Titres/Résumés
Max Carter (Louvain-La-Neuve)
- Titre : Harmonic analysis on contraction groups and their weighted Orlicz *-algebras
- Résumé : It is a classical question in harmonic analysis and Banach algebra theory to determine which locally compact groups are Hermitian, Wiener and CCR. These properties are formulated in terms of the \(L^1\) group algebra of a locally compact group, however, the Hermitian and Wiener properties can be formulated more generally for any Banach *-algebra. These properties have been investigated extensively in the past, particularly for Lie groups and discrete groups. On the other hand, for non-discrete totally disconnected locally compact (tdlc) groups, it is not well understood, in general, which of these groups satisfy these properties. In this talk I will discuss a variety of recent projects working on these properties for tdlc groups. The focus will be on the class of contraction groups (an analogue of unipotent groups in tdlc group theory) and their semi-direct products, and also a broader class of Banach *-algebras on these groups which are of the form a weighted Orlicz space.
Viola Giovannini (Luxembourg)
- Titre : Renormalized Volume for Convex Co-Compact Hyperbolic 3-Manifolds with Compressible Boundary
- Résumé : Given a hyperbolizable 3-manifold N with boundary components of genus \(g\geq 2\), the renormalized volume is a real-analytic function on the Teichmüller space of its boundary. When the boundary of \(N\) is incompressible the renormalized volume is always non-negative; otherwise, its infimum is \(-\infty\). We will present these differences, focusing on the second setting.
Efthymia Papageorgiou (Paderborn)
- Titre : \(L^p\) asymptotics for the heat equation on symmetric spaces for non-symmetric solutions
- Résumé : On Euclidean space \(\mathbb{R}^n\), it is known that if \(h_t\) is the heat kernel and \(u_0\) is a continuous, compactly supported function, then for all \(p\in[1,\infty]\) it holds
\[\|h_t\|_{L^p(\mathbb{R}^n)}^{-1}\,\|u_0\ast h_t \, - \,M\,h_t\|_{L^p(\mathbb{R}^n)} \rightarrow 0 \quad \text{as} \quad t\rightarrow \infty,\]
where \(M=\int_{\mathbb{R}^n}u_0.\)
However, the analogous long-time convergence of solutions to the heat equation on noncompact symmetric spaces \(G/K\) of non-compact type, turns out to be drastically different. For a continuous, compactly supported function \(u_0\), we introduce a mass {function} \(M_p(u_0)(\cdot)\) which {varies with \(p\)}, and we show that
\[\|h_t\|_{L^p(G/K)}^{-1}\,\|u_0\ast h_t \, - \,M_p(u_0)(\cdot)\,h_t\|_{L^p(G/K)} \rightarrow 0 \quad \text{as} \quad t\rightarrow \infty.\]
Interestingly, the \(L^p\) heat concentration leads to completely different expressions of the mass function for \(1\leq p < 2\) and \(2\leq p\leq \infty\). If we further assume that the initial datum is bi-\(K\)-invariant, then our mass function boils down to the constant \(\int_{G/K}u_0\) in the case \(p=1\), and more generally to the spherical transforms \(\mathcal{H}{u_0}(i\rho(2/p-1))\) if \(1\leq p< 2\),
and to \(\mathcal{H}{u_0}(0)\) if \(2\leq p \leq \infty\). This result generalizes and improves upon works by Vazquez, Anker et al, Naik et al.
Aurélie Paull (IECL-Metz)
- Titre : The Weil representation for a finite field of characteristic two
- Résumé : The Weil representation is a beautiful mathematical object with applications in many domains, such as harmonic analysis, number theory, and quantum physics.
It is relatively well understood for local or finite fields of odd characteristics and for adelic rings. This representation arises from the Heisenberg group and precisely from one of its main features: the uniqueness of a specific irreducible unitary representation. This is the Stone-von Neumann theorem. The action of the symplectic group then leads to the construction of the Weil representation of either the metaplectic two-fold covering of the symplectic group or the symplectic group itself. In characteristic two, the situation is quite different, and even in the case of a finite field, there is a lack of clear understanding and explicit formulas. In this talk, we will discuss the challenges of studying the Weil representation for a finite field of characteristic two and present an explicit construction. After defining the Heisenberg group in these settings, we construct the Weil representation of a two-fold covering of the pseudo-symplectic group. In particular, we obtain explicit formulas for the associated intertwining operators, the cocycle and the character of this representation. Unfortunately, one drawback of this construction is that the pseudo-symplectic group is not related to the entire symplectic group. Therefore, for the field \(F_2\), using the ring \(Z/4Z\), we will provide a second construction of the Weil representation, which will be defined this time on a covering of the affine symplectic group. We will also make comments about the link between the two constructions, which is determined by the size we choose for the coverings. All along, we will illustrate our results with the example of a two-dimensional vector space over the field of two elements.
Ralf Schiffler (Connecticut)
- Titre : From continued fractions to knot invariants and cluster algebras
- Résumé : A combinatorial interpretation of continued fractions in terms of perfect matchings of so-called snake graphs was given in [1]. In this talk, we will present this interpretation and explain how it leads to interesting connections between knot theory and cluster algebras.
[1] I. Canakci and R. Schiffler, Cluster algebras and continued fractions, Compos. Math. 154 (3) (2018) 565-593.
Nguyen Viet Dang (Strasbourg)
- Titre : The Phi43 theory on de Sitter space
- Résumé : In this talk, I would like to explain how a certain construction of a probability measure on distributions over the 3-sphere, gives rise to a relativistic quantum field theory on de Sitter space dS3.
The construction relies on analytic continuation of representation of the orthogonal group \(SO(4)\) into representation of the Lorentz group \(SO(3,1)\). I will try to convey as much as possible how the Wick rotation and the notions of quantum field theory can emerge naturally from representation theory. This is joint work with Bonthonneau, Ferdinand and Lin.