Tue, 03 Dec 2019

15:45 - 16:45
L4

Combinatorial Lefschetz theorems beyond positivity

Karim Adiprasito
(Hebrew University)
Abstract

The hard Lefschetz theorem is a fundamental statement about the symmetry of the cohomology of algebraic varieties. In nearly all cases that we systematically understand it, it comes with a geometric meaning, often in form of Hodge structures and signature data for the Hodge-Riemann bilinear form.

Nevertheless, similar to the role the standard conjectures play in number theory, several intriguing combinatorial problems can be reduced to hard Lefschetz properties, though in extreme cases without much geometric meaning, lacking any existence of, for instance,  an ample cone to do Hodge theory with.

I will present a way to prove the hard Lefschetz theorem in such a situation, by introducing biased pairing and perturbation theory for intersection rings. The price we pay is that the underlying variety, in a precise sense, has itself to be sufficiently generic. For instance, we shall see that any quasismooth, but perhaps nonprojective toric variety can be "perturbed" to a toric variety with the same equivariant cohomology, and that has the hard Lefschetz property.

Finally, I will discuss how this applies to prove some interesting theorems in geometry, topology and combinatorics. In particular, we shall see a generalization of a classical result due to Descartes and Euler: We prove that if a simplicial complex embeds into euclidean 2d-space, the number of d-simplices in it can exceed the number of (d-1)-simplices by a factor of at most d+2.

Thu, 24 Jan 2019
11:00
L6

Kim-independence in NSOP1 theories

Itay Kaplan
(Hebrew University)
Abstract

NSOP1 is a class of first order theories containing simple theories, which contains many natural examples that somehow slip-out of the simple context.

As in simple theories, NSOP1 theories admit a natural notion of independence dubbed Kim-independence, which generalizes non-forking in simple theories and satisfies many of its properties.

In this talk I will explain all these notions, and in particular talk about recent progress (joint with Nick Ramsey) in the study of Kim-independence, showing transitivity and several consequences.

 

Mon, 02 May 2005
17:00
L1

On a class of quasilinear parabolic equations

Matania Ben-Artzi
(Hebrew University)
Abstract

An important class of nonlinear parabolic equations is the class of quasi-linear equations, i.e., equations with a leading second-order (in space) linear part (e.g., the Laplacian) and a nonlinear part which depends on the first-order spatial derivatives of the unknown function. This class contains the Navier-Stokes system of fluid dynamics, as well as "viscous" versions (or "regularized") of the Hamilton-Jacobi equation, nonlinear hyperbolic conservation laws and more. The talk will present various recent results concerning existence/uniqueness (and nonexistence/nonuniqueness) of global solutions. In addition, a new class of "Bernstein-type" estimates of derivatives will be presented. These estimates are independent of the viscosity parameter and thus lead to results concerning the "zero-viscosity" limit.

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