Seminars from until

What really happens when we shuffle a deck of cards — and what mathematics lies behind this seemingly simple act?

Persi Diaconis explores the mathematics of three ways that real people shuffle real cards.

Remarkably, these three methods call for three quite different kinds of mathematics.

Riffle shuffling leads to elementary combinatorics; overhand shuffling involves sophisticated coupling methods; and smooshing leads all the way to stochastic partial differential equations.

Behind these familiar ways of mixing cards lie challenging questions in analysis and probability.

Yet the problems themselves are easy to understand, making them an engaging gateway to surprisingly deep mathematics.

Along the way, the course will reveal connections to card tricks, gambling, probability, and other areas of mathematics.

In the study of topological symmetry, defects are embedded submanifolds labelled by algebraic data; the evaluation of defect TQFTs (i.e., TQFTs which make sense for bordisms supporting defects) implements topological symmetry. The (2+1) TQFT of Reshetikhin and Turaev can be enhanced to a defect TQFT, using a construction called orbifolding, due to work of Carqueville–Runkel–Schaumann. This construction is a generalized state sum, and requires that the defects are labelled by semisimple data; the state spaces for surfaces are given by a projector from the state space of the original RT theory. On the other hand, given a 3-manifold whose boundary is a surface with defects, the usual definitions of skein theory can be extended to define a defect skein module. In joint work with Ingo Runkel, we define the defect skein module and prove that it is isomorphic to the defect RT state space of its boundary. This generalizes the well-known fact that the state spaces of RT theory are skein modules. Our work points to non-semisimple generalizations of defect RT theories using skein methods, while in the other direction it gives a state sum construction for defect skein modules labelled by semisimple data.

In this talk, we consider partial differential equations of parabolic $p$-Laplace type in noncylindrical domains $U = \bigcup_{t \in [0,T]} \{t\} \times \Omega_t$, where the underlying spatial set $\Omega_t \subset \mathbb{R}^n$ may change in time. Our goal is to characterize time-dependent boundary values that yield the existence of a solution, focusing on domains $U$ that satisfy a Lipschitz condition with respect to a parabolic metric. In particular, we discuss the extension to $U$ of certain functions that are only defined on the parabolic boundary of $U$, and conversely the trace space for functions defined on $U$ and contained in a suitable parabolic function space. The talk is based on joint work with Daniel Campbell and Sebastian Schwarzacher.

The goal of this talk is to present the construction of new families of complete Calabi-Yau metrics with maximal volume growth on the small resolutions of the ordinary double point singularity in dimension 3. These metrics have tangent cone $\mathbb{C} ×(\mathbb{C}^2/\mathbb{Z}_2)$ at infinity and are parametrised by their Kähler class. As the Kähler class degenerates, the metrics converge in the pointed Gromov-Hausdorff sense to a singular Calabi-Yau metric with an isolated conical singularity modelled on the Stenzel metric at the ordinary double point, thereby providing a new metric realisation of the Atiyah flop. As we will explain, this construction is meant to provide a local adiabatic model for certain degenerations of compact Calabi-Yau manifolds.

Convex extensions are natural ways to extend functions defined on the extreme points of a convex set to the whole set. They rely on Choquet theory, which aims at representing points of convex sets by probability measures over their extreme points.

We focus on the case of a product of convex sets, where we find a correspondence between convex extensions and Kantorovich optimal transport problems. In particular, "classical" Kantorovich costs between probability measures are recovered as convex extensions in the simplex case, and entanglement-free quantum Wasserstein distances in the (non-simplex) case of a quantum state space. Conversely, convex extensions can themselves be recovered as suitable quotients of classical Kantorovich costs.

This correspondence provides a geometric interpretation of optimal transport and allows for transferring tools between this field and the theory of convex extensions.

Baxter's classical theorem for orthogonal polynomials on the unit circle establishes that the linear Fourier coefficients of a measure are in $\ell^1$ if and only if the nonlinear coefficients, i.e., the so-called Verblunsky coefficients, are in $\ell^1$. We prove an analog of Baxter's theorem for the SU(2)-valued nonlinear Fourier transform. The proof of Baxter's theorem is qualitative in nature. We explore possible formulations of such quantitative theorems with norm and Lipschitz estimates. In particular, we construct a counterexample for certain such estimates