Events

We will compute the index of the deformation operator of a BPS monopole on R^3. During this computation, we will encounter how classical index results fall short in the presence of symmetry breaking. A hybridization of the b-calculus and scattering calculus will be the remedy to bypass the symmetry breaking and perform the computation.

MC 5417

We discuss several noncommutative generalizations of the classical Wasserstein distance of two probability measures on R^d, or more general metric spaces, which measures the minimal L^2 distance of two random variables with the given distributions. The first motivation comes from free probability theory, where ideas from optimal transport theory have been used to show isomorphism of von Neumann algebras and appear in functional inequalities with free entropy, as well as providing a general notion of "perturbing generators" of a von Neumann algebra. The second motivation comes from quantum information theory, where optimal transport theory provides a way to assign distances for states on a von Neumann algebra. As we will see, Duvenhage's version of quadratic Wasserstein distance relates quite closely to the Biane and Voiculescu's free Wasserstein distance. We discuss these distances in a common framework and demonstrate several properties that make the noncommutative setting more challenging, including non-separability of the Wasserstein space, and a stark difference between the Wasserstein and weak-* topologies on certain state spaces.

MC 5403

We will continue working through Arulseelan and Hanson's recent paper on AW*-algebras.

MC 5301

We will begin with a proof of the Ash-Knight-Manasse-Slaman and Chisholm theorem. We will then discuss some properties of effective bi-interpretability, and start the proof of the first jump inversion theorem for structures. If time permits, we will also show several consequences of the theorem.

MC 5403

There is an interesting interplay between model theory and topological dynamics. Newelski suggested using the machinery of topological dynamics to study definable groups and the related dynamics on spaces of types. For example, he proposed a conjecture stating that, under suitable model-theoretic assumptions, the Ellis group of a flow should be isomorphic to the compact group (G/G^{00}). This conjecture was confirmed for definably amenable groups in NIP by Chernikov and Simon (2018), but was also refuted in NIP for SL(2,R) by Gismatullin, Penazzi and Pillay (2015). Krupiński and Pillay formulated a weaker version, the so-called revised Newelski conjecture, stating that, under NIP, the Ellis group associated with definable group dynamics has a Hausdorff (\tau)-topology. This revised conjecture was confirmed by Chernikov, Gannon and Krupiński for definable group dynamics in a metrizable setting. Basso and Zucker then extended this result to the non-metrizable setting via forcing; in fact, they generalized some parts of Glasner’s structural results.

Krupiński and I observed that an abstract variant of Newelski’s original conjecture follows from Glasner’s results for amenable, tame, minimal and metrizable flows. Our goal was then to remove the metrizability assumption, as had been done in the case of the revised conjecture. This led to results on the amenability of the Ellis semigroup and on the measurable structure of amenable minimal tame flows. These results will be presented in two papers.

The first is already available on arXiv: https://arxiv.org/abs/2607.24381.

MC 5403

A projective toric variety \(X\Sigma\) comes with a moment map \(\mu\), whose image is a convex polytope that encodes a lot of the geometry of \(X\Sigma\). These moment maps and their associated polytopes have played a fundamental role in quantization programs and mirror symmetry. One natural question when studying these maps is whether the image \(\mu(X)\) also encodes geometric information of a subvariety \(X\subset X_\Sigma\). In this talk, we introduce combinatorial objects that let us study this image, called the (compactified) amoeba of \(X\), in the case where \(X\) is a linear subspace of projective space.

MC 5403

The Hilbert-Kamke problem is a generalization of Waring’s problem to a system of equations, in which the sums of the first n powers of the same variables are given. By the circle method, the existence of solutions depends on a positive lower bound for the singular series. In this talk, we study this problem over a number field. We set up the circle method, following Siegel and Wang, and write the singular series as a product of local densities. Under a necessary arithmetic condition, and with enough variables, we prove that the singular series is bounded below by a positive constant depending only on the number of equations and the degree of the field. The proof is given first for quadratic fields and then for every number field.

MC 5501