This short course is an introduction to the nascent field of Fourier analysis on polytopes and cones. There is a rapidly growing number of applications of these methods, so it is appropriate to invite students, as well as professionals, to the field.
Of the many applications of these techniques, we will focus on the following, as time permits:
The Fourier transform of a polytope, given its vertex description
Minkowski and Siegel's theorems in the geometry of numbers
Tilings and multi-tilings of Euclidean space by translations of a polytope
Discrete volumes of polytopes (Ehrhart theory)
The Fourier transform of a polytope, given its hyperplane description. Here we iterate the divergence theorem.
We assume familiarity with linear algebra, calculus and infinite series. Throughout, we introduce the topics gently, by giving examples and exercises.
In this talk, we investigate the regularity properties of solutions to a class of fully nonlinear nonlocal equations, both elliptic and degenerate. For the elliptic case, we establish optimal regularity results for equations with a right-hand side in $L^p$, where the regularity space depends on the value of $p$. In the degenerate case, we prove the existence of at least one viscosity solution belonging to the class $C_{loc}^{1, \alpha}$ for some constant $\alpha \in (0,1)$. Additionally, when the order of the operator is sufficiently close to 2, we derive regularity estimates in Hölder spaces for the gradient of any viscosity solution.
A lack of coordinates makes differentiating vector fields on manifolds depending on a choice of connection. We discuss how most connections introduce curvature and experience it first hand. We discuss how curvature leads to holonomy. For Riemannian manifolds we introduce the classification of holonomy groups due to Berger and deRham. If time permits we conclude with submersion results in the case of non-vanishing skew-torsion.
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 $(M,\varphi,\xi,\eta,g)$, locally fibering along the 1-dimensional foliation generated by $\xi$ 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 $Sp(n)\times {1\}$-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).
References 1. D. Di Pinto, On anti-quasi-Sasakian manifolds of maximal rank J. Geom. Phys. 200 (2024), Paper no. 105174, 10 pp. 2. D. Di Pinto, G. Dileo, Anti-quasi-Sasakian manifolds, Ann. Global Anal. Geom. 64 (1), Article no. 5 (2023), 35 pp.
A fundamental question in mathematical analysis, with consequences to fields such as partial differential equations, mathematical physics, signal processing and many others, is that of simultaneous time-frequency concentration of signals. More specifically: given a set $\Omega = I \times J \subset \mathbb{R}^2$, how much information can be inferred about $f$ from its measurements on $I$, and measurements of its Fourier transform on $J$?
As classically known, the Uncertainty Principle prevents sharp localization, but one may hope to obtain suitably `almost-localized' signals if the underlying sets $\Omega$ considered are large enough. The topic of how much of such a localization is achievable has been extensively studied by several authors, such as I. Daubechies, Y. Meyer, D. Donoho, C. Fefferman and many others, leading to the development of the theory of Wavelets and that of the Short-time Fourier transform.
In this talk, we shall highlight recent developments on concentration results for time-frequency representations, with emphasis on the cases of the Short-time Fourier and Wavelet transforms. Our main goal will be to understand the Nicola-Tilli theorem on sharp concentration for the Gabor transform, its recent stability version, and other generalizations thereof, with a focus on open problems and possible future directions.
In this talk, I will overview works on random walks in dynamical random environments. I will recall a result obtained in collaboration with Hilario and Teixeira and then I will focus on a work with Conchon--Kerjan and Rodriguez. Our main interest is to investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, with density in [0,1]. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole. We prove that the speed of the walk, seen as a function of the density, exists for all density but at most one, and that it is strictly monotonic. We will explain how this can be seen as a sharpness result and provide an outline of the proof, whose general strategy is inspired by techniques developed for studying the sharpness of strongly-correlated percolation models.
There are at least three ways to calculate the eigenvalues of quantum graphs: Through solving eigenvalue equations directly, using a secular equation based on the vertices (nodes), and using a secular equation based on the edges (connections). I'll describe and relate these approaches, and will discuss some algebraic methods and averaging techniques that simplify and clarify the problem of finding the eigenvalues.
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.
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.
