Seminar

Polarizations of Powers of Maximal Ideals in a Polynomial Ring

Abstract:

We give a complete combinatorial characterization of all possible polarizations of powers of the maximal ideal $(x_1,x_2,\dotsc,x_n)$ in a polynomial ring of $n$ variables. We also give a combinatorial description of the Alexander duals of such polarizations. In the three variable case we show that every polarization defines a (shellable) simplicial ball. We conjecture that any polarization of an artinian monomial ideal defines a simplicial ball. This is joint work with Gunnar Fløystad.

Large deviations for a class of semilinear stochastic partial differential equations in any space dimension

Abstract:

We prove the large deviation principle for the law of the solutions to a class of parabolic semilinear stochastic partial differential equations driven by multiplicative noise, in C[0,T] : Lρ(D) where D ⊂ Rd with d 1 is a bounded convex domain with smooth boundary and ρ is any real, positive and large enough number. The equation has nonlinearities of polynomial growth of any order, the space variable is of any dimension, and the proof is based on the weak convergence method.