In this talk I will describe a class of minimization problems arising in modeling shape memory alloys.  I will start with a shape memory    material illustration, followed with a simple one dimensional model. Its extensions in 2D and 3D will help us understand the energetic mechanism behind the formation of patterns in physical experiments.

In this talk we'll consider a variational problem involving the Wasserstein distance between equal volume sets. After a short introduction to optimal transport theory and the Wasserstein metric, we will show that the maximizer of this particular problem is a ball, up to translations. This is a joint work with Almut Burchard and Davide Carazzato.

Integrable PDEs arise as leading-order models in an astoundingly wide range of physical scenarios, such as fluid dynamics, ferromagnetism, and nonlinear optics. For rapidly decaying solutions, the inverse scattering method provides a powerful tool for understanding their dynamics. However, due to the requirement of rapid decay, inverse scattering techniques cannot be used to analyze even the set of all finite energy solutions. In this talk, I will survey some recent work on the dynamics of solutions to integrable PDEs in translation-invariant spaces. In particular, I will discuss several optimal well-posedness results obtained in collaboration with Rowan Killip, Maria Ntekoume, and Monica Visan.

One of the strangest consequences of the Axiom of Choice is the following Hardin-Taylor 2008 result: there is a “predictor” such that for every function $f$ from the reals to the reals—even nowhere continuous $f$—the predictor applied to $f \restriction (-\infty,t)$ correctly predicts $f(t)$ for *almost every* $t \in R$. They asked how robust such a predictor could be, with respect to distortions in the time (input) axis; more precisely, for which subgroups $H$ of Homeo^+(R) do there exist $H$-invariant predictors? Bajpai-Velleman proved an affirmative answer when H=Affine^+(R), and a negative answer when H is (the subgroup generated by) C^\infty(R). They asked about the intermediate region; in particular, do there exist analytic-invariant predictors? We have partially answered that question: assuming the Continuum Hypothesis (CH), the answer is “no”. Regarding other subgroups of Homeo^+(R), we have affirmative answers that rely solely on topological group-theoretic properties of the subgroup. But these properties are very restrictive; e.g., all known positive examples are metabelian. So there remain many open questions. This is joint work with Aldi, Buffkin, Cline, Cody, Elpers, and Lee.

The talk is devoted to the global well-posedness questions of the Cauchy problem for the nonlocal nonlinear Schrödinger (NNLS) equation. It is known that the NNLS equation is integrable and it has soliton solutions with smooth initial profiles, which can have isolated finite time blow-up points. Our goal is to propose a suitable concept for continuation of local solutions of the general Cauchy problem (particularly, those admitting long-time soliton resolution) beyond possible singularities. The main tool is the inverse scattering transform method in the form of the Riemann-Hilbert problem combined with the PDE existence theory for nonlinear dispersive equations.

This talk is an introduction to supersymmetric quantum mechanics. We discuss the origin of this subject in quantum field theory and some interesting applications to other fields of mathematics and physics. We focus on the finite-dimensional case where everything can be formulated in terms of linear algebra. Our main applications are to a tantalizing framework relating graph theory, Lie theory and quantum computational complexity theory. 

Some cells, e.g., keratocytes, exhibit a spontaneous transition from rest to steady, persistent motion due to the myosin protein they contain. The 2D free-boundary Keller-Segel PDE model of this motion with Hele-Shaw boundary conditions was introduced by L. Berlyand, V. Rybalko, and A. Safsten. They found a family of asymmetric traveling wave solutions to this model bifurcating from a family of symmetric stationary solutions via direct pitchfork. Recently, physicists introduced nonlinear concentration dependent diffusion coefficient to this model to capture subcritical bifurcation that can be observed experimentally. We discuss challenges arising in active matter systems, such as non-self-adjointness of operators leading to novel spectral analysis techniques. We propose an analytical approach to capture different bifurcations in the nonlinear diffusion model, that is, we formulate a conjecture that describes the transition from supercritical to subcritical pitchfork depending on the diffusion coefficient. We derive the formal asymptotic of the travelling wave solutions for small velocities this conjecture.

The Tukey order finds its origins in the concept of Moore-Smith convergence in topology, and is especially important when restricted to ultrafilters with reverse inclusion. The Tukey order of ultrafilters over $\omega$ was studied intensively by Blass, Dobrinen, Isbell, Raghavan, Shelah, Todorcevic and many others, but still contains many fundamental unresolved problems. In the first part of this talk, I will present a recent development in the theory of the Tukey order restricted to ultrafilters on measurable cardinals, and explain how different the situation is when compared to ultrafilters on $\omega$. Moreover, we will see an important application to the Galvin property of ultrafilters.
In the second part of the talk we will demonstrate how ideas and intuition from ultrafilters over measurable cardinals lead to new results on the Tukey order restricted to ultrafilters over $\omega$. This is joint with Natasha Dobrinen.

