In this talk, we consider operator-valued polynomials in Gaussian Unitary Ensemble random matrices. I will explain a new strategy to bound its $L^p$-norm, up to an asymptotically small error, by the operator norm of the same polynomial evaluated in free semicircular variables as long as $p=o(N^{2/3})$. As a consequence, if the coefficients are $M$-dimensional matrices with $M=exp(o(N^{2/3}))$, then the operator norm of this polynomial converges towards the one of its free counterpart. In particular this provides another proof of the Peterson-Thom conjecture thanks to the result of Ben Hayes. The approach that we take in this paper is based on an asymptotic expansion obtained in a previous paper combined with a new result of independent interest on the norm of the composition of the multiplication operator and a permutation operator acting on a tensor of $C^*$-algebras.

By the way, can I assume that the people attending the seminar will be familiar with notions of free probability?

Binary search trees (BSTs) are a fundamental data structure, optimized for data retrieval, entry, and deletion as needed for priority queue tasks. A fundamental statistic that controls each of these operations is the height of the tree with $n$ nodes, $h_n$, which returns the maximal depth of a node within the tree. Devroye established the height of a random BST, generated using a uniform permutation of length $n$, has height that limits to $c^*\approx 4.311$ when scaled by $\log n$. We are interested in studying the  heights of random block BSTs, $h_{n,m}$, which correspond to uniform permutations combined using Kronecker or wreath products of permutations of lengths $n$ and $m$, that thus arise naturally in the setting of a BST data structure implemented using parallel architecture. We show using one such product of two permutations suffices to increase the asymptotic height of a random BST, while maintaining a logarithmic scaling with respect to the length of the generated permutation. We then explore the question of how much can the height increase when repeatedly using such products. These butterfly trees correspond to block BSTs formed using uniform butterfly permutations, that include a particular 2-Sylow subgroup of the symmetric group of $N = 2^n$ objects formed by taking $n$-iterated wreath products of $S_2$. In this setting, we show the expected heights for the corresponding block BSTs are now polynomial, with a lower bound of $N^\alpha$ for $\alpha \approx 0.585$. We provide exact nonasymptotic and asymptotic distributional descriptions for the case of simple butterfly permutations, which also include connections to other well-studied permutations statistics (e.g., the longest increasing subsequence, number of cycles, left-to-right maxima), while for nonsimple butterfly permutations we provide power-law bounds on the expected heights. This project is joint with Chenyang Zhong.

Nowadays prismatic crystals are gathering an increasing interest as they unify various coefficients in $p$-adic cohomology theories. Recently, attached to any $p$-adic formal scheme $X$, Drinfeld and Bhatt-Lurie constructed certain ring stacks, including the prismatization of $X$, on which quasi-coherent complexes correspond to various crystals on the prismatic site of $X$. While such a stacky approach sheds some new light on studying prismatic crystals, little is known outside of the Hodge-Tate locus. In this talk, we will introduce our recent work on studying quasi-coherent complexes on the prismatization of $X$ via various charts.

[pre-talk at 3:00PM]

A dilation surface is roughly a surface made up of polygons in the complex plane with parallel sides glued together by complex linear maps. The action of $\mathrm{SL}(2, \mathbb R)$ on the plane induces an action of $\mathrm{SL}(2, \mathbb R)$ on the collection of all dilation surfaces, aka the moduli space, which possesses a natural manifold structure. As with translation surfaces, which can be identified with holomorphic $1$-forms on Riemann surfaces, dilation surfaces can be thought of as “twisted” holomorphic $1$-forms.

The first result that I will present, joint with Nick Salter, produces an $\mathrm{SL}(2, \mathbb R)$-invariant measure on the moduli space of dilation surfaces that is mutually absolutely continuous with respect to Lebesgue measure. The construction fundamentally uses the group cohomology of the mapping class group with coefficients in the homology of the surface. It also relies on joint work with Matt Bainbridge and Jane Wang, showing that the moduli space of dilation surfaces is a $K(\pi,1)$ where $\pi$ is the framed mapping class group. 

The second result that I will present, joint with Bainbridge and Wang, is that an open and dense set of dilation surfaces have Morse-Smale dynamics, i.e. the horizontal straight lines spiral towards a finite set of simple closed curves in their forward and backward direction. A consequence is that any fully supported $\mathrm{SL}(2, \mathbb R)$ invariant measure on the moduli space of dilation surfaces cannot be a finite measure.

In 2003, P. Biane characterized a free semi-circular system in terms of free Poincaré inequality, which is an inequality related to the non-commutative L^2-norm of free difference quotients. In this talk, we will generalize his result to $B$-valued semi-circular system using a “natural” $B$-valued free Poincaré inequality. If time permits, we will also give a counterexample to Voiculescu’s conjecture related to $B$-valued free Poincaré inequality.

This talk will focus on systems-theoretic and control theory tools that help characterize the responses of nonlinear systems to external inputs, with an emphasis on how network structure “motifs” introduce constraints on finite-time, transient behaviors.  Of interest are qualitative features that are unique to nonlinear systems, such as non-harmonic responses to periodic inputs or the invariance to input symmetries. These properties play a key role as tools for model discrimination and reverse engineering in systems biology, as well as in characterizing robustness to disturbances. Our research has been largely motivated by biological problems at all scales, from the molecular (e.g., extracellular ligands affecting signaling and gene networks), to cell populations (e.g., resistance to chemotherapy due to systemic interactions between the immune system and tumors; drug-induced mutations; sensed external molecules triggering activations of specific neurons in worms), to interactions of individuals (e.g., periodic or single-shot non-pharmaceutical “social distancing'” interventions for epidemic control). Subject to time constraints, we'll briefly discuss some of these applications.

Many data-driven problems in the modern world involve solving nonconvex optimization problems. The large-scale nature of many of these problems also necessitates the use of first-order optimization methods, i.e., methods that rely only on the gradient information, for computational purposes. But the first-order optimization methods, which include the prototypical gradient descent algorithm, face a major hurdle for nonconvex optimization: since the gradient of a function vanishes at a saddle point, first-order methods can potentially get stuck at the saddle points of the objective function. And while recent works have established that the gradient descent algorithm almost surely escapes the saddle points under some mild conditions, there remains a concern that first-order methods can spend an inordinate amount of time in the saddle neighborhoods. It is in this regard that we revisit the behavior of the gradient descent trajectories within the saddle neighborhoods and ask whether it is possible to provide necessary and sufficient conditions under which these trajectories escape the saddle neighborhoods in linear time. The ensuing analysis relies on precise approximations of the discrete gradient descent trajectories in terms of the spectrum of the Hessian at the saddle point, and it leads to a simple boundary condition that can be readily checked to ensure the gradient descent method escapes the saddle neighborhoods of a class of nonconvex functions in linear time. The boundary condition check also leads to the development of a simple variant of the vanilla gradient descent method, termed Curvature Conditioned Regularized Gradient Descent (CCRGD). The talk concludes with a convergence analysis of the CCRGD algorithm, which includes its rate of convergence to a local minimum of a class of nonconvex optimization problems.

For a smooth complex projective variety, the Lefschetz standard conjectures of Grothendieck predict the existence of algebraic self-correspondences that provide inverses to the hard Lefschetz isomorphisms. These conjectures have broad implications for Hodge theory and the theory of motives. In this talk, we describe recent progress on the Lefschetz standard conjectures for hyper-Kähler varieties of generalized Kummer deformation type.