The border rank of tensors is a widely studied topic with practical applications to theoretical computer science and algebraic statistics. New lower bounds were obtained for the matrix multiplication tensor using techniques from representation theory and algebraic geometry. In this talk, we will prove non-trivial border rank lower bounds for a class of GL(V)-invariant tensors using Young flattenings. We will see how this comes down to proving results on ranks of certain maps between schur functors, the proofs of which surprisingly uses deep results in representation theory and commutative algebra.

Let $T: X\to X$ be a continuous map, where $X$ is a topological space. Fix also a family $\mathsf{I}\subseteq 𝒫(\mathbb{N})$ of "small" sets of nonnegative integers (for instance, the family $\mathrm{Fin}$ of finite sets, or the family $\mathsf{Z}$ of asymptotic density zero sets). A point $x \in X$ is said to be $\mathsf{I}$-hypercyclic if 
$$
\{n \in \mathbb{N}: T^nx \in U\}\notin \mathsf{I}
$$
for each nonempty open $U\subseteq X$. On a similar direction, a point $x \in X$ is said to be $\mathsf{I}$-strong hypercyclic if for each $y \in X$ there exists a subsequence $(T^nx: n \in A)$ of its orbit which is convergent to $y$ and, in addition, the set of indexes $A$ is \textquotedblleft not small,\textquotedblright\, that is, $A\notin \mathsf{I}$. In both cases, if $\mathsf{I}=\mathrm{Fin}$ and $X$ is a sufficiently nice topological space, then $x$ is $\mathsf{I}$-hypercyclic iff it is $\mathsf{I}$-strong hypercyclic iff its orbit is dense. 

We provide several structural relationships between the above notions and the related ones of recurrence with respect to $\mathsf{I}$. None of our results relies on the full linearity of $T$. As applications, we show that if $T$ is a homomorphism on a Fréchet space $X$ and there exists a dense set of vectors with orbits convergent to $0$, then for each $y \in X$ the set of all vectors $x \in X$ such that $\lim_{n \in A}T^nx=y$ for some $A\notin \mathsf{Z}$ is either empty or comeager. In a special case of bounded linear operators on Banach spaces, we obtain that $T$ is $\mathsf{Z}$-hypercyclic if and only if there exists a hypercyclic vector $x \in X$ for which $\lim_{n \in A}T^nx=0$ for some $A\notin \mathsf{Z}$. We conclude with several open questions. 

Reduced pipe dreams are combinatorial objects that encode some of the algebraic, enumerative, geometric, and probabilistic properties of Schubert and Grothendieck polynomials. They were introduced in 1993 by Bergeron and Billey, who showed that the set of all reduced pipe dreams for a fixed permutation w has a natural poset structure, with covering relations given by simple local operations called chute and ladder moves. In 2011, Rubey generalized chute and ladder moves on the set of reduced pipe dreams for a permutation w, and he conjectured that the induced poset on reduced pipe dreams is a lattice. We prove this conjecture and give simple recursive formulas for joins and meets in Rubey's lattice. This talk is based on joint work with Sara Billey and Clare Minnerath.

The fractional GJMS operators form a one-parameter family of conformally invariant operators defined on the conformal infinity of a Poincare-Einstein manifold. We mainly focus on operators of order between 0 and 2. First, I will show that the associated fractional Yamabe constants provide lower bounds for the relative volume of geodesic balls in the interior. Then, I will present some monotonicity properties for this family of fractional Yamabe constants. Finally, I will introduce some recent progress on the positive mass theorem for these fractional GJMS operators. This is joint work with Huihuang Zhou.

Graph rigidity is one of the most classical subjects in graph theory, studying geometric properties of graphs. Formally, a graph $G=(V,E)$ is $d$-rigid if a generic embedding of its vertex set $V$ into $R^d$ is rigid, namely, every continuous motion of its vertices preserving the lengths of the edges of $G$ necessarily preserves all pairwise distances between the vertices of $G$.

We develop a new sufficient condition for d-rigidity, formulated in graph theoretic terms. This condition allows us to obtain several newresults about rigidity of random graphs. In particular, we argue that for edge probability $p>2\ln(n)/n$, a random graph $G(n,p)$ is with high probability (whp) $cnp$-rigid, for $c>0$ being an absolute constant. We also show that a random r-regular graph $G_{n,r}$, $r>=3$, is whp $cr$-rigid. Another consequence is a sufficient condition for $d$-rigidity based on the minimum co-degree of the graph.   

The talk should be accessible to a general graph theoretic audience, no previous experience (whether positive or negative) with graph rigidity will be assumed.
 

A joint work with Alan Lew and Peleg Michaeli.

Optimizing intrathecal drug delivery procedures requires a deeper understanding of flow and transport in the spinal canal. Numerical modeling of drug dispersion is challenging because of the strong separation of time scales: dispersion occurs over approximately one hour, whereas cerebrospinal fluid pulsations driven by cardiac motion occur on a time scale of about one second. Patient-specific predictions in clinical settings therefore call for simplified descriptions that focus on dispersion time scales while bypassing the rapid concentration oscillations induced by cyclic motion. We show how asymptotic methods that exploit this separation of time scales can be used to derive a reduced transport equation in which convective transport driven by the mean Lagrangian drift governs drug dispersion. The model is validated through comparisons with MRI-informed direct numerical simulations of drug dispersion in a cervical spinal canal geometry that includes nerve rootlets and denticulate ligaments. These comparisons demonstrate that the reduced model accurately captures drug transport while enabling dispersion predictions at a fraction of the computational cost required by direct numerical simulations.

Random matrix theory is a very broad and well-developed research area at the intersection of physics, statistics, probability and combinatorics (and arguably others). Applications range from numerical analysis to engineering, to biology, economics, signal processing and machine learning. Classically, the type of matrices that have been studied have certain invariance properties (e.g., orthogonal), and therefore are mostly dense; however, the last decade has been marked by a tremendous increase in sparse applications, particularly related to the ubiquity of sparse networks and graphs. This, in turn, has led to rapid development of sparse random matrix theory. Some spectral properties (eigenstatistics) of sparse matrices turn out to match the dense ones, but others generate new and interesting phenomena. I will provide a high-level perspective on this rapidly evolving field, and describe some applications of interest.

Quantum information is encoded in a state vector of a tensor product of Hilbert spaces. Quantum error correction codes are useful for correcting errors when transporting quantum information through a noisy channel. In this talk, we will discuss a family of 2D "local topological" quantum error correction codes which use the robustness of topology to deformation to protect quantum information. We will then explain how operator algebra and tensor category techniques can be used to analyze quasi-particle excitations called anyons.

The proximal bundle method (PBM) is a powerful and widely used approach for minimizing nonsmooth convex functions. For smooth objectives, its best-known convergence rate has remained suboptimal. We present the first accelerated proximal bundle method to achieve the optimal O(1/sqrt(epsilon)) iteration complexity. The proposed method is conceptually simple, differing from Nesterov's accelerated gradient descent by a single line, and preserving the key structural properties of the classical PBM. As a result, we obtain an accelerated algorithm with much broader stepsize selection than what is allotted by accelerated gradient descent. The talk will include many pictures and numerical simulations to motivate algorithm design and illustrate fast convergence, respectively.