Department of

Mathematics


Seminar Calendar
for events the day of Monday, November 28, 2016.

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Monday, November 28, 2016

4:00 pm in 241 Altgeld Hall,Monday, November 28, 2016

Toric Poisson Structures of type (1,1) and their cohomology

Arlo Caine (Notre Dame, Cal Poly Pomona)

Abstract: We show that the real Poisson structures on complex toric manifolds $(M,T_{\mathbb C})$ which are invariant under the complex torus action are in bijection with the hermitian forms on $\mathfrak t_{\mathbb C}$. In holomorphic coordinates in which the torus action is equivalent to the product action of $\mathbb T_{\mathbb C}^n$ on $\mathbb C^n$, the components of such tensors are homogeneous quadratic functions of the coordinates and their conjugates with coefficients given by the matrix $[B_{pq}]$ representing the associated hermitian form $B$. We investigate the dependence of the Poisson cohomology on the associated form, focusing on examples where $B$ is integral. In the algebraic category, the computation of the Poisson cohomology on such a chart $\mathbb C^n$ admits a complete solution using representation theory. We prove that, with respect to the wedge product, the cohomology is a module over the exterior algebra of $\mathfrak t_\mathbb C$ with generators which are monomial decomposable multi-vector fields corresponding to solutions of Diophantine inequalities determined by the form $B$. This is joint work Berit Givens of Cal Poly Pomona.

4:00 pm in 245 Altgeld Hall,Monday, November 28, 2016

J-holomorphic curves

Ely Kerman (Department of Mathematics, University of Illinois)

Abstract: Since they were introduced by Gromov in 1985, J-holomorphic curves have become a crucial tool in symplectic topology and many related fields. In this talk I will describe what J-holomorphic curves are and what makes them so useful.