March 28, Wednesday, 15:30, room 211

shinymath, 27.03.2018 · Categories: Без рубрики

Artem Galkin

Mahler conjecture and Blaschke-Santalo inequality

Mahler volume of a centrally symmetric convex body is a dimensional quantity that is associated with the body and is invariant under linear transformations. It is known that the shapes with the largest possible Mahler volume are the balls and solid ellipsoids; this is now known as the Blaschke–Santaló inequality. The still-unsolved Mahler conjecture states that the minimum possible Mahler volume is attained by a hypercube. I will make a brief overview of the known results obtainted towards the complete solution of Mahler conjecture and provide the proof of the Blaschke-Santalo inquality for symmetric convex bodies. If time permits, we will cover some stability estimates and discuss the relationship between Mahler conjecture and Faber-Krahn inequality for the Cheeger constant.

March 21, Wednesday, 15:30, room 211

shinymath, 20.03.2018 · Categories: Без рубрики

Evgeny Goncharov

Kouchnirenko’s theorem and its proof via Hilbert’s theorem

Kouchnirenko’s theorem states that the number of roots in the complex torus (С-0)^n of a generic system of equations P_1= …=P_n=0, where P_i’s are Laurent polynomials with a common Newton polytope \Delta is equal to n!Vol(\Delta). Askold Khovanskii has found around 15 different proofs of this theorem and we are going to review the one based on Hilbert’s Theorem. I also aim to provide some motivation and examples. A few general theorems of complex algebraic geometry will be used. If time permits I will also say something about a more elementary variation of the proof.

March 14, Wednesday, 15:30, room 211

shinymath, 12.03.2018 · Categories: Без рубрики

Valeriya Ignatovskaya

Three-dimensional Generalizations of Pick’s Formula and the Khovanskii-Pukhlikov theorem

It is known that classical Pick’s formula can not be transferred to higher dimensions. In the first part of my talk, I will show another approach to express the area of convex integer polygons. This way can be easily generalized to higher dimensions so the volume of 3-dimensional polytopes can be found. After that, I will introduce the Todd operator and prove the Khovanskii-Pukhlikov theorem about relations between the continuous and discrete volume of an integral polytope.
No prerequisites required.

March 7, Wednesday, 15:30, room 211

shinymath, 06.03.2018 · Categories: Без рубрики

Dmitry Korshunov

Line bundles, the degree of a projective variety, Hilbert polynomial

We will survey basic definitions and properties of line bundles on projective varieties, as well as their relations to embeddings into projective spaces. Our aim will be to find an intrinsic definition of degree of a projective variety. The computable way to do it is via Hilbert polynomial. Consider the generating function counting dimensions of homogeneous components of the coordinate ring of a variety. It turns out that for large degrees this function coincides with some well defined polynomial over rationals, containing a lot of intrinsic information about the variety. In particular the dimension and degree can be read off from the leading monomial.