The main aim of this talk is to discuss the existence of nontrivial (and non semi-trivial) least energy solutions for a Neumann elliptic system with a critical nonlinearity, characterized by a cooperative-competitive behaviour, namely
\[ \begin{cases} -\Delta u+\lambda_1 u=u^3+β uv^2 & \text{ in } \Omega\\ -\Delta v+\lambda_2 v=v^3+\beta u^2v & \text{ in } \Omega\\ \frac{\partial u}{\partial\nu}=\frac{\partial v}{\partial \nu}=0 &\text{ on } \partial\Omega,\\ \end{cases} \label{eq:pbeta} \tag{$\mathcal P_\beta$} \] where $\Omega\subset \mathbb{R}^4$ is a $C^2$ bounded domain, and $\lambda_1,\lambda_2>0$ and the parameter $\beta\in\mathbb{R}$ captures the essence of cooperation-competition, assuming positive or negative values respectively.

The approach is variational and the idea is to minimize the energy functional on a suitable manifold of the Nehari type. In addition, to deal with the critical power, we estimate the energy level, using the solutions of $-\Delta w=w^3$ in $ \mathbb{R} ^4$ and the solution for the scalar equations $-\Delta u_i+\lambda_iu_i=u_i^3$ in $\Omega$, to establish a compactness condition based on the classical Cherrier's inequality: if $\partial\Omega\in C^1$ then for each $\varepsilon\gt 0$ there exists $M_{\varepsilon}>0$ such that $$\|u\|_{2^*}\le\left(\frac{2^{2/N}}{S}+\varepsilon\right)^{1/2}\|\nabla u\|_2+M_{\varepsilon}\|u\|_2,\ \ ∀u\in H^1(\Omega),$$ where $S$ is the best Sobolev constant. Additionally, I will discuss the more difficult cases in which $\lambda_1,\lambda_2\le0$, that I have started to study recently.

Topological quantum field theories (TQFTs) have attracted much attention from the physics and mathematical communities over the last thirty years, and for good reason: in low dimensions they let simple topology inform less-understood algebraic constructions. In the first half of this talk we will introduce factorization homology, a powerful procedure for constructing TQFTs out of homotopical gadgets, called $\mathsf{E}_n$-algebras. We explore this in dimension $n=1$, by using string-nets. In the second half of the talk, we introduce TQFTs with defects and factorization homology for stratified spaces and for an appropriate notion of stratified $\mathsf{E}_n$-algebra. Once again, we focus on dimension $n=1$, and will end by offering a conjectural connection between string-nets on stratified cylinders, Drinfel'd centres for bimodule categories and quasiparticles in a topological quantum computer.

A what?!

We will give a gentle introduction to gerbes and other assorted "higher structures" from topology and mathematical physics. Gerbes are a generalization of line bundles. For a line bundle, the space of sections forms a vector space, and a little extra geometric structure can make it into a Hilbert space, beloved by quantum physicists everywhere. After introducing gerbes, we will ponder the analogous construction: how do we define a "Hilbert space of sections" for a gerbe?

This expository talk is based on the work of Bunk and Szabo.

A mathematical model for the pitch of a plate is given by the first eigenvalue of the bilaplacian over the domain representing the plate at rest. Therefore, the question in title amounts to finding the domain for which the first eigenvalue of the bilaplacian is minimal. To address this shape optimization problem, we will introduce classic tools such as symmetrization techniques and comparison principles.