# PDE and Applied Mathematics Seminar[ Edit ]

## Moderator: Amy Novick-Cohen

*Abstract:*

Special MSc Seminar

The Laplacian eigenvalue problem on a bounded domain admits an increasing sequence of eigenvalues and a basis of eigenfunctions. The nodal domains of an eigenfunction are the connected components on which the function has a fixed sign. Courant's theorem asserts that the number of nodal domains of the n'th eigenfunction is bounded by n. In this work, we determine the eigenfunctions and eigenvalues which attain Courant's bound in some specific domains in R^d. Our analysis involves interesting symmetry properties of the eigenfunctions and surprising lattice counting arguments.

Supervisor: Assistant Professor Ram Band

*Abstract:*

A common mechanism for intramembrane cavitation bioeffects is presented and possible bioeffects, both delicate and reversible or destructive and irreversible, are discussed. Two conditions are required for creating intramembrane cavitation in a bi-layer sonophore (BLS) *in vivo*: low peak pressure of a pressure wave and an elastic wave of liquid removal from its surroundings. Such elastic waves may be generated by a shock wave, by motion of a free surface, by radiation pressure, by a moving beam of focused ultrasound or any other source of localized distortion of the elastic structure. Soft, cell laden tissues such as the liver, brain and the lung, are more susceptible to irreversible damage. Here, we show the similarity between ultrasound, explosion and impact, where the driving force is negative pressure, and decompression, induced by imbalance of gas concentration. Based on this unified mechanism, one can develop a set of safety criteria for cases where the above driving forces act separately or in tandem, (e.g., ultrasound and decompression). Supporting histological evidence is provided to show locations prone to IMC-related damage; where the damaging forces are relatively high and the localized mechanical strength is relatively poor.

*Abstract:*

Typically, when semi-discrete approximations to time-dependent partial differential equations (PDE) or explicit multistep schemes for ordinary differential equation (ODE) are constructed they are derived such that they are stable and have a specified truncation error $\tau$. Under these conditions, the Lax--Richtmyer equivalence theorem assures that the scheme converges and that the error is, at most, of the order of $||\tau||$. In most cases, the error is in indeed of the order of $||\tau||$.

We demonstrate that schemes can be constructed, whose truncation errors are $\tau$, however, the actual errors are much smaller. This error reduction is done by constructing the schemes such that they inhibit the accumulation of the local errors, therefore they are called Error Inhibiting Schemes (EIS).

ADI DITKOWSKI, School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel. email: adid@post.tau.ac.il

*Abstract:*

A nonlocal nonlinear Schrödinger (NLS) equation was recently introduced in Phys.Rev.Lett. 110, 064105 (2013) and shown to be an integrable infinite dimensional Hamiltonian evolution equation. In this talk we present a detailed study of the inverse scattering transform of this nonlocal NLS equation. The direct and inverse scattering problems are analyzed. Key symmetries of the eigenfunctions and scattering data and conserved quantities are discussed. The inverse scattering theory is developed by using a novel left-right Riemann–Hilbert problem. The Cauchy problem for the nonlocal NLS equation is formulated and methods to find pure soliton solutions are presented; this leads to explicit time-periodic one and two soliton solutions. A detailed comparison with the classical NLS equation is given and brief remarks about nonlocal versions of the modified Korteweg–de Vries and sine-Gordon equations are made.

*Abstract:*

One of the main characteristics of infinite-dimensional dissipative evolution equations, such as the Navier-Stokes equations and reaction-diffusion systems, is that their long-time dynamics is determined by finitely many parameters -- finite number of determining modes, nodes, volume elements and other determining interpolants. In this talk I will show how to explore this finite-dimensional feature of the long-time behavior of infinite-dimensional dissipative systems to design finite-dimensional feedback control for stabilizing their solutions. Notably, it is observed that this very same approach can be implemented for designing data assimilation algorithms of weather prediction based on discrete measurements. In addition, I will also show that the long-time dynamics of the Navier-Stokes equations can be imbedded in an infinite-dimensional dynamical system that is induced by an ordinary differential equations, named *determining form*, which is governed by a globally Lipschitz vector field. Remarkably, as a result of this machinery I will eventually show that the global dynamics of the Navier-Stokes equations is be determining by only one parameter that is governed by an ODE. The Navier-Stokes equations are used as an illustrative example, and all the above mentioned results equally hold to other dissipative evolution PDEs, in particular to various dissipative reaction-diffusion systems and geophysical models.

