Introductory School - IES Cargèse

Introductory School - IES Cargèse

Introductory School - IES Cargèse

Introductory School - IES Cargèse

Introductory School - IES Cargèse

Lecture series: A class of II$_1$ factors with no non-trivial crossed product decompositions

• Adrian Ioana (UCSD)

Room: Amphitheater Darboux/Hermite In this minicourse, I will present the first examples of separable II$_1$ factors $M$ that admit no non-trivial crossed product decompositions: $M\not\cong B\rtimes\sigma G$, for any trace-preserving action of an infinite countable group $G$ on a tracial von Neumann algebra $(B,\tau)$. Our approach relies on constructing separable II$_1$ factors $M$ with the strong rigidity property that every embedding of $M$ into $M\overline{\otimes}M$ arises from the canonical embeddings $x\mapsto x\otimes 1$ and $x\mapsto 1\otimes x$. I will also discuss the main deformation/rigidity and coarseness techniques used in the proof. This is joint work with Adriana Fernández Quero and Hui Tan.

Lecture series: A class of II$_1$ factors with no non-trivial crossed product decompositions

• Adrian Ioana (UCSD)

Room: Amphitheater Darboux/Hermite In this minicourse, I will present the first examples of separable II$_1$ factors $M$ that admit no non-trivial crossed product decompositions: $M\not\cong B\rtimes\sigma G$, for any trace-preserving action of an infinite countable group $G$ on a tracial von Neumann algebra $(B,\tau)$. Our approach relies on constructing separable II$_1$ factors $M$ with the strong rigidity property that every embedding of $M$ into $M\overline{\otimes}M$ arises from the canonical embeddings $x\mapsto x\otimes 1$ and $x\mapsto 1\otimes x$. I will also discuss the main deformation/rigidity and coarseness techniques used in the proof. This is joint work with Adriana Fernández Quero and Hui Tan.

Lecture series: A class of II$_1$ factors with no non-trivial crossed product decompositions

• Adrian Ioana (UCSD)

Room: Amphitheater Darboux/Hermite In this minicourse, I will present the first examples of separable II$_1$ factors $M$ that admit no non-trivial crossed product decompositions: $M\not\cong B\rtimes\sigma G$, for any trace-preserving action of an infinite countable group $G$ on a tracial von Neumann algebra $(B,\tau)$. Our approach relies on constructing separable II$_1$ factors $M$ with the strong rigidity property that every embedding of $M$ into $M\overline{\otimes}M$ arises from the canonical embeddings $x\mapsto x\otimes 1$ and $x\mapsto 1\otimes x$. I will also discuss the main deformation/rigidity and coarseness techniques used in the proof. This is joint work with Adriana Fernández Quero and Hui Tan.

Workshop 1: OA Groups and Dynamics Room: Amphitheater Darboux/Hermite

Workshop 1: OA Groups and Dynamics Room: Amphitheater Darboux/Hermite

Workshop 1: OA Groups and Dynamics Room: Amphitheater Darboux/Hermite

Workshop 1: OA Groups and Dynamics Room: Amphitheater Darboux/Hermite

Workshop 1: OA Groups and Dynamics Room: Amphitheater Darboux/Hermite

Lecture series: An introduction to higher (T)

• Uri Bader (University of Maryland)

Room: Amphitheater Darboux/Hermite We will study unitary cohomology, that is group cohomology with unitary representations coefficients, with special emphasis on criteria for their vanishing. This topic brings together group theory, homological algebra and operator theory. We will emphasize operator algebraic techniques.

Lecture series: An introduction to higher (T)

• Uri Bader (University of Maryland)

Room: Amphitheater Darboux/Hermite We will study unitary cohomology, that is group cohomology with unitary representations coefficients, with special emphasis on criteria for their vanishing. This topic brings together group theory, homological algebra and operator theory. We will emphasize operator algebraic techniques.

Junior seminar - Operator Algebras Room: Amphitheater Darboux/Hermite

Lecture series: An introduction to higher (T)

• Uri Bader (University of Maryland)

Room: Amphitheater Darboux/Hermite We will study unitary cohomology, that is group cohomology with unitary representations coefficients, with special emphasis on criteria for their vanishing. This topic brings together group theory, homological algebra and operator theory. We will emphasize operator algebraic techniques.

Seminar - Operator Algebras - Diameters and box spaces

• Alain Valette (Université de Neuchatel)

Room: Amphitheatre Darboux

Lecture series: Stability for C*-algebras and groups

• Tatiana Shulman (University of Gothenburg)

Room: Amphitheater Darboux/Hermite Stability asks whether approximate solutions of algebraic relations can be perturbed to exact solutions. In C*-algebra theory, stability questions have a long history, motivated in part by classical problems in operator theory, and have become useful tools for studying the structure of C*-algebras. For groups, stability asks whether approximate representations are close to genuine representations. Different notions arise depending on the target groups and the chosen metrics, such as the Hilbert–Schmidt or operator norm metrics on unitary groups and the Hamming metric on symmetric groups. In recent years, interest in group stability has grown considerably, partly because of its connections with approximation problems for groups. This minicourse will introduce stability in C*-algebras and groups and explore the connections between these settings, with particular emphasis on the role of operator algebraic methods in the study of group stability.

