I will review the notion of a topological (or gapped) domain wall between topological quantum field theories and illustrate an equivalence between domain walls and oplax natural transformations. I will show how this provides a reformulation of Lurie's cobordism hypothesis with singularities.

Since a Riemannian manifold is locally similar to the Euclidean space, it is easy to see that isoperimetric sets of small volume in such a manifold are very close to balls, and in particular they are connected. Much less is known for the case of minimal clusters. In this talk, we will describe the general situation and we will present a recent result showing that also small minimal clusters are connected if the ambient space is a compact Riemannian manifold. In addition, we will discuss also the situation for Finsler manifolds, showing that a small minimal m-cluster can have at most m connected components. While it might seem reasonable that also in this case small minimal clusters are connected, we will present an example showing that this is not true. We will conclude by listing some open problems. (Most of the presented results are based on joint works with D. Carazzato and S. Nardulli).

In this talk we will discuss recent advancements on $G_2$-instantons on 7-dimensional 2-step nilpotent Lie groups endowed with a left-invariant coclosed $G_2$-structures. I will present necessary and sufficient conditions for the characteristic connection of the $G_2$-structure to be an instanton, in terms of the torsion of the $G_2$-structure, the torsion of the connection and the Lie group structure. These conditions allow to show that the metrics corresponding to the $G_2$-instantons define a naturally reductive structure on the simply connected 2-step nilpotent Lie group with left-invariant Riemannian metric. This is a joint work with Andrew Clarke and Andrés Moreno.

The classical non-abelian Hodge Correspondence is a gauge-theoretic construction that has allowed for the use of complex geometric methods in the study of representations of the fundamental group of a closed surface. The conformal limit was introduced by Gaiotto as a parameterized variation of this classical correspondence. In this talk, we will explore how, in the case of representations into $\operatorname{SL}(2,\mathbb{C})$, this limit is related to complex projective structures. We will also use this relation to further our geometric understanding of the limiting process. This is joint work with Peter B. Gothen.

A long-standing topic in the analysis of partial differential equations, and mathematical physics, is the study of how the spectrum (eigenvalues) of a differential operator such as the Laplacian or a Schrödinger operator depends on the geometry or other structural properties of the domain on which they are defined. A classic example of such shape optimisation is the 19th Century conjecture of Lord Rayleigh, proved by Faber and Krahn in the 1920s, that the ball minimises the lowest eigenvalue of the Laplacian with Dirichlet conditions, among all domains of given volume. Informally, this amounts to saying that “heat loss is slowest in the ball”.

In the last 10 to 20 years, the study of such eigenvalue optimisation problems has become popular in the specific context of differential operators defined on metric graphs, also known as quantum graphs. Despite being essentially one-dimensional objects, these can display surprisingly rich behaviour, and are often useful as toy models for higher-dimensional problems.

In this talk I will give an informal introduction, first to some of the original ideas behind shape optimisation of domains, then quantum graphs, discussing what kinds of questions one can ask, what techniques are used, and how our understanding of these problems (and what questions are interesting) has developed over the last couple of decades.

In the present talk I will introduce a new class of almost contact metric manifolds, called anti-quasi-Sasakian (aqS for short). They are non-normal almost contact metric manifolds $(M, φ, ξ, η, g)$, locally fibering along the 1-dimensional foliation generated by $ξ$ onto Kähler manifolds endowed with a closed 2-form of type $(2,0)$. Various examples of anti-quasi-Sasakian manifolds will be provided, including compact nilmanifolds, $\mathbb{S}^1$-bundles and manifolds admitting a $\operatorname{Sp}(n) × \{1\}$-reduction of the structural group of the frame bundle. Then, I will discuss some geometric obstructions to the existence of aqS structures, mainly related to curvature and topological properties. In particular, I will focus on compact manifolds endowed with aqS structures of maximal rank, showing that they cannot be homogeneous and they must satisfy some restrictions on the Betti numbers. This is based on joint works with Giulia Dileo (Bari) and Ivan Yudin (Coimbra).

We consider and resolve the gap problem for global automorphisms of complex or real parabolic geometries. Concretely, the automorphism group of a parabolic geometry of type $(G, P)$ is largest for the flat model $G/P$. The symmetry dimension is maximal in this case and is equal to $\operatorname{dim} G$. We prove that the next realizable, so-called submaximal dimension of the automorphism group of a $(G, P)$ type geometry is the dimension of a (specific) maximal parabolic subgroup in G. We also discuss maximal and submaximal dimensions of the automorphism group of compact models and provide several examples. Joint work with B. Kruglikov.

Geometric quantization is an attempt to use the differential-geometric properties of a classical phase pace assumed to be a symplectic manifold M in order to define a corresponding quantum theory. In this talk, I will give an introduction to geometric quantization on symplectic manifold. In particular, I will focus on Kähler manifold endowed with T-symmetry. This is a joint work with Conan Leung.

In applied mathematics, effective problem solving begins with precise problem formulation, highlighting the importance of the initial problem-setting phase. Without a clearly defined problem, identifying suitable tools and techniques for resolution becomes arduous and often futile. This transition from problem setting to problem solving is pivotal within the broader framework of knowledge advancement. Despite the remarkable progress of AI tools, they remain reliant on the groundwork laid by human intelligence. Mathematicians, leveraging their adeptness in discerning patterns and relationships within data and variables, play a crucial role during this phase. This lecture will introduce fundamental mathematical concepts encompassing both traditional machine learning and scientific machine learning. The latter offers an optimal platform for the harmonious fusion of problem setting and problem solving, bolstered by profound domain expertise.

We derive the general structure for returning to the steady macroscopic nonequilibrium condition, assuming a first-order relaxation equation obtained as zero-cost flow for the Lagrangian governing the dynamical fluctuations. The main ingredient is local detailed balance from which a canonical form of the time-symmetric fluctuation contribution (aka frenesy) can be obtained. That determines the macroscopic evolution. As a consequence, the linear response around stationary nonequilibrium gets connected with the small dynamical fluctuations, leading to fluctuation-response relations. Those results may be viewed as nonequilibrium extension of the well-known structure where the relaxation to equilibrium is characterized by a (dissipative) gradient flow on top of a Hamiltonian motion.