*Abstract:*

We discuss the question of global regularity for a general class of Eulerian dynamics driven by a forcing with a commutator structure.

The study of such systems is motivated by the hydrodynamic description of agent-based models for flocking driven by alignment.

For commutators involving bounded kernels, existence of strong solutions follows for initial data which are sub-critical, namely -- the initial divergence is “not too negative” and the initial spectral gap is “not too large”. Singular kernels, corresponding to fractional Laplacian of order 0<s<1, behave better: global regularity persists and flocking follows. Singularity helps! A similar role of the spectral gap is found in our study of two-dimensional pressure-less equations, corresponding to the formal limit s=0. Here, we develop a new BV framework to prove the existence of weak dual solutions for the 2D pressure-less Euler equations as vanishing viscosity limits.

*Abstract:*

In his famous 1900 ICM address Hilbert proposed his famous list of problems for the 20th century. Among these was his 6th problem which was less clearly formulated than the others but dealt with a rigorous derivation of the macroscopic equations of continuum mechanics from the available microscopic theory of his time, i.e. statistical mechanics and specifically Boltzmann's kinetic theory of gases. The problem has drawn attention from analysts over the years and even Hilbert himself made a contribution. In this talk I will note how an exact summation of the Chapman-Enskog expansion for the Boltzmann equation due to Ilya Karlin ( ETH) and Alexander Gorban (Leicester) can be used to represent solutions of the Boltzmann equation and then show that these solutions CANNOT converge the classical balance laws of mass, momentum, and energy associated the Euler equation of compressible gas dynamics. Hence alas Hilbert's program (at least with respect to gas dynamics) has a negative outcome.

Some references:

1) Gorban, Alexander N.; Karlin, Ilya Hilbert's 6th problem: exact and approximate hydrodynamic manifolds for kinetic equations. *Bull. Amer. Math. Soc. (N.S.)* 51 (2014), no. 2, 187–246.

2) Famous Fluid Equations Are Incomplete, in Quanta Magazine, https://www.quantamagazine.org/20150721-famous-fluid-equations-are-incomplete/

3) A.N. Gorban, I.V. Karlin Beyond Navier–Stokes equations: capillarity of ideal gas, Contemporary Physics, 58(1) (2016), 70-90.

4)The Mathematician's Shiva by Stuart Rojstaczer

*Abstract:*

The mathematical problem of group synchronization deals with the question of how to estimate unknown group elements from a set of their mutual relations. This problem appears as an important step in solving many real-world problems in vision, robotics, tomography, and more. In this talk, we present a novel solution for synchronization over the class of Cartan motion groups, which includes the special important case of rigid motions. Our method is based on the idea of group contraction, an algebraic notion origin in relativistic mechanics.

*Abstract:*

In this talk I will discuss a model for auto-ignition of fully developed free round turbulent jets consisting of oxidizing and chemically reacting components.I will present the derivation of the model and present results of its mathematical analysis.

The derivation of the model is based on well established experimental fact that the fully developed free round turbulent jets, in a first approximation, have the shape

of a conical frustum. Moreover, the velocity as well as concentrations fields within such jets, prior to auto-ignition, assume self-similar profiles and can be viewed as prescribed. Using these facts as well as appropriately modified

Semenov-Frank-Kamenetskii theory of thermal explosion I will derive an equation that describes initial stage of evolution of the temperature field within the jet.

The resulting model falls into a general class of Gelfand type problems.

The detailed analysis of the model results in a sharp condition for auto-ignition of free round turbulent jets in terms of principal physical and geometric parameters involved in this problem. This is a joint work with M.C. Hicks and U.G. Hegde of NASA Glenn Research Center.