Junior seminar - Operator Algebras Room: Amphitheater Darboux/Hermite

Lecture series: Stability for C*-algebras and groups

• Tatiana Shulman (University of Gothenburg)

Room: Amphitheater Darboux/Hermite Stability asks whether approximate solutions of algebraic relations can be perturbed to exact solutions. In C*-algebra theory, stability questions have a long history, motivated in part by classical problems in operator theory, and have become useful tools for studying the structure of C*-algebras. For groups, stability asks whether approximate representations are close to genuine representations. Different notions arise depending on the target groups and the chosen metrics, such as the Hilbert–Schmidt or operator norm metrics on unitary groups and the Hamming metric on symmetric groups. In recent years, interest in group stability has grown considerably, partly because of its connections with approximation problems for groups. This minicourse will introduce stability in C*-algebras and groups and explore the connections between these settings, with particular emphasis on the role of operator algebraic methods in the study of group stability.

Lecture series: Stability for C*-algebras and groups

• Tatiana Shulman (University of Gothenburg)

Room: Amphitheater Darboux/Hermite Stability asks whether approximate solutions of algebraic relations can be perturbed to exact solutions. In C*-algebra theory, stability questions have a long history, motivated in part by classical problems in operator theory, and have become useful tools for studying the structure of C*-algebras. For groups, stability asks whether approximate representations are close to genuine representations. Different notions arise depending on the target groups and the chosen metrics, such as the Hilbert–Schmidt or operator norm metrics on unitary groups and the Hamming metric on symmetric groups. In recent years, interest in group stability has grown considerably, partly because of its connections with approximation problems for groups. This minicourse will introduce stability in C*-algebras and groups and explore the connections between these settings, with particular emphasis on the role of operator algebraic methods in the study of group stability.

Seminar - Operator Algebras Room: Amphitheatre Darboux

Lecture series: Extremal geometries and convergence of unitary representations

• Michael Magee (Durham University)

Room: Amphitheater Darboux/Hermite I will discuss three interrelated problems about Riemann surfaces: How large can the spectral gap of the Laplacian be? (Old, e.g. Selberg) Do there exist negatively curved minimal surfaces in Euclidean spheres? (Yau) How good (in terms of conformal factors) can holomorphic immersions of Riemann surfaces in flat complex manifolds be? Progress on all these questions comes from understanding notions of convergence of unitary representations of both PSL_2(R) and the fundamental groups of closed surfaces. I'll give an introduction to (induced) representations of PSL_2(R), and an introduction to the notions of convergence we need. To give attributions, on question 1 I will discuss my results with Hide and Louder but perhaps not with the same proofs. On question 2 I report on the works of Song and Ancona, Labourie, Roig-Sanchis, and Toulisse. The last topic is a variation of a theme of Kazhdan, Rhodes, and McMullen, from a new perspective, which raises some interesting questions about group C*-algebras.

Junior seminar - Operator Algebras Room: Amphitheater Darboux/Hermite

Comment répartir au mieux les ressources ? Les mathématiques du transport optimal Room: Amphithéâtre Perrin Conférence carte blanche à la Maison Poincaré Orateur: Alain Vallette (l’Université de Neuchâtel) Réservez votre billet gratuit https://billetterie-maison-poincare.ihp.fr/evenements-maison-poincare/conferences-cartes-blanche/carte-blanche-les-mathematiques-du-transport-optimal

Lecture series: Extremal geometries and convergence of unitary representations

• Michael Magee (Durham University)

Room: Amphitheater Darboux/Hermite I will discuss three interrelated problems about Riemann surfaces: How large can the spectral gap of the Laplacian be? (Old, e.g. Selberg) Do there exist negatively curved minimal surfaces in Euclidean spheres? (Yau) How good (in terms of conformal factors) can holomorphic immersions of Riemann surfaces in flat complex manifolds be? Progress on all these questions comes from understanding notions of convergence of unitary representations of both PSL_2(R) and the fundamental groups of closed surfaces. I'll give an introduction to (induced) representations of PSL_2(R), and an introduction to the notions of convergence we need. To give attributions, on question 1 I will discuss my results with Hide and Louder but perhaps not with the same proofs. On question 2 I report on the works of Song and Ancona, Labourie, Roig-Sanchis, and Toulisse. The last topic is a variation of a theme of Kazhdan, Rhodes, and McMullen, from a new perspective, which raises some interesting questions about group C*-algebras.

