This talk is intended as a very gentle introduction to the classical subject of enumerative geometry concerned with problems of counting algebraic curves with prescribed properties.

Derived deformation theory studies the formal neighborhoods of moduli spaces. The classical work of Kodaira-Spencer on deformation of complex manifolds tells that the dg Lie algebra A^(0,*)(X,TX) controls the deformations of the complex manifold. The example illustrates the following general principle: A dg Lie algebra gives rise to a deformation functor by considering the  Maurer-Cartan elements of the dg Lie algebra tensoring with the maximal ideal of an Artin local ring. This idea is explored by many people, including  Quillen, Deligne, Drinfeld,  Kapranov and Kontesevich. Later, Lurie and Pridham independently proved that the category of dg Lie algebra is indeed equivalent to the category of Formal Moduli problem. I will state their results and explain the idea of the proof.

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).

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.

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.

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.

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. 

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.