*Abstract:*

Given a closed smooth Riemannian manifold M, the Laplace operator is known to possess a discrete spectrum of eigenvalues going to infinity. We are interested in the properties of the nodal sets and nodal domains of corresponding eigenfunctions in the high energy limit. We focus on some recent results on the size of nodal domains and tubular neighbourhoods of nodal sets of such high energy eigenfunctions. (joint work with Bogdan Georgiev)

*Abstract:*

When time-narrow wave-packets scatter by complex target, the field is trapped for some time, and emerges as a time broadened pulse, whose shape reflects the distribution of the delay (trapping)-times. I shall present a comprehensive framework for the computation of the delay-time distribution, and its dependence on the scattering dynamics, the wave-packet envelope (profile) and the dispersion relation. I shall then show how the well-known Wigner-Smith mean delay time and the semi-classical approximation emerge as limiting cases, valid only under special circumstances. For scattering on random media, localization has a drastic effect on the delay-time distribution. I shall demonstrate it for a particular one-dimensional system which can be analytically solved.

*Abstract:*

Adoption of new products that mainly spread through word-of-mouth is a classical problem in Marketing. In this talk, I will use agent-based models to study spatial (network) effects, temporal effects, and the role of heterogeneity, in the adoption of solar PV systems**. **

*Abstract:*

Abstract within link...

*Abstract:*

In the last 15 years, there has been much progress on higher dimensional solutions to the Einstein equation, much of it from the physics community. They are particularly interesting as, unlike 4 dimensional spacetimes, the horizon is no longer restricted to being diffeomorphic to the sphere, as demonstrated by the celebrated black ring solution of Emparan and Reall. Using the Weyl-Papapetrou coordinates and harmonic map, we show the existence of stationary solutions to the 5 dimensional vacuum Einstein equation, which are bi-axisymmetric solutions with lens space horizons. This is a joint project with Marcus Khuri and Sumio Yamada.

*Abstract:*

We consider a general class of sparse graphs which includes for example graphs that satisfy a strong isoperimetric inequality. First, we characterize these graphs in a functional analytic way by means of the form domain of Schrödinger operators. Furthermore, we study spectral bounds and characterize discreteness of the spectrum. As a particular consequence we obtain estimates on the eigenvalue asymptotics in this case. (This is joint work with Michel Bonnefont and Sylvain Golénia.)

*Abstract:*

One-dimensional Toeplitz words generalize periodic sequences and are therefore used as model for quasicrystals. They are constructed from periodic words with "holes" (that is, undetermined positions) by successively filling the holes with other periodic words. In this talk, the subclass of so called simple Toeplitz words is considered. We will discuss combinatorial properties of subshifts associated them. In addition to describing certain aspects of how ordered the word is, these properties are important tools for other questions as well. We will apply them to answer questions concerning the spectrum of Schrödinger operators and Jacobi operators on the subshift.

*Abstract:*

We study global solutions $u:{\mathbb R}^3\to{\mathbb R}^2$ of the Ginzburg-Landau equation $-\Delta u=(1-|u|^2)u$ which are local minimizers in the sense of De Giorgi. We prove that a local minimizer satisfying the condition $\liminf_{R\to\infty}\frac{E(u;B_R)}{R\ln R}<2\pi$ must be constant. The main tool is a new sharp $\eta$-ellipticity result for minimizers in dimension three that might be of independent interest. This is a joint work with Etienne Sandier (Universit\'e Paris-Est).

*Abstract:*

This is a special seminar in Mathematical Physics, please note the special time and place.

We consider a quantum mechanical system, which is modeled by a Hamiltonian acting on a finite dimensional space with degenerate eigenvalues interacting with a field of relativistic bosons. Provided a mild infrared assumption holds, we prove the existence of the ground state eigenvalues and ground state eigenvectors using an operator theoretic renormalization. We show that the eigenvectors and eigenvalues are analytic functions of the coupling constant in a cone with apex at the origin.

*Abstract:*

In this talk we discuss asymptotic relations between sharp constants of approximation theory in a general setting. We first present a general model that includes a circle of problems of finding sharp or asymptotically sharp constants in some areas of univariate and multivariate approximation theory, such as inequalities for approximating elements, approximation of individual elements, and approximation on classes of elements. Next we discuss sufficient conditions that imply limit inequalities and equalities between various sharp constants. Finally, we present applications of these results to sharp constants in Bernstein-V. A. Markov type inequalities of different metrics for univariate and multivariate trigonometric and algebraic polynomials and entire functions of exponential type.

*Abstract:*

Earlier and recent one-dimensional estimates and asymptotic relations for the cosine and sine Fourier transform of a function of bounded variation are refined in such a way that become applicable for obtaining multidimensional asymptotic relations for the Fourier transform of a function with bounded Hardy variation.

