Geometry Seminar

This weekly seminar aims to explore various types of modern geometries and the intricate relations between them, Mirror Symmetry being a prototypical example of such, from overviews of classical results to state-of-the-art breakthroughs, with an emphasis on the open problems and current gaps of knowledge. The scope for the first few months includes topics in algebraic, symplectic, tropical, non-Archimedean, and derived geometries. Our seminar operates with a flexible and dynamic structure, without a strict long-term program, where the subject of the oncoming session is decided on the spot by participants. A typical length of a presentation is expected to be from one to two hours, with the possibility of an extension for a free discussion.

Talks list

Topics in non-archimedean analytic geometry (II), by Jiachang Xu

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. 

Categories: Geometry Seminar|

Topics in non-archimedean analytic geometry (I), by Jiachang Xu

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. 

Categories: Geometry Seminar|

Tropicalizations II, a talk by M. Shkolnikov

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.

Categories: Geometry Seminar|

Tropicalizations, a talk by Mikhail Shkolnikov

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.

Categories: Geometry Seminar|

Tropical structures in sandpile model, talk by Mikhail Shkolnikov, IMI-BAS

I will tell how tropical curves arise in the scaling limit of the sandpile model in the vicinity of the maximal stable state and explain two major consequences inspired by this fact. The first one is that there is a continuous model for self-organized criticality, the only known model of a kind, defined in the realm of tropical geometry. The second is that the totality of recurrent states in the original sandpile model, the sandpile group, approximates a continuous group, a tropical Abelian variety, which is functorial with respect to inclusions of domains, allowing to compute its scaling limit as a space of circle-valued harmonic functions on the whole lattice.

Categories: Geometry Seminar, ICMS Seminar, News|
Go to Top