Exposés à venir

-Mardi 5 mars: Oleg Aristov, Moscow.
Duality for Hopf holomorphically finitely generated algebras

Inspired by the duality theory of locally compact quantum groups, we discuss a class of topological Hopf algebras that can be considered as algebras of 'holomorphic functions on quantum complex Lie groups'. A Hopf holomorphically finitely generated (HFG) algebra is introduced as a topological Hopfalgebra that is a quotient of Taylor's algebra of free entire functions. For every Hopf HFG algebra $H$, the dual topological Hopf algebra $H^\circ $ can be defined. We talk over conditions under which $H^\circ $ is HFG. The natural commutative example of a Hopf HFG algebra is $\mathcal O(G)$, the algebra of holomorphic functions on a complex Lie group $G$. It is shown, under the assumption that $G$ is connected, that $\mathcal O(G)^{\circ \circ }\cong \mathcal O(G)$ iff $G$ is linear, i.e., admits a faithful finite-dimensional holomorphic representation.

-Jeudi 7 mars à 13h30: Oleg Aristov, Moscow.
Commutative and noncommutative $C^\infty $-functional calculus

In Noncommutative Differential Geometry, some dense subalgebras of $C^*$-algebras are considered. But there is still no an axiomatic definition of 'function algebras' on noncommutative $C^\infty $-manifolds or, more generally, $C^\infty $-spaces. One of main requirement to a candidate for the title is stability under $C^\infty $ -functional calculus for several commuting self-adjoint elements. We exhibit conditions that guarantee existence of a $C^\infty $-functional calculus on the joint spectrum of a commuting tuple of self-adjoint elements on a $C^*$-spectral Arens-Michael $*$-algebra and compare them to the Kissin-Shulman differential norm condition. We also discuss how to construct an enveloping functor that maps algebras of polynomials to algebras of $C^\infty $-functions and that is compatible with Pontryagin duality for abelian real Lie groups. The more distant goal is to extend Pontryagin duality from abelian Lie groups to all Lie groups.

-Mardi 12 mars: Uwe Franz, LMB.
Monotone Increment Processes, Classical Markov Processes and Loewner Chains

We prove a one-to-one correspondence between certain decreasing Loewner chains in the upper half-plane, a special class of real-valued Markov processes, and quantum stochastic processes with monotonically independent additive increments. This leads us to a detailed investigation of probability measures on the real line with univalent Cauchy transform. We discuss several subclasses of such measures and obtain characterizations in terms of analytic and geometric properties of the corresponding Cauchy transforms. Joint work with Takahiro Hasebe and Sebastian Schleissinger.

-Mardi 19 mars: Emiel Lorist, Delft.
Singular stochastic integral operators

Singular stochastic integrals of the form$$ S_K G(t) :=\int_0^\infty K(t,s) G(s) ,\mathrm d W_H(s), \qquad t\in \mathbb R_+,$$appear naturally in questions related to stochastic maximal regularity. Here $G$ is an adapted process, $W_H$ is a cylindrical Brownian motion and $K$ is allowed be singular.In this talk I will introduce Calder\'on--Zygmund theory for such singular stochastic integrals with operator-valued kernel $K$.I will first discuss $L^p$-extrapolation under a H\"ormander condition on the kernel. Afterwards I will treat sparse domination and sharp weighted bounds under a Dini condition on the kernel, leading to a stochastic analog of the solution to the $A_2$-conjecture. The developed theory implies $p$-independence and weighted bounds for stochastic maximal $L^p$-regularity both in the complex and real interpolation scale. This leads to mixed $L^p(L^q)$-theory for several stochastic partial differential equations, of which I will give a few examples. This talk is based on joint work with Mark Veraar.

-Mercredi 20 mars à 16:30: Frédéric Patras, Nice.


Jeudi 21 - Vendredi 22 mars: Julien Bichon, Malte Gerhold Anna Kula, Martin Lindsay, Michael Schuermann, Adam Skalski, Moritz Weber
Journées Thématiques : Cohomology of Compact Quantum Groups and Related Topics

See the dedicated web-page

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