Mathematics

Construction of the Poincare sheaf on the stack of Higgs bundles

Mao Li (University of Wisconsin)

Abstract: An important part of the Langlands program is to construct the Hecke eignsheaf for irreducible local systems. Conjecturally, the classical limit of the Hecke eignsheaves should correspond to the Poincare sheaf on the stack of Higgs bundles. The Poincare sheaf for the compactified Jacobian of reduced planar curves have been constructed in the pioneering work of Dima Arinkin. In this talk I will present the construction of the Poincare sheaf on the stack of rank two Higgs bundles for any smooth projective curve over the entire Hitchin base, and it turns out to be a maximal Cohen-Macaulay sheaf. This includes the case of nonreduced spectral curves, and thus provides the first example of the existence of the Poincare sheaf for nonreduced planar curves.

Some rigorous results on a long-range infection model on infinite lattices

Partha Dey   [email] (University of Illinois at Urbana–Champaign)

Abstract: We consider a long range infection model on finite dimensional lattices with no recovery and L\'evy interaction with exponent $\alpha$. Using monotonicity, one can prove that the growth rate decreases as $\alpha$ increases. We prove existence of four different growth regimes with thresholds depending on the dimension:
a) for $\alpha$ < d instantaneous growth,
b) for d < $\alpha$ < 2d stretched exponential growth,
c) for 2d< $\alpha$ < 2d+1 super linear growth and
d) for $\alpha$>2d+1 linear growth,
where $d$ is the dimension. In one-dimension we characterize the asymptotic distributional limits, which shows existence of a new ``fluctuation'' transition threshold. Finally, we will mention partial results and conjectures in higher dimension and when recovery is added to the model. Prior knowledge about epidemics models is not required for this talk.