No Convex and Algebraic Geometry seminar on March 27

There will be no seminar on Wednesday, March 27, due to the spring examination period.

Wednesday, March 20, 18:30, room 306

Evgeny Goncharov

Symplectic resolutions of certain hypersurfaces in A^5

The hypersurfaces we are going to consider are obtained by the general procedure of constructing Coulomb branches (an object of interest to physicists that was rigorously defined mathematically in a series of papers by A. Braverman, M. Finkelberg and H. Nakajima). The general procedure is quite involved but for the case we will consider (quiver with one vertex and l loops with dim V = 2, dim W = 1) there is a theorem that describes the generators (and gives a way to find relations between them) of the coordinate ring of the Coulomb branch (it is an algebraic variety). We will apply the theorem to find the Coulomb branch which turns out to be a hypersurface in A^5.
Then I will explain how to (naturally) define a symplectic form on the smooth locus of this hypersurface. For l=1 the Coulomb branch turns out to be Sym^2 A^2 and its symplectic resolution (resolution of singularities such that the pullback of the symplectic form on the smooth locus extends to define a symplectic form on the resolution) has been known for a long time (it is given by the Hilbert scheme Hilb^2 A^2). I will show that a naive attempt to construct a symplectic resolution for a general l does not work and discuss the general properties of our hypersurface (compute the Hilbert polynomial, etc).
We will then note that a certain construction in a Lie algebra sp_{2l} (Slodowy slice to the subsubregular orbit of the nilpotent of type (2l-2, 1, 1)) has precisely the same properties and compute the Slodowy slice explicitly to see that the resulting variety is the same. The symplectic resolution of the Slodowy slice, however, is easy to describe. This argument gives a new family of symplectic hypersurfaces (apart from the known Kleinian singularities in A^3). Time permitting we will discuss what happens if one varies the dimensions of the vector spaces V and W in the quiver we started with.

Wednesday, March 13, 17:30, room 110

This week, instead of a regular talk, we invite you to the Mathematical physics seminar (note the change of time and place):

17:30, room 110

Xin Fang (University of Cologne)

Toric degenerations of flag varieties and their applications

Toric degenerations translate geometric properties of varieties to combinatorial ones of polytopes or polyhedral cones. In this talk I will introduce a general framework to construct toric degenerations of flag varieties via birational sequences and Newton-Okounkov bodies. Most of the known toric degenerations turn out to be concrete examples of this construction. As an application, I will explain how to get the exact value of the Gromov width of a coadjoint orbit.

March 6, Wednesday, 18:30, room 306

Fedor Selyanin

LATTICE VERTEX POLYTOPES WITH INTERIOR LATTICE POINTS

Consider a convex polytope with lattice vertices and at least one interior lattice point. We will prove that the number of boundary lattice points is bounded from above by a function of the dimension and the number of interior lattice points. We will find the boundaries for zero symmetric convex polytopes, then for arbitrary lattice polytopes and compare them.
This is the starting point to many results on classification of small lattice polytopes that will be presented in subsequent talks.