Polynomial optimization is a special case of dc (Difference of Convex functions) programming, however representing a multivariate polynomial into a dc function is a difficult task. We propose some new results on dc programming formulations for polynomial optimization. We are interested in polynomial decomposition techniques for representing any multivariate polynomial into difference-of-sums-of-squares (DSOS) and difference-of-convex-sums-of-squares (DCSOS) polynomials. We firstly prove that the set of DSOS and DCSOS polynomials are vector spaces and equivalent to the set of real valued polynomials. We also show that the problem of finding DSOS and DCSOS decompositions are equivalent to semidefinite programs (SDPs). Then, we focus on establishing several practical algorithms for DSOS and DCSOS decompositions without solving SDPs. Some examples illustrate how to use our methods.

Following Smirnov’s proof of Cardy’s formula and Schramm’s discovery of SLE, a thorough understanding of the scaling limit of critical percolation on the regular triangular lattice the has been achieved. Smirnorv’s proof in fact gives a discrete approximation of the conformal embedding which we call the Cardy embedding. In this talk I will present a joint project with Nina Holden where we show that the uniform triangulation under the Cardy embedding converges to the Brownian disk under the conformal embedding. Moreover, we prove a quenched scaling limit result for critical percolation on uniform triangulations. Time permitting, I will also explain how this result fits in the the larger picture of random planar maps and Liouville quantum gravity

We investigate the influence of a stochastically fluctuating step-barrier potential on bimolecular reaction rates by analytical theory and stochastic simulations. We demonstrate that the system exhibits a ``resonant reaction'' behavior with rate enhancement if an appropriately defined fluctuation decay length is of the order of the system size. Importantly, we find that in the proximity of resonance, the standard reciprocal additivity law for diffusion and surface reaction rates is violated due to the dynamical coupling of multiple kinetic processes. Together, these findings may have implications on the correct interpretation of various kinetic reaction problems in complex systems, as, e.g., in biomolecular association or catalysis.

In his seminal work from 1979,
Joseph J. Kohn invented
his theory of multiplier ideal sheaves
connecting a priori estimates for the d-bar problem
with local boundary invariants
constructed in purely algebraic way.

I will explain the origin and motivation of the problem,
and how Kohn's algorithm reduces it
to a problem in local geometry
of the boundary of a domain.

I then present my work with Sung Yeon Kim
based on the technique of jet vanishing orders,
and show how it can be used to
control the effectivity of multipliers in Kohn's algorithm,
subsequently leading to precise a priori estimates.

We consider the family of normal octic fields with Galois group $D_4$, ordered by their discriminant. In forthcoming joint work with Arul Shankar, we verify the strong Malle conjecture for this family of number fields, obtaining the order of growth as well as the constant of proportionality. In this talk, we will discuss and review the combination of techniques from analytic number theory and geometry-of-numbers methods used to prove these results.