tbranzov

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.