Seminars from until

In 1961, Jovan Karamata proved a remarkable identity involving the Dilogarithm function and intersection of diagonals of regular polygons. We reframe the problem and give different proofs for the result. We also investigate what happens when we consider different approaches to it.

In this talk, we explore étale groupoids $G$ with a locally compact Hausdorff unit space $X$, where $G$ itself may not be globally Hausdorff. For such groupoids, the essential $C^*$-algebra $C_{\operatorname{ess}}^*(G)$ offers a more suitable framework than the reduced $C^*$-algebra $C_r^*(G)$, as it captures additional structural nuances. Specifically, $C_{\operatorname{ess}}^*(G)$ arises as a proper quotient of $C_r^*(G)$.

We introduce the concept of essential amenability for groupoids, a condition that is strictly weaker than (topological) amenability yet sufficient to guarantee the nuclearity of $C_{\operatorname{ess}}^*(G)$. To establish this, we define a maximal version of the essential $C^*$-algebra and show that any function with dense cosupport must be supported within the set of "dangerous arrows”, that is, arrows that cannot be topologically separated.

This essential amenability framework characterizes the nuclearity of $C_{\operatorname{ess}}^*(G)$ and establishes its isomorphism to the maximal essential $C^*$-algebra. Our results offer new insights into the interplay between groupoid structure and operator algebras, extending the utility of $C_{\operatorname{ess}}^*(G)$ in non-Hausdorff settings. This is based on joint work with Diego Martinez.

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.

Perhaps the most (mathematically) well understood 3d TQFT is that of Reshetikhin–Turaev. Famously, the input data for their construction is that of a semisimple modular tensor category (MTC). Attempts at generalizing this construction to the non-semisimple case date back to the 90's with work of Hennings, Lyubashenko and Kerler–Lyubashenko. However, only partial results were achieved. This was until De Renzi et al. defined a 3d TQFT from such non-semisimple modular categories. Importantly, they had to impose an admissibility condition on the cobordism categories they use. My work has been in the direction of defining a once-extended 3d TQFT from this data. However, Bartlett et al. proved that such TQFTs are classified by semisimple modular categories. We will investigate the most natural method of circumventing this. This will lead to the notion of noncompact TQFT. I will then proceed to talk about my work on constructing such a TQFT from the data of a (potentially) non-semisimple MTC, with an emphasis on the key ingredients of this construction. Time permitting, I will also discuss how to extract 3-manifold invariants and a modified trace from such a noncompact TQFT.

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.

We present some recent results on the asymptotic behavior of almost periodic solutions to stochastic conservation laws and, more generally, degenerate parabolic-hyperbolic equations. Two types if stochastic perturbations are considered: forcing and rough-flux. The part concerning the forcing stochastic source is from joint works with Claudia Espitia and Daniel Marroquin. The part concerning stochastic rough-flux is from a joint project with Rui Jin Yachun Li and João Nariyoshi.