*Abstract:*

*Abstract:*

In plain words chaos refers to extreme dynamical instability and unpredictability.Yet in spite of such inherent instability, quantum systems with classically chaotic dynamics exhibit remarkable universality. In particular, their energy levels often display the universal statistical properties which can be effectively described by Random Matrix Theory. From the semiclassical point of view this remarkable phenomenon can be attributed to the existence of pairs of classical periodic orbits with small action differences. So far, however, the scope of this theory has, by and large, been restricted to low dimensional systems. I will discuss recent efforts to extend this program to hyperbolic coupled map lattices with a large number of sites. The crucial ingredient of our approach are two-dimensional symbolic dynamics which allow an effective representation of periodic orbits and their pairings. I will illustrate the theory with a specific model of coupled cat maps, where such symbolic dynamics can be constructed explicitly.

*Abstract:*

Lecture Series : Coffee 9:30, L1 10:00-10:50 (intro), L2 11:00-11:40, L3 10:50-12:30. In equilibrium systems there is a long tradition of modelling systems by postulating an energy and identifying stable states with local or global minimizers of this energy. In recent years, with the discovery of Wasserstein and related gradient flows, there is the potential to do the same for time-evolving systems with overdamped (non-inertial, viscosity-dominated) dynamics. Such a modelling route, however, requires an understanding of which energies (or entropies) drive a given system, which dissipation mechanisms are present, and how these two interact. Especially for the Wasserstein-based dissipations this was unclear until rather recently. In these talks I will discuss some of the modelling arguments that underlie the use of energies, entropies, and the Wasserstein gradient flows. This understanding springs from the common connection between large deviations for stochastic particle processes on one hand, and energies, entropies, and gradient flows on the other. In the first talk I will describe the variational structure of gradient flows, introduce generalized gradient flows, and give examples. In the second talk I will enter more deeply into the connection between gradient flows on one hand and stochastic processes on the other, in order to explain `where the gradient-flow structures come from. Organizers: Amy Novick-Cohen and Nir Gavish

*Abstract:*

We discuss the main ideas in the derivation of two-sided estimates of Green functions for a class of Schroedinger operators defined on Lipschitz bounded domains. An important ingredient is the Boundary Harnack Principle which in smooth domains is closely related to Hopf's lemma. Except for some special cases, these estimates seem to be new even in the case of smooth domains. In Lipschitz domains the estimates are known for the Laplacian and for Schroedinger equations provided that the potential has no strong singularity.

*Abstract:*

The first quasicrystals where discovered by D. Shechtman in the year 1984. From the mathematical point of view, the study of the associated Schrödinger operators turns out to be a challenging question. Up to know, we can mainly analyze one-dimensional systems by using the method of transfer matrices. In 1987, A. Tsai et al. discovered a quasicrystalline structure in an Aluminum-Copper-Iron composition. By changing the concentration of the chemical elements, they produce a stable quasicrystaline structure by an approximation process of periodic crystals. In light of that it is natural to ask whether Schrödinger operators related to aperiodic structures can be approximated by periodic ones while preserving spectral properties. The aim of the talk is to provide a mathematical foundation for such approximations.

In the talk, we develop a theory for the continuous variation of the associated spectra in the Hausdorff metric meaning the continuous behavior of the spectral gaps. We show that the convergence of the spectra is characterized by the convergence of the underlying dynamics. Hence, periodic approximations of Schrödinger operators can be constructed by periodic approximations of the dynamical systems which we will describe along the lines of an example.

*Abstract:*

Realistic physical models represented by elliptic boundary value problems are of immense importance in predictive science and engineering applications. Effective solution of such problems, essentially, requires accurate numerical discretization that take into account complexities such as irregular geometries and unstable interfaces. This typically leads to large-scale (1M unknowns or more) ill-conditioned linear systems, that can only be resolved by iterative methods combined with multilevel preconditioning schemes. The class of hierarchical matrix approximations is a multilevel scheme which offers unique advantages over other traditional multilevel methods, e.g., multigrid. Essentially, a hierarchical matrix is a perturbed version of the input linear system. Thus, in principle, the magnitude of the perturbation needs to be smaller than the smallest modulus eigenvalue of the system matrix. For many problems, the perturbation may have to be chosen quite small, generally, leading to less efficient preconditioners. In this talk we will present a new strong hierarchical preconditioning scheme that overcomes the perturbation limit. We will start with an overview on hierarchical matrices, and continue with theoretical results on optimal preconditioning in the symmetric positive definite case. The effectiveness of the new method which outperforms other classical techniques will be illustrated through numerical experiments. In the final part of the talk we will also suggest directions towards extending the theory to indefinite and non-symmetric linear systems.