In this talk, the rigorous periodic homogenization for a coupled system, which models a suspension of magnetizable rigid particles in a non-conducting carrier viscous Newtonian fluid is discussed. Both one-way and two-way coupling between the fluid and particles are considered. As the size of the particles approaches zero, it is shown that the suspension’s behavior is governed by a generalized homogenized magnetohydrodynamic system, whose parameters are explicitly derived. The two-scale convergence is utilized to justified obtained homogenized behavior of the original heterogeneous system. A number of stand-alone results that could be utilized in future contributions to the topic of periodic homogenization for nonlinear electrostatic and/or elastic composite materials under minimal regularity assumptions of the original system is also presented.

In this talk nonlocal operators with nonsingular integral terms will be discussed. Evolution and spectral problems for such operators arise, e.g., in models of population dynamics. We consider operators corresponding to strongly inhomogeneous environments with either periodic or locally periodic structure and study the asymptotic behavior of their spectra. Using viscosity solutions techniques we derive homogenized additive eigenvalue problem describing limit of the bottom point of the spectrum. In the periodic case, via a factorization trick  we obtain more precise two-term asymptotics. 

Given an uncountable cardinal k, the uncountable product Z^k of -many copies of the integers  consists of all sequences of integers of length k, where two sequences can be added together by adding corresponding components. The direct sum of k-many copies of Z is the subgroup of Z^k consisting of just those sequences which are equal to () in all but finitely many coordinates. So, in a sense, sequences in the direct sum are equal to () in all but a negligible set of coordinates. In set theory, there are many different notions of negligibility for subsets of an uncountable cardinal k other than finiteness, each of which corresponds to a particular proper subgroup of Z^k that contains the direct sum. In this talk we will discuss new techniques, involving homological algebra, for distinguishing such subgroups from the full product Z^k.

This is joint work with Chris Lambie-Hanson and Jan Saroch. 

Viscous liquid films coating the interior of a tube occur in a variety of applications. If the film is thick enough, it may pinch off and form a plug, occluding the tube. In this talk I will discuss recent work examining the impact of surfactant, viscoelasticity, and viscosity stratification on plug formation in a model for film flow.  Implications for understanding occlusion in human airways will be discussed, and potential applications of these results in more comprehensive models will be explored. 

The Aviles-Giga functional is a classical problem in the calculus of variations. The energy functional in 2D was used to model smectic liquid crystals and thin film blisters.

The Gamma-convergence of the Aviles-Giga functional remains a challenging open problem, which requires better understanding of the fine structure of the underlying function space, namely the space of \emph{finite entropy solutions} of the Eikonal equation. In this talk, I will give an introduction to the above mentioned problem, and present our recent progress towards understanding finite entropy solutions of the Eikonal equation in 2D. Our result supports the conjecture that entropy measures of finite entropy solutions concentrate on 1D sets, and completes the picture on the rigidity/flexibility threshold of the 2D Eikonal equation in the Besov space scaling. This is joint work with Xavier Lamy and Andrew Lorent.

One of the simplest examples of Lie group is given by motions of the ordinary Euclidean space, obtained by composing translations, rotations and reflections. In 1971, George Mackey observed a certain amount of resemblance between the (elementary) representation theory of such groups and that (more intricate) of general semisimple groups. The purpose of this talk will be to present various aspects of this "analogy", recent developments and connections with other problems. No previous knowledge of representation theory will be assumed.

Monotone rearrangement is a useful tool in investigation of extremal problems on various functional classes. It is known that the monotone rearrangement does not increase the BMO-norm of a function defined on an interval, and the $A_p$-characteristic of a Muckenhoupt weight on an interval. Moreover, the same result holds for more general classes of functions that are defined in terms of averages. The proof uses the scale invariance of the norm (or characteristic) on such classes of functions.
In a recent paper joint with Leonid Slavin, we proved that the monotone rearrangement does not increase the Campanato-type norm of a VMO-function defined on an interval. The main difficulty and difference with the mentioned above results is that the Campanato-type norm is not scale invariant. The result has immediate applications in a Bellman function theory for extremal problems on VMO.

Networks are primarily based on Backpropagation and (Stochastic) Gradient Descent variations like ADAM. They require  differentiable Loss Functions and struggle in handling vanishing or exploding gradients. Some "tricks" have been introduced, such as dropout, batch normalization, layer normalization, gradient clipping, etc. but they remain mysterious from a mathematical perspective. The goal of the talk is to introduce a few ideas about Random Search optimization, which does not need derivatives, and to explore its potential to function as an alternative (or support) to canonical optimization algorithms for deep networks. I will discuss a simple random algorithm as an example.

Monotone rearrangement is a useful tool in investigation of extremal problems on various functional classes. It is known that the monotone rearrangement does not increase the BMO-norm of a function defined on an interval, and the $A_p$-characteristic of a Muckenhoupt weight on an interval. Moreover, the same result holds for more general classes of functions that are defined in terms of averages. The proof uses the scale invariance of the norm (or characteristic) on such classes of functions.