ORGANIZER: Centre of Pure Mathematics (Russian Federation)
TIME: July 1, 2024 - July 5, 2024
LOCATION: Centre of Pure Mathematics, MIPT, Dolgoprudny, Moscow Region
SPEAKERS: Alexey Bondal, Chris Brav, Vladimir Zhgoon
REGISTRATION LINK: http://puremaths.org/CPM-Summer-school/
SCHEDULE:
Lecture 1
Classical algebraic geometry
Speaker: Vladimir Zhgoon
Abstract: Ample invertible sheaves play an important role in the study of the geometry of algebraic varieties. For example, Lefschetz's theorem about the cohomology of the zero locus of sections of these bundles (hyperplane sections) holds for them. Also in the case of line bundles, Kodaira's theorem gives a criterion for the ampleness of line bundles in terms of the positivity of curvature. In the case of vector bundles, the situation becomes much trickier. There are at least two concepts of positivity according to Griffiths and Nakano, and the definition of mobility itself is not as natural as in the case of line bundles. At the same time, there are many interesting geometric properties of such bundles, which we will discuss.
Lecture 2
Derived algebraic geometry
Speaker: Alexey Bondal
Abstract: Reflexive coherent sheaves are considered as a natural generalization of vector bundles. They have a particularly nice behavior on normal varieties. It will be explained how to characterize reflexive sheaves in categorical terms and which consequence this characterization imply. Also I will describe the reconstruction of normal varieties from their categories of reflexive sheaves. In the case of normal surface, this allows us to construct a saturated model of the surface. These models can be characterized in geometric terms via Nagata compactification and the boundary divisor. All required mathematical notions will be explained and relevant open problems will be formulated.
Lecture 3
Condenced algebraic geometry
Speaker: Chris Brav
Abstract: A convenient synthesis of analysis and homological algebra has long been needed for applications in analytic geometry, infinite dimensional algebraic geometry, representation theory of Lie groups, and other parts of mathematics in which various modules are endowed with some kind of ‘topology’. Recently, Clausen and Scholze have introduced a tensor abelian category of ‘condensed abelian groups’ that has excellent formal properties and that faithfully contains the category of discrete abelian groups as well as all classically occurring topological abelian groups. We give a short introduction to this theory and some its applications in geometry.