*Abstract:*

We study the properties of the set S of non-differentiable points of viscosity solutions of the Hamilton-Jacobi equation, for a Tonelli Hamiltonian.The main surprise is the fact that this set is locally arc connected—it is even locally contractible. This last property is far from generic in the class of semi-concave functions.We also “identify” the connected components of this set S. This work relies on the idea of Cannarsa and Cheng to use the positive Lax-Oleinik operator to construct a global propagation of singularities (without necessarily obtaining uniqueness of the propagation).

This is a joint work with Piermarco Cannarsa and Wei Cheng.

*Abstract:*

The mean curvature flow appears naturally in the motion of interfaces in material science, physics and biology. It also arises in geometry and has found its applications in topological classification of surfaces. In this talk I will discuss recent results on formation of singularities under this flow. In particular, I will describe the 'spectral' picture of singularity formation and sketch the proof of the neck pinching results obtained jointly with Zhou Gang and Dan Knopf.

*Abstract:*

The space of smoothly embedded n-spheres in R^{n+1} is the quotient space M_n:=Emb(S^n,R^{n+1})/Diff(S^n). In 1959 Smale proved that M_1 is contractible and conjectured that M_2 is contractible as well, a fact that was proved by Hatcher in 1983.For n\geq 3, even the simplest questions regarding M_n are both open and central. For instance, whether or not M_3 is path connected is an equivalent form of one of the most important open questions in differential topology - the smooth Schoenflies conjecture. In particular, if M_3 is not path connected, the smooth 4 dimensional Poincare conjecture can not be true. In this talk, I will explain how mean curvature flow, a geometric analogue of the heat equation, can assist in studying the topology of geometric relatives of M_n.I will first illustrate how the theory of 1-d mean curvature flow (aka curve shortening flow) yields a very simple proof of Smale's theorem about the contractibility of M_1.I will then describe a recent joint work with Reto Buzano and Robert Haslhofer, utilizing mean curvature flow with surgery to prove that the space of 2-convex embedded spheres is path connected.

*Abstract:*

Liouville's rigidity theorem (1850) states that a map $f:\Omega\subset R^d\to R^d$ that satisfies $Df \in SO(d)$ is an affine map. Reshetnyak (1967) generalized this result and showed that if a sequence $f_n$ satisfies $Df_n \to SO(d)$ in $L^p$, then $f_n$ converges to an affine map.

In this talk I will discuss generalizations of these theorems to mappings between manifolds, present some open questions, and describe how these rigidity questions arise in the theory of elasticity of pre-stressed materials (non-Euclidean elasticity).

If time permits, I will sketch the main ideas of the proof, using Young measures and harmonic analysis techniques, adapted to Riemannian settings.

Based on a joint work with Asaf Shachar and Raz Kupferman.

*Abstract:*

Atomic systems are regularly studied as large sets of point-like particles, and so understanding how particles can be arranged in such systems is a very natural problem. However, aside from perfect crystals and ideal gases, describing this kind of “structure” in an insightful yet tractable manner can be challenging. Analysis of the configuration space of local arrangements of neighbors, with some help from the Borsuk-Ulam theorem, helps explain limitations of continuous metric approaches to this problem, and motivates the use of Voronoi cell topology. Several short examples from materials research help illustrate strengths of this approach.

*Abstract:*

Brezis raised the question of uniqueness of positive radial solutions for critical exponent problems in a ball. Long back this was affirmatively solved in dimensions greater than two using the clever use of Pohozaev's identity. In dimension two, the critical nonlinearity is of exponential nature and the Pohozaev's identity is not effective. Using the Asymptotic analysis, I would like to show that Large solutions are unique.

