P. Alonso-Ruiz: Sobolev embeddings in Dirichlet spaces
One of the classical Sobolev embeddings in \( \mathbb{R}^n\) asserts that functions in the Sobolev space \(W^{1,p}(\mathbb{R}^n)\) with \(1\leq p < n\) belong to a suitable \( L^q \)-space with an explicit optimal exponent \(q\) that depends on \(n\) and \(p\).
This talk focuses on this embedding in the more general framework of Dirichlet spaces with (sub-)Gaussian heat kernel estimates. In particular, we discover how the optimal exponent depends on the Hausdorff dimension, the walk dimension, and also on a further invariant of the space. To this end, we will follow a recent approach to \( (1,p)\)-Sobolev spaces via heat semigroups inspired by ideas going back to de Giorgi and Ledoux.
If time permits, we will outline some results and conjectures concerning critical exponents that include other dimensions of interest in the theory of metric measure spaces. Besides heat kernel estimates, the main assumption on the underlying space is a non-negative curvature type condition that we call weak Bakry-Émery that is satisfied in classical settings as well as in fractals like (infinite) Sierpinski gaskets and carpets.
The talk is based on joint work with F. Baudoin.

C. Biswas: Sharp Fourier restriction estimates onto curves
How can we make sense of the value of the Fourier transform of a function on a subset, possibly of measure zero (such as the sphere, a hyperplane, a curve etc) of the Euclidean space? This is one of the fundamental problems in modern analysis. In this talk we will discuss the existence of optimizers for Lebesgue space bounds when we restrict the Fourier transform to the moment curve, \( (t, t^2, \ldots, t^d) \).
This is based on our work with Betsy Stovall.

L. Bowen: Symmetric spaces for von Neumann algebras with applications to the invariant subspace problem and the multiplicative ergodic theorem
For each semi-finite von Neumann algebra \(M\), we associate the group \(GL^2(M)\) of log square-integrable operators and the space \(P(M)\) of positive definite log square integrable operators. These are analogous to the general linear group \(GL(n,\mathbb{R})\) and the symmetric space of positive definite matrices on which \(GL(n,\mathbb{R})\) acts transitively by isometries. This symmetric space determines the von Neumann algebra up to isomorphism (assuming trivial center). We apply this to give a new proof of Haagerup-Schultz's result on the convergence of \(\|T^n\|^{1/n}\) (where \(T\) is in the algebra) and to give a new infinite dimensional version of the Multiplicative Ergodic Theorem (MET) which allows for continuous Lyapunov spectrum. No von Neumann algebra background will be assumed.

M. Duchin: New directions in Heisenberg geometry
Nilpotent groups have been fruitfully studied with tools from several fields that don't always overlap in their methods: geometric group theory, Lie theory, and analysis. I'll describe a research program (starting with Pansu's thesis) to build more bridges between the analytic and group theoretic study of nilpotent groups, with the Heisenberg group as the central example.

S. Evington: Symmetries of Operator Algebras
I will begin by discussing Cones' classification of automorphisms of the hyperfinite II\(_1\) factor \(R\) and its subsequent generalisations (e.g. group actions and actions of unitary fusion categories).
I will then switch to the \(C^*\)-setting and explain the new obstructions that occur (spoiler: algebraic \(K_1\)) and the result of adapting the von Neumann constructions to the \(C^*\)-setting (spoiler: different descriptions of \(R\) lead to different classifiable \(C^*\)-algebras).
The talk is based on joint work with Sergio Giron Pacheco.

P. Ganesan: Quantum graphs
Quantum graphs are an operator space generalization of classical graphs. In this talk, I will present the different notions of quantum graphs that arise in operator systems theory, non-commutative topology and quantum information theory. I will then introduce a non-local game with quantum inputs and classical outputs, that generalizes the graph homomorphism game for classical graphs.
This is based on joint work with Michael Brannan and Samuel Harris.

J. Kuan: Joint moments of multi-species \(q\)-Boson
The Airy\(_2\) process is a universal distribution which describes fluctuations in models in the Kardar-Parisi-Zhang (KPZ) universality class, such as the asymmetric simple exclusion process (ASEP) and the Gaussian Unitary Ensemble (GUE). Despite its ubiquity, there are no proven results for fluctuations of multi-species models. Here, we will discuss one model in the KPZ universality class, the \(q\)-Boson. We will show that the joint multi-point fluctuations of the single-species \(q\)-Boson match the single-point fluctuations of the multi-species \(q\)-Boson. Therefore the single-point fluctuations of multi-species models in the KPZ class ought to be the Airy\(_2\) process.

M.-J. Kuffner: Boundedness of commutators on weighted Hardy spaces
It is known that boundedness of the commutator \([b, H]\) on weighted \(L^p\)-spaces for \(1< p <\infty\) is characterized by \(b\) being in a certain \(BMO\) space adapted to the given weights. In this talk, we present the case \(p=1\) and discuss the space that characterizes boundedness of \([b, H]\) on the weighted Hardy space \(H^1(w)\) for certain \(A_p\) weights.

S. Volberg: Orthogonality in Banach spaces
All spaces below are not Hilbert spaces. Given two finite dimensional subspaces \(L, K\) of a normed space \(X\) we call \(K\) orthogonal to \(L\) if for every unit vector in \(K\) the distance of this vector to \(L\) is \(1\). This usually does not mean that \(L\) is orthogonal to \(K\). We consider several questions:
1) Let \(E, F\) be two finite dimensional subspaces of a normed space \(X\) and let \(dim F= dim E+m\). Can we always find a subspace \(K\) in \(F\) such that \(E\) is orthogonal to \(K\)?
2) Can we always find a subspace \(K\) in \(F\) such that \(K\) is orthogonal to \(E\)?
3) Can we always choose \(K\) of dimension \(m\)?
4) If not, what is the maximal possible dimension?
Some of these questions seem to have been answered 60-80 years ago, and in fact, some of them were answered by Krein-Krasnoselski-Milman. But it looks like not all of them were answered.

K. Wrobel: Orbit equivalence of wreath products
Let F be a nonabelian free group. We show that, for any two nontrivial finite groups, there are natural actions of the wreath product groups A\(\wr\)F and B\(\wr\)F that are orbit equivalent. In the process, we introduce the notion of a cofinitely equivariant map. On the other hand, we show that these actions are not even stably orbit equivalent if F is replaced with any ICC sofic group with property (T), and A and B have different cardinalities.
This is joint work with Robin Tucker-Drob.