Support varieties for Hopf algebras (and more general tensor categories) give a way of associating geometry to finite-dimensional modules. The support variety of a module is empty if and only if the module is projective. We give a method for extending a support variety theory from the finite-dimensional modules to the infinite-dimensional ones, and give conditions under which the theory still detects projectivity. This talk will include joint work with Nakano—Yakimov and with Cai.

A countable discrete group is called Choquet-Deny if for every non-degenerate probability measure on the group, the corresponding space of bounded harmonic functions is trivial. Recently a complete characterization of Choquet-Deny groups was obtained by Frisch, Hartman, Tamuz and Ferdowsi. In this talk, we will look at the extension of the Choquet-Deny property to the framework of discrete measured groupoids. Our main result gives a complete characterisation of this property in terms of the associated measured equivalence relation and the isotropy groups of the groupoid. This talk is based on a joint work with Tey Berendschot, Milan Donvil, Mario Klisse and Se-Jin Kim.

Nonlinearly Partitioned Runge--Kutta (NPRK) methods are a newly proposed class of time integration schemes which target differential equations in which different scales, stiffnesses or physics are coupled in a nonlinear way. In this talk, I will provide a broad overview of this new class of methods. First, I will motivate these methods as a nonlinear generalization of classical Runge--Kutta (RK) and Additive Runge--Kutta (ARK) methods. Subsequently, I will discuss order conditions for NPRK methods; we obtain the complete order conditions using an edge-colored rooted tree framework. Interestingly, NPRK methods have nonlinear order conditions which have no classical additive counterpart. We will show how these nonlinear order conditions can be used to obtain embedded estimates of state-dependent nonlinear coupling strength and present a numerical example to demonstrate these embedded estimates. I will then discuss how these methods yield efficient semi-implicit time integration of numerical partial differential equations; numerical examples from radiation hydrodynamics will be presented. Finally, I will discuss our recent work on multirate NPRK methods, which target problems with nonlinearly coupled processes occurring on different timescales. We will discuss properties of these multirate methods such as timescale coupling, stability and efficiency, and conclude with several numerical examples, such as a fast-reaction viscous Burgers’ equation and the thermal radiation diffusion equations.

The Riemann Mapping Theorem is a fundamental result in classical complex analysis in one variable: If $\Omega\subset \mathbb C$ is a simply connected domain, $\Omega\neq \mathbb C$, then there is a biholomorphic map $F\colon \Omega\to\mathbb D:=\{|z|<1\}$. One of the first things we teach students in several complex variables is that the analogous fails miserably for domains in $\mathbb C^n$ for $n\geq 2$, as was already discovered by Poincaré; There is no biholomorphic map from the bidisk $\mathbb D^2:=\{(z_1,z_2)\colon |z_1|<1, |z_2|<1\}$ to the unit ball $\mathbb B^2=\{|z_1|^2+|z_2|^2<1\}$. There are clearly no topological obstructions to the existence, which is essentially the only obstruction to a Riemann map in one variable (but what about $\Omega\neq \mathbb C$?). As a first reaction, one might then give up and exclaim "if this example doesn't work, there is no hope for a reasonable Riemann Mapping Theorem in higher dimensions". Well, I intend to convince the audience that one would be wrong, and one would then miss an extremely rich theory that blends real and complex geometry, partial differential equations, and, of course, real and complex analysis.

The continuous random energy model (CREM) is a Gaussian process indexed by a binary tree of depth T, introduced by Derrida and Spohn (1988) and Bovier and Kurkova (2004) as a toy model of a spin glass. In this talk, I will present recent results on hardness thresholds for algorithms that search for low-energy states. I will first discuss the existence of an algorithmic hardness threshold x_*: finding a state of energy lower than -x T is possible in polynomial time if x < x_*, and takes exponential time if x > x_*, with high probability. I will then focus on the transition from polynomial to exponential complexity near the algorithmic hardness threshold, by studying the performance of a certain beam-search algorithm of beam width N depending on T — we believe this algorithm to be natural and asymptotically optimal. The algorithm turns out to be essentially equivalent to the time-inhomogeneous version of the so-called N-particle branching Brownian motion (N-BBM), which has seen a lot of interest in the last two decades. Studying the performance of the algorithm then amounts to investigating the maximal displacement at time T of the time-inhomogeneous N-BBM. In doing so, we are able to quantify precisely the nature of the transition from polynomial to exponential complexity, proving that the transition happens when the log-complexity is of the order of the cube root of T. This result appears to be the first of its kind and we believe this phenomenon to be universal in a certain sense.

