A Ferrers diagram is a graphical way of representing an integer partition. A q-series is a series in which the ratio of the nth term to the next is a rational function of qn. With reference to the origins of the subject in the work of Sylvester, I will present a short introduction to the use of Ferrers diagrams in giving combinatorial interpretations of q-series identities. I will then move on to more recent developments involving a sort of generalized partition, called an overpartition. Finally, I will describe some further generalizations and related open problems.
We present a systematic study of kinematic formulas in convex geometry. We first give a classical presentation of kinematic formulas for integration with respect to the rotation group $SO(n)$, where Steiner's Formula, the intrinsic volumes and Hadwiger's Characterization Theorem play a crucial role. Then we will show a new extension to integration along the general linear group $GL(n)$. Using the bijection of matrix polar decomposition and a Gaussian measure to integrate along positive definite matrices, a new formula is obtained, for which the classical $SO(n)$ formula is a particular case. We also reference the unitary group $U(n)$ case and its corresponding extension to the symplectic group $Sp(2n,\mathbb{R})$.
We present a random interface model on the one-dimensional torus of size $N$ with a weak perturbation, i.e. an asymmetry $\sim N^{-\gamma}$ of the direction of growth that switches from up to down based on the sign of the area underneath. The evolution of the interface can be studied in terms of the density field of an underlying, non-Markovian exclusion process. We compute the order of the correlation functions of this process for the invariant measure of the interface model, and investigate the stationary fluctuations of the density field: we establish the convergence to an Ornstein-Uhlenbeck equation for $\gamma>\frac{8}{9}$, and discuss the limit for $\frac{1}{2}\leq \gamma<\frac{8}{9}$. Based on joint work with Martin Hairer and Patrícia Gonçalves.
Reaction-diffusion equations arise naturally when modelling multi-component systems of interacting populations. These equations are widely employed to describe pattern formation phenomena across various biological, chemical and physical processes. The kinetic theory of statical mechanics provides a powerful framework to describe different types of interactions at multiple spatial or temporal scales. Through appropriate hydrodynamic limits of the kinetic systems, macroscopic equations can be derived, describing observable quantities and explaining how macroscopic phenomena emerge from the underlying microscopic dynamics. In this talk, I will apply these tools to study the evolution and interactions of competing bacterial populations on a leaf surface. Specifically, I will consider self and cross diffusion effects and investigate Turing instability properties leading to the formation and persistence of stationary spatial patterns.
This work is a collaboration with D. Cusseddu (University of Minho), M. Bisi and R. Travaglini (University of Parma, Italy).
3d mirror symmetry is a mysterious duality for certain pairs of hyperkähler manifolds, or more generally complex symplectic manifolds/stacks. In this talk, we will describe its relationships with 2d mirror symmetry. This could be regarded as a 3d analog of the paper Mirror Symmetry is T-Duality by Strominger, Yau and Zaslow which described 2d mirror symmetry via 1d dualities.
I will discuss some recent results obtained in collaboration with A. Figalli, S. Kim and H. Shahgholian. We consider minimizers of the Dirichlet energy among maps constrained to take values outside a smooth domain $O$ in $\mathbb{R}^m$. These minimizers can be thought of either as solutions of a vectorial obstacle problem, or as harmonic maps into the manifold-with-boundary given by the complement of $O$. I will discuss results concerning the regularity of the minimizers, the location of their singularities, and the structure of the free boundary.
