December 13, 2014

UIUC-WUSTL Symplectic Geometry Seminar :: Fall 2014

Hosts: Rui L. Fernandes (UIUC) & Xiang Tang (WUSTL)

This is the second joint UIUC-WUSTL meeting. The first meeting was held at WUSTL on April 5, 2014.


SCHEDULE

10:00 - 10:50
Location: Altgeld Hall, Room 245

On the persistence of multiple closed Reeb orbits

Ely Kerman, University of Illinois at Urbana-Champaign

Abstract: In this talk I will describe a Floer theoretic proof that certain clusters of simple closed Reeb orbits must persist over surprisingly long distances as measured by a Hofer-type metric for contact forms. This allows one to reprove and generalize a classic result of Ekeland and Lasry concerning multiple closed characteristics on convex hypersurfaces pinched between spheres of radius 1 and $\sqrt{2}$.


Coffee Break

Location: Altgeld Hall, Room 321

 

11:10- 12:00
Location: Altgeld Hall, Room 245

Differential geometry of rigid body collisions

Renato Feres, Washington University in St. Louis

Abstract: Systems of rigid bodies with impact interactions are very classical examples of Hamiltonian mechanical systems on manifolds with boundary. Boundary conditions correspond to assignments of a (linear) map, at each boundary configuration, specifying the post-collision state of the system as a function of the pre-collision state. Our main result, which is joint work with Chris Cox and Will Ward, is a classification of boundary conditions compatible with the main physical conservation laws. We then briefly explore simple examples of generalized billiard dynamical systems suggested by this classification.

 

14:00 - 14:50
Location: Altgeld Hall, Room 245

The momentum map for Caine's toric Poisson variety

Songhao (Travis) Li, Washington University in St. Louis

Abstract: The Caine's toric Poisson variety is a real Poisson structure on a complex toric variety that is invariant under the complex torus action. We may blow it up along its C* fixed point loci to produce a toric log symplectic manifold with corners. Conversely, we define, in analogy with symplectic cuts, the operation of elliptic cuts; by cutting along the circle fixed point loci, we recover the Caine's toric Poisson variety from the toric log symplectic manifold. Moreover, both operations behave well with respect to the momentum map, which coincides with the tropicalization map as in tropical geometry.

 

15:00 - 15:50
Location: Altgeld Hall, Room 245

Symplectic cohomology of log Calabi-Yau manifolds

James Pascaleff, University of Illinois at Urbana-Champaign

Abstract: Log Calabi-Yau manifolds are obtained by taking the complement of an anticanonical divisor in a compact complex algebraic manifold. They have a rich symplectic geometry, for example, (in complex dimension 2), we have shown that their symplectic cohomology is highly nontrivial. I will discuss this result and its connection to mirror symmetry and canonical bases in representation theory (after Gross-Hacking-Keel).


Coffee Break

Location: Altgeld Hall, Room 321

 

16:00 - 16:50
Location: Altgeld Hall, Room 245

Taut submanifolds are algebraic

Quo-Shin Chi, Washington University in St. Louis

Abstract: Hamiltonian systems of hydrodynamic type is an important class of nonlinear PDEs associated with anisentropic gas dynamics. As Ferapontov pointed out, the solutions to such integrable nondiagonalizable systems have strong ties with compact Dupin hypersurfaces in Euclidean space. In fact, in the 3-by-3 case, integrability of the hydrodynamic type is equivalent to, up to Lie sphere transformations, solving three integrable equations on the Cartan hypersurface in R^4 (or equivalently in S^4). It has been an outstanding open problem, to understand its structure, as to whether a compact Dupin hypersurface is taut in the ambient Euclidean space, i.e., whether the Euclidean distance functions are perfect Morse-Bott (the converse is true). On the other hand, Kuiper raised the question in the 1980s as to whether a compact taut submanifold is algebraic, i.e., whether it is a connected component of a real irreducible variety in the same ambient Euclidean space. In this talk, I will sketch a proof of the affirmation of Kuiper's question, which appears to also point to a proof that a compact Dupin hypersurface is algebraic, which could be a first step toward settling the above open problem.

 

 


Contact Rui Loja Fernandes for more details.

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