We show that the Bergman metric on ball quotients $\mathbb{B}^2/\Gamma$ is Kähler-Einstein if and only if $\Gamma$ is trivial, leading to a characterization of the unit ball among certain two-dimensional Stein spaces, confirming a version of Cheng’s conjecture. We also relate the boundary type of two-dimensional Stein spaces to the local algebraic degree of their Bergman kernel, characterizing ball quotients via the local rationality of the Bergman kernel. Finally, we derive the rotational symmetry properties of certain domains in $\mathbb{C}^n$ from the orthogonality of holomorphic monomials in their Bergman spaces.

Let  M  be a smooth real codimension two compact submanifold in a Stein manifold. We will prove the following theorem: Suppose that  M  has two elliptic complex tangents and that CR points are non-minimal. Assume further that  M  is contained in a bounded strongly pseudoconvex domain. Then  M  bounds a unique smoothly up to  M  Levi-flat hypersurface  $\widehat{M}$  that is foliated by Stein hyper-surfaces diffeomorphic to the ball. Moreover,  $\widehat{M}$  is the hull of holomorphy of M . This subject has a long history of investigation dating back to E. Bishop and Harvey-Lawson. I will discuss both the historical context and the techniques used in the proof of the aforementioned theorem.

The regulation and maintenance of a tissue’s shape and structure is a major outstanding question in developmental biology and plant biology. In this talk, through iterations between experiments and multiscale model simulations that include a mechanistic description of interkinetic nuclear migration, we will show that the local curvature, height, and nuclear positioning of cells in the Drosophila wing imaginal disc are defined by the concurrent patterning of actomyosin contractility, cell-ECM adhesion, ECM stiffness, and interfacial membrane tension. The biologically calibrated model describing both tissue growth and morphogenesis incorporates the spatial patterning of fundamental subcellular properties. Additionally, the model implements for the first time the dynamics of interkinetic nuclear migration within the simulated pseudostratified epithelium. This includes the basal to apical motion of the nucleus, mitotic rounding, and cell division dynamics. Key characteristics of global tissue architecture, such as the local curvature of the basal wing disc epithelium, cell height, and nuclear positioning, serve as metrics for model calibration. The experiments have shown how these physical features are jointly regulated through spatiotemporal dynamics in the localization of pMyoII, β-Integrin, and ECM stiffness. As the disc grows, there are progressive changes in the patterning of key subcellular features such as actomyosin contractility. The predictions made by the model simulations agree with the observed changes in contractility and cell-ECM adhesion during wing disc morphogenesis. Multiscale modeling approach combined with experiments was also applied to studying stem cell maintenance in multilayered shoot apical meristems (SAMs) of plants which requires strict regulation of cell growth and division. In this talk, the combined approach will be demonstrated through testing three hypothesized mechanisms for the regulation of cell division plane orientation and the direction of anisotropic cell expansion in the corpus.

A fundamental task in linear algebra is that of trace estimation: Suppose we have a PSD matrix A that can be accessed only by matrix-vector products. Then, with as few matrix-vector products as possible, estimate the trace of A to relative error with high probability. This is an essential subroutine in all sorts of applications, for instance in efficiently estimating the log-determinant of a matrix.

In the first part of the talk, I'll rigorously introduce this problem, the prior state-of-the-art algorithm (the Girard-Hutchinson Estimator), and our improvement upon it (the Hutch++ Estimator), which we show to have asymptotically optimal matrix-vector complexity. In the second part of the talk, I'll introduce a Kronecker-structured variant of this problem with applications for tensorized data, alongside the only known algorithm that solves this problem.

However, we'll see that this algorithm converges very slowly. We will show this is a result of this Kronecker-structured computational model, which elicits strange computational properties. We will see that good design decisions in the non-Kronecker case can cause catastrophic failure in the Kronecker case, that using complex random variables leads to exponential speedups over reals, and that subgaussianity does not suffice to understand the performance of randomized algorithms here.

Joint work with Haim Avron, David Woodruff, and William Swartworth.

I will discuss some recent results about relative entropies in QFT, with particular emphasis on the singular limits of such entropies.

We give an explicit geometric computation of the quantum K rings of symplectic Grassmannians of lines, which are deformations of their Grothendieck rings of vector bundles and refinements of their quantum cohomology rings. We prove that their Schubert structure constants have signs that alternate with codimension (just like in the Grothendieck ring) and vanish for degrees at least 3. We also give closed formulas that characterize the multiplicative structure of these rings. This is based on joint work with V. Benedetti and N. Perrin.