*Abstract:*

The celebrated Faber-Krahn inequality yields that, among all domainsof a fixed volume, the ball minimizes the lowest eigenvalue of the Dirichlet Laplacian. This result can be viewed as a spectral counterpart of the well known geometric isoperimetric inequality. The aim of this talk is to discuss generalizations of the Faber-Krahn inequality for optimization of the lowest eigenvalues for:

- Schrodinger operators with $\delta$-interactions supported on conical surfaces and open arcs [1,3];
- Robin Laplacians on exterior domains and planes with slits [2,3].

Beyond a physical relevance of $\delta$-interactions and Robin Laplacians,a purely mathematical motivation to consider these optimization problems stems from the fact that standard methods, going back to the papers of Faber and Krahn, are not applicable anymore. Another interesting novel aspect is that in some cases the shape of the optimizer bifurcates as the boundary parameter varies while in the othercases no optimizer exists. The results in the talk are obtained in collaboration with P. Exner and D. Krejcirik.

Bibliography

- P. Exner and V. Lotoreichik, A spectral isoperimetric inequality for cones, \emph{to appear in Lett. Math. Phys., arXiv:1512.01970.
- D. Krejcirik and V.Lotoreichik, Optimisation of the lowest Robin eigenvalue in the exterior of a compact set, submitted, arXiv:1608.04896.
- V. Lotoreichik, Spectral isoperimetric inequalities for $\delta$-interactions on open arcs and for the Robin Laplacian on planes with slits, submitted, arXiv:1609.07598.

*Abstract:*

I present a new approach to classify the asymptotic behavior of certain types of wave equations, supercritical and others, with large initial data. In some cases, as for Nirenberg type equations, a fairly complete classification of the solutions (finite time blowup or global existence and scattering) is proved.

New results are obtained for the well known monomials wave equations in the sub/critical/super critical cases.

This approach, developed jointly with M. Beceanu, is based on a new decomposition into incoming and outgoing waves for the wave equation, and the positivity of the fundamental solution of the wave equation in three dimensions.

*Abstract:*

*Abstract:*

We investigate the dynamics of a two-layer system consisting of a thin liquid film and an overlying gas layer, sandwiched between an asymmetric corrugated surface and a flat upper plate held at a constant temperature. The flow in question is driven by the Marangoni instability induced, in one case, by thermal waves propagating along a flat, solid substrate, and in another case, by the asymmetric topographical structure of the substrate, uniformly heated from below. We propose different methods for flow-rate amplification and rupture prevention, both of great importance for transport problems in microfluidic devices.The talk is based on the speaker’s PhD thesis which was carried out under the supervision of Professor Alexander Oron.

*Abstract:*

We study the influence of a compactly supported magnetic field on spectral-threshold properties of the Schrodinger operator and the large-time behaviour of the associated heat semigroup. We derive new magnetic Hardy inequalities in any space dimension d and develop the method of self-similar variables and weighted Sobolev spaces for the heat equation.

A careful analysis of the heat equation in the self-similar variables shows that the magnetic field asymptotically degenerates to a singular Aharonov-Bohm magnetic field, which in turn determines the large-time behaviour of the solutions in the physical variables. We deduce that in d=2 there is an improvement of the decay rate of the heat semigroup by a polynomial factor with power proportional to the distance of the total magnetic flux to the discrete set of flux quanta, while there is no extra polynomial decay rate in higher dimensions.

*Abstract:*

We consider the initial value problem for the inviscid Primitive equations in three spatial dimensions. We recast the system to an abstract Euler-type system. We use an addaptation of the method of convex integration for Euler equations (following works of L. Sz\ekelyhidi, C. De Lellis and Feireisl). As a result, we obtain the existence of infinitely many global weak solutions for large initial data. We also introduce an appropriate notion of dissipative solutions and show the existence of an initial data from which emanate infinitely many dissipative solutions. This is a joint work with E. Chiodaroli (EPFL, Switzerland).

*Abstract:*

Scaling transformations (translations and dilations) are known to define wavelet bases, give equivalent definitions of important functional spaces, and prove optimal inequalities. We will summarize some known results in the Euclidean case and on nilpotent Lie groups, and discuss the work in progress dealing with analogous transformations on manifolds, where scaling is defined via the Green's function of Laplace-Beltrami operator. Preliminary results include sharp inequalities of Caffarelli-Kohn-Nirenberg type on the hyperbolic space. The work involves collaborations with L. Skrzypczak and K. Sandeep.

*Abstract:*

We consider the drifting Laplacian over a noncompact, smooth, weighted manifold. We associate to the weighted manifold a family of higher dimensional Riemannian manifolds in warped product form. We show that various geometric analysis results on the weighted manifold are closely related to those on the warped product, by directly relating the geometry of the two spaces. In particular, we can demonstrate Gaussian heat kernel estimates for the drifting Laplacian over the weighted manifold whenever its Bakry-Emery Ricci tensor is bounded below. These are obtained effortlessly from the respective heat kernel bounds on the warped product. The proofs reveal the strong geometric connection of the weighted space to the warped product spaces. At the same time, they further illustrate the fact that the drifting Laplacian and Bakry-Emery Ricci tensor are projections (in some sense) of the Laplacian and Ricci tensor of a higher dimensional space. We then use these results to study the spectrum of the drifting Laplacian on the weighted manifold. This is joint work with Zhiqin Lu.

*Abstract:*

I will introduce the Ginzburg-Landau (GL) equations and give a very brief discussion of solutions with a single vortex per lattice cell. The focus of this talk, however, will be on the general case of multi-vortex solutions. We attempt to bifurcate a branch of such solutions from the normal state solution with constant magnetic field. A main difficulty is the reduction of dimension of solutions of the linearized problem. One can transfer this problem onto a suitable space of theta functions and use more algebraic methods to study the problem. I will discuss low flux (per lattice cell) results and give a brief sketch of the proof by exploiting symmetries of the underlying Abrikosov lattice.

*Abstract:*

I'll review the basic notions of optimal transportation (Monge-Kantorovich theory), and introduce some limit theorems and their relation to Sobolev embedding and geometry of tangent spaces associated with the cone of probability measures. These results leads naturally to a new notion of "optimal teleportation", which I'll introduce.

*Abstract:*

Delays, arising in nonoscillatory and stable ordinary differential equations, can induce oscillation and instability of their solutions. That is why the traditional direction in the study of nonoscillation and stability of delay equations is to establish a smallness of delay, allowing delay differential equations to preserve these convenient properties of ordinary differential equations with the same coefficients. In this paper, we find cases in which delays, arising in oscillatory and asymptotically unstable ordinary differential equations, induce nonoscillation and stability of delay equations. We demonstrate that, although the ordinary differential equation x"(t)+c(t)x(t)=0 can be oscillating and asymptoticaly unstable, the delay equation x"(t)+a(t)x(t-h(t))-b(t)x(t-g(t))=0, where c(t)=a(t)-b(t), can be nonoscillating and exponentially stable. Results on nonoscillation and exponential stability of delay differential equations are obtained. On the basis of these results on nonoscillation and stability, the new possibilities of non-invasive (non-evasive) control, which allow us to stabilize a motion of single mass point, are proposed. Stabilization of this sort, according to common belief requires damping term in the second order differential equation. Results proposed in this talk refute this delusion.

*Abstract:*

The Becker-Döring equations are a fundamental set of equations that describe the kinetics of first order phase transition such as crystallisation, vapour condensation and aggregation of lipids.Much like many other kinetic equations, the Becker-Döring equations have a state of equilibrium which any reasonable solution to the equations converge to as the time goes to infinity. While the existence, uniqueness and proof of convergence to equilibrium is known since the late 80’s, the question of finding the rate of the convergence to equilibrium is one that has received much focus in the last 10 years.In our talk we will present the Becker-Döring model and resolve the question of the rate of convergence by means of the so-called ‘entropy method’: finding an appropriate functional inequality that connects between the appropriate ‘entropy’ of the problem and its dissipation under the flow of the equation. We will discuss the optimality of our result, and the underlying relative log-Sobolev inequality.

*Abstract:*

This is joint work with Daniel Waltner (Duisburg-Essen)building on previous work with Uzy Smilansky (Weizmann Institute) andStanislav Derevyanko (now Ben Gurion University).I will consider solutions to the stationary nonlinearSchrödinger equation on a metric graph with `standard' matchning conditions.I will summarise the framework and show how the coupled differentialequations reduce to a finite number of algebraic nonlinear equations.In the low intensity limit these equations reduce to well-know linearequations for (linear) quantum graphs. A particularly interesting limitis te short wavelength limit as it allows for a regime with locally weaknonlinearity but strong global effects. These effects can be captured inthe leading order in a canonