second talk about tropical modifications. During the first I explained the motivation for using tropical modifications while studying tropical curves and maps between them. In the second talk I will define modifications of the tropical plane and explain how tropical planar curves are changed. Also I show how this can be used in studying inflection points (following the work of E. Brugalle and L. Lopez de Medrano).
We will define tropical modifications for the case of tropical curves and tropical hypersurfaces. Also we consider applications of tropical modifications.
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.
In the last few decades, Berkovich’s theory of k -analytic space has extended classic rigid geometry. Since k -analytic space has good topological properties, and the Berkovich analytification of algebraic varieties could be also dealt with via the geometry of the model and it has a strong connection with tropical geometry, this allows us to use the combinatorial techniques to study algebraic varieties. Our lectures will mainly discuss the contents of Berkovich space and its application in other aspects of mathematics, we plan to go over the basic properties of Berkovich space in both algebraic and topological ways for the first lecture.
I will describe a new geometric method for constructing and controlling shifted symplectic structures on the moduli of vector bundles along the fibers of a degenerating family of Calabi-Yau varieties. The method utilizes bubbling modifications of the boundaries of limiting moduli spaces to extend the symplectic structure on the general fiber to a relative symplectic structure defined on the whole family. As a proof of concept we show that this produces a universal relative symplectic structure on the moduli of Gieseker Higgs bundles along a semistable degeneration of curves. We also check that the construction works globally over the moduli stack of stable curves and show that the Hitchin map has the expected behavior in the limit. This is a joint work with Oren Ben-Bassat and Sourav Das.
In the last few decades, Berkovich’s theory of k -analytic space has extended classic rigid geometry. Since k -analytic space has good topological properties, and the Berkovich analytification of algebraic varieties could be also dealt with via the geometry of the model and it has a strong connection with tropical geometry, this allows us to use the combinatorial techniques to study algebraic varieties. Our lectures will mainly discuss the contents of Berkovich space and its application in other aspects of mathematics, we plan to go over the basic properties of Berkovich space in both algebraic and topological ways for the first lecture.
I will start by briefly recalling the two approaches to tropicalizing subvarieties of algebraic tori. We then transition to the compactified version, substituting the torus with a toric variety, and formalizing the concept of embedded tropical varieties together with a notion of their smoothness. Notably, not all such varieties can be realized as tropicalizations of classical varieties, and different embeddings of classical varieties yield combinatorially distinct tropicalizations, which we equate through so-called modifications. We will conclude by addressing the question of recovering information about classical varieties from their tropicalizations, exemplified by the phase tropical limit having the same topology as corresponding complex hypersurfaces provided that the usual tropicalization is smooth.
Tropical geometry is perhaps the most elementary of the geometries we are interested in. A tropicalization is often described as a combinatorial shadow of an algebraic variety. I will explain two approaches to this tropicalization procedure, in the context of complex and non-Archimedean varieties, and how they are related. The basics of tropical geometry and (co)amoebas will be covered as well.
International Center for Mathematical Sciences
Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
Acad. G. Bonchev St. bl.8
1113 Sofia, Bulgaria
Tel. +359 2 9793822
icms-office@math.bas.bg
Copyright 2020 ICMS | All Rights Reserved