Seminar - Operator Algebras

• N. Christopher Phillips (University of Oregon)

Room: Amphitheatre Darboux Nonexistence of homomorphisms between matrix algebras over spheres which are injective on K-theory. This is joint work with Ilan Hirshberg. In 1993, Blackadar asked for which $n$, $k$, $d$, and $r$ there is a unital homomorphism $C (S^{2 n}, M_r) \to C (S^{2 k}, M_{d r})$ which is injective on $K_0$. There are "obvious" examples constructed as follows. For $m \in {\mathbb{N}}$, $K^1 (S^{2 m - 1})$ is generated by the class of a unitary in $C (S^{2 m - 1}, M_m)$, and from this one gets homomorphisms $C (S^{1}) \to C (S^{2 m - 1}, M_m)$ which are injective on $K_1$. One can then take (odd) suspensions, increase the matrix size in the codomain, and tensor everything with $M_r$. Ordered $K_0$ suggests that many more might exist. In particular, Blackadar asked whether there is a unital homomorphism $C (S^{2 n}, M_{2 n}) \to C (S^{4 n}, M_{4 n})$ which is injective on $K_0$. The answer is no. We prove that the only choices of $n$, $k$, $d$, and $r$ for which such a homomorphism exists are those which occur in the examples above. The proof uses indecomposability in the rational cohomology ring of the space of homomorphisms from $C (S^{2 n})$ to $M_d$. This material is based upon work partially supported by the US National Science Foundation under Grant DMS-2400332.

OpenAI’s proof of isomorphism of free group factors, and further developments

• Dimitri Shlyakhtenko (UCLA)

Room: Amphitheater Darboux/Hermite We will present a version of OpenAI’s proof of free group factor isomorphism. A strengthening of the proof also shows that the reduced group C*-algebras of free groups have strong reduced free products absorption properties. Moreover, C*-algebras generated by n=2,3,… or countably many semicircular elements are all isomorphic, and the same is true for reduced free products of copies of the Jiang-Su algebra. A common ingredient in all of these proofs is a certain automorphism approximation property; a relative version of that property appeared in our recent work on identification of von Neumann algebras of Fuchsian groups.

Lecture series: Extremal geometries and convergence of unitary representations

• Michael Magee (Durham University)

Room: Amphitheater Darboux/Hermite I will discuss three interrelated problems about Riemann surfaces: How large can the spectral gap of the Laplacian be? (Old, e.g. Selberg) Do there exist negatively curved minimal surfaces in Euclidean spheres? (Yau) How good (in terms of conformal factors) can holomorphic immersions of Riemann surfaces in flat complex manifolds be? Progress on all these questions comes from understanding notions of convergence of unitary representations of both PSL_2(R) and the fundamental groups of closed surfaces. I'll give an introduction to (induced) representations of PSL_2(R), and an introduction to the notions of convergence we need. To give attributions, on question 1 I will discuss my results with Hide and Louder but perhaps not with the same proofs. On question 2 I report on the works of Song and Ancona, Labourie, Roig-Sanchis, and Toulisse. The last topic is a variation of a theme of Kazhdan, Rhodes, and McMullen, from a new perspective, which raises some interesting questions about group C*-algebras.

Workshop 2: OA Matricial Approximations Room: Amphitheater Darboux/Hermite

Workshop 2: OA Matricial Approximations Room: Amphitheater Darboux/Hermite

Workshop 2: OA Matricial Approximations Room: Amphitheater Darboux/Hermite

Workshop 2: OA Matricial Approximations Room: Amphitheater Darboux/Hermite

Workshop 2: OA Matricial Approximations Room: Amphitheater Darboux/Hermite

Junior seminar - Operator Algebras Room: Amphitheater Darboux/Hermite

Seminar - Kirchberg's embedding problem Room: Amphitheatre Darboux

Reading seminar on Connes rigidity conjecture

• Cyril Houdayer (Ecole normale supérieure)

Room: Amphitheater Darboux/Hermite

Reading seminar - The generator problem for von Neumann algebras

• Sorin Popa (UCLA)

Room: Amphitheater Darboux/Hermite

Junior seminar - Operator Algebras Room: Amphitheater Darboux/Hermite

Reading seminar - Connes rigidity conjecture Room: Amphitheater Darboux/Hermite

Seminar - Operator Algebras

• Jorge Castillejos (UNAM)

Room: Amphitheatre Darboux

Junior seminar - Operator Algebras Room: Amphitheater Darboux/Hermite

Seminar - Kirchberg's embedding problem Room: Amphitheatre Darboux

Seminar - Operator Algebras

• Arturo Jaime (IMJ-PRG)

Room: Amphitheatre Darboux

Junior seminar - Operator Algebras Room: Amphitheater Darboux/Hermite

Seminar - Kirchberg's embedding problem Room: Amphitheatre Darboux

Seminar - Operator Algebras

• Kristin Courtney

Room: Amphitheatre Darboux

Workshop 3: OA Rigidity and Classification Room: Amphitheater Darboux/Hermite

Workshop 3: OA Rigidity and Classification Room: Amphitheater Darboux/Hermite

Workshop 3: OA Rigidity and Classification Room: Amphitheater Darboux/Hermite

Workshop 3: OA Rigidity and Classification Room: Amphitheater Darboux/Hermite

Workshop 3: OA Rigidity and Classification Room: Amphitheater Darboux/Hermite