Séminaires

If we have a flat vector bundle E over a compact manifold M, then we can form the de Rham cohomology H(M; E) of the M twisted by E. Given a Riemannian metric on M, this cohomology can be realised as the kernel of a Laplacian, via the Hodge theorem. Using the same Laplacian, In 1971, Ray and Singer defined the analytic torsion of M (now also assumed oriented), twisted by E. If H(M; E) vanishes, then analytic torsion is independent of the Riemannian metric used to define the Laplacian, and therefore a smooth invariant. Ray and Singer’s motivation for this definition was to give an analytic way to realise Reidemeister-Franz torsion, a combinatorial invariant of finite cell complexes. In 1978, Cheeger and Müller proved independently that these two notions of torsion are indeed equal. Fried’s conjecture from 1986 is a relation between analytic torsion and the Ruelle dynamical zeta function of a flow on M with suitable properties. That is a quantity involving the lengths of closed flow curves. So analytic torsion gives us a link between analysis, topology and dynamical systems. Most of this talk is a survey of analytic torsion. I will also mention some recent work with Hemanth Saratchandran on an equivariant version of analytic torsion for proper group actions.

Soit P un polynôme à coefficients entiers, unitaire, irréductible, de degré 4 et de groupe de Galois diédral ou cyclique.
Il existe c = c(P) > 0, tel que P(n) ait un facteur premier supérieur à n^1+c pour une proportion positive d’entiers n.

Il s’agit d’un travail  avec James Maynard.

Let I and J be two orthogonal respectively left and right coideal $*$-subalgebras of a dual pair of Hopf $*$-algebras. We define the Drinfeld double coideal of J and I. Next, we define for $0<q<1$ the unital $*$-algebra $U_q(sl(2,R))$, quantizing the universal enveloping $*$-algebra of $sl(2, R)$, as the Drinfeld double of the equatorial Podles sphere coideal $*$-subalgebra of $O_q(SU(2))$ and its associated orthogonal coideal $*$-subalgebra of $U_q(su(2))$. We finally classify its admissible irreducible $*$-representations.

Répétition du séminaire Bourbaki du vendredi 1er avril.

(Travail commun avec P. Delorme). Soit $G$ un groupe de Lie réductif réel, muni d’une involution $\sigma$ et $\Gamma$ un sous-groupe discret cocompact. Nous établissons une formule des traces relative, en lien avec $\Gamma$ et $H=G^\sigma$, exprimant la somme de  certaines intégrales orbitales de $f\in C_c^\infty(G)$ en terme de  coefficients généralisés de représentations unitaires irréductibles de $G$. Lorsque $G/H$ admet une série discrète relative $\pi_0$, l’existence de pseudocoefficient relatif pour $\pi_0$ à support « petit » implique, via la formule des traces relative,  que $\pi_0$ intervient dans la décomposition spectrale de $L^2(\Gamma\backslash G)$. Nous étudions l’existence de tels pseudocoefficients pour les espaces hyperboliques et les espaces symétriques de type $G(\mathbb{C})/G(\mathbb{R})$.

I will present several applications of the Dirac inequality to the determination of unitary representations of p-adic groups and associated « spectral gaps ». The method works particularly well in order to attach irreducible unitary representations to the large nilpotent orbits (e.g., regular, subregular) in the Langlands dual complex Lie algebra. These results can be viewed as a p-adic analogue of Salamanca-Riba’s classification of irreducible unitary (g,K)-modules with strongly regular infinitesimal character.

Chebyshev famously observed empirically that more often than not, there are more primes of the form 3 mod 4 up to x than primes of the form 1 mod 4. This was confirmed theoretically much later by Rubinstein and Sarnak in a logarithmic density sense. Our understanding of this is conditional on the generalized Riemann Hypothesis as well as Linear Independence of the zeros of L-functions.

We investigate similar questions for sums of two squares in arithmetic progressions. We find a significantly stronger bias than in primes, which happens for almost all integers in a natural density sense. Because the bias is more pronounced, we do not need to assume Linear Independence of zeros, only a Chowla-type Conjecture on non-vanishing of L-functions at 1/2.

We’ll aim to be self-contained and define all the notions mentioned above during the talk. We shall review the origin of the bias in the case of primes and the work of Rubinstein and Sarnak. We’ll explain the main ideas behind the proof of the bias in the sums-of-squares setting.