In this talk I will describe a class of minimization problems arising in modelling shape memory alloys. I will start with a shape memory material illustration, followed by a simple one dimensional model of it. Its extensions in 2D and 3D will help us understand the energetic mechanism behind the formation of twin patterns in physical experiments. By means of sharp upper and lower bounds, we show that the experimentally observed structures provide optimal energy scaling law. Furthermore, under certain assumptions we will see that some of these patterns are true energy minimizers - a rare find in problems of this type! I will also discuss the extension of the linear elastic models for two twins to model multiscale patterns, which appear in recent physical experiments and involve mixing of four twins.

We consider particle interaction energies defined via a pairwise interaction kernel. These energies are directly related to, and can be considered as, discrete versions of continuous interaction energies defined over probability measures. In their discrete or continuum form, such pairwise interaction energies appear in many biological or physical applications, ranging from swarming models to models of molecular structure. Under rather general assumption on the interaction kernel we prove that the discrete energies admit minimizers for sufficiently large number of particles, they converge to their continuum counterpart in the weak-* topology of probability measures, and minimizer of discrete energies converge to the minimizer of the continuum energy. This is a joint work with Davide Carazzato and Aldo Pratelli.

Diffuse domain methods (DDMs) approximate partial differential equations on complex geometries by replacing the sharp boundary interface with a diffuse layer of thickness ε. This approach reformulates the original equation on an extended regular domain, incorporating boundary conditions through singular source terms. In this work, we conduct a matched asymptotic analysis of a DDM approximation for a two-sided problem with transmission Robin boundary conditions. Our results show that, in one dimension, the solution of the DDM approximation asymptotically converges to the solution of the original problem, with exactly first-order accuracy in ε. Furthermore, for the Neumann boundary condition case, we show that the energy functional of the DDM approximation Γ-converges to the energy functional of the original problem, and the solution of the DDM approximation strongly converges, up to a subsequence, to the solution of the original problem in H¹, as ε approaches 0. We also provide numerical simulations that validate and illustrate the analytical result.

I've successfully used set-theoretic ``almost everywhere" reasoning in recent years in a variety of settings (homological algebra, actions of monoids, Weak Factorization Systems in homotopy theory).  Most recently, I used it to prove the Flat Cover Conjecture in categories of monoid actions over right-reversible monoids.  Here,  "almost everywhere" is defined in terms of a certain boolean algebra depending on the setting, and can be modeled using Shelah's Stationary Logic.  For example, if P is a projective module over a ring, X is a subset of P, and <X> is the submodule generated by X, it's possible that the quotient P/<X> fails to be projective (e.g., Z is a projective Z-module but Z/2Z is not).  But, P/<X> is projective for "almost every" subset X of P, and this is enough to (re)-prove a classic theorem of Kaplansky published in the Annals of Mathematics in the 1950s, that every projective module is a direct sum of countably generated modules.   I will discuss how this reasoning works, and discuss some recent applications involving so-called cofibrantly-generated Weak Factorization Systems (a concept originating with Quillen's Small Object Argument).