# Faculty Activities

*Abstract:*

**Advisor:** Naama Brenner

**Abstract: **Cellular networks exhibit pre-designed responses to many challenges, but also endow the cell with the ability to adapt and display novel phenotypes in the face of unforeseen challenges. In this seminar we will present a computational framework which attempts to describe such plasticity in random networks. We show that the convergence of this exploration in the high-dimensional space of network connections depends crucially on network topology. For large networks, convergence is most efficient for networks with scale-free out going degree distributions which are typical of cellular networks.

In order to investigate the dynamics and convergence properties of such networks we develop an approximation for scale-free networks, the *STAR network,* which is based on the crucial role hubs play in network dynamics.

We show that *STAR networks* retain many of the properties of scale-free networks and enable analytical understanding of the convergence properties exhibited in our model.

*Abstract:*

A group invariant is called profinite if it concides for groups with the same finite quotients. We discuss the (non-)profiniteness of Euler characteristic and other invariants. This is based on joint work with Holger Kammeyer, Steffen Kionke and Jean Raimbault.

*Abstract:*

T.B.A.

*Abstract:*

Let f: R^n \to R^n be a Sobolev map. Suppose that the k-minors of df are smooth. What can we say about the regularity of f? This question arises naturally in the context of Liouville's theorem, which states that every weakly conformal map is smooth. I will explain the connection of the minors question to the conformal regularity problem, and describe a regularity result for maps with regular minors. If time permits, I will discuss these questions in the context of mappings between Riemannian manifolds.

*Abstract:*

**Abstract: **The notion of roots is absolutely central to Lie theory and, in its classical version, very much tied to groups generated by reflections. My goal in this talk it to explore how this notion may be broadened to incorporate what is currently happening on the frontiers of Lie theory. One interesting new phenomenon, which I am hoping to discuss, is the possibility that roots and dual roots live in lattices of different rank. This makes the Langlands-like duality that exchanges them a particularly dramatic operation.

*Light refreshments on the 8^{th} floor (faculty lounge) at 16:30*

*Abstract:*

Since its establishment, the field of interpolation theory (and interpolation spaces) has proved to be a very useful tool in several branches of Analysis. Crudely speaking, interpolation theory concerns itself with the finding of "intermediate" spaces $X$ lying "between" two given Banach spaces, $X_0$ and $X_1$, with the property that every linear operator which is bounded on both $X_0$ and $X_1$ is also bounded on $X$. One fundamental result in the early history of this field was the work of Riesz and Thorin who showed that for any numbers $p<q<r$, the space $L^q$ is an interpolation space "between" $L^p$ and $L^r$. Later, in 1958, Stein and Weiss extended this result to cases where the various $L^p$ spaces are weighted, with possibly different weights, so that it also makes sense to consider the case where $p=q=r$. It is natural to ask whether analogues of the Stein-Weiss results hold when, instead of weighted $L^p$ spaces, one considers weighted Sobolev spaces on $R^n$. Indeed some authors have obtained such analogous results for special choices of weight functions via Fourier transforms and multiplier theorems. In our talk we will show that, at least when $p=q=r$, the Calderon complex interpolation method enables us to obtain such results for a large class of weight functions which apparently cannot be treated by Fourier methods. We will also briefly discuss how these results can be helpful in the study of time asymptotic behaviour of solutions to evolution equations. This talk is based on joint work with Michael Cwikel. Details can be found in our paper on the arXiv, which is to appear in J. Funct. Analysis.

*Abstract:*

We discuss Lie-type algebraic operations - brackets, cobrackets, and double brackets - in the module generated by free homotopy classes of loops in a surface. This subject was initially inspired by the study of the Atiyah-Bott Poisson brackets on the moduli spaces of surfaces. Recently, the algebraic operations on loops were related to the Kashiwara-Vergne equations on automorphisms of free Lie algebras.

Lecture 1: Monday, March 25, 2019 at 15:30

Lecture 2: Wednesday, March 27, 2019 at 15:30

Lecture 3: Thursday, March 28, 2019 at 15:30

*Abstract:*

T.B.A.

*Abstract:*

The prediction of interactions between nonlinear laser beams is a longstanding open problem. A traditional assumption is that these interactions are deterministic. We have shown, however, that in the nonlinear Schrodinger equation (NLS) model of laser propagation, beams lose their initial phase information in the presence of input noise. Thus, the interactions between beams become unpredictable as well. Not all is lost, however. The statistics of many interactions are predictable by a universal model.

Computationally, the universal model is efficiently solved using a novel spline-based stochastic computational method. Our algorithm efficiently estimates probability density functions (PDF) that result from differential equations with random input. This is a new and general problem in numerical uncertainty-quantification (UQ), which leads to surprising results and analysis at the intersection of probability and approximation theory.

*Abstract:*

A single round soap bubble provides the least-perimeter way to enclose a given volume of air, as was proved by Schwarz in 1884. The Double Bubble Problem seeks the least-perimeter way to enclose and separate two given volumes of air. Three friends and I solved the problem in Euclidean space in 2000. In the latest chapter, Emanuel Milman and Joe Neeman recently solved the problem in Gauss space (Euclidean space with Gaussian density). The history includes results in various spaces and dimensions, some by undergraduates. Many open questions remain.

*Abstract:*

T.B.A.

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T.B.A.

*Abstract:*

T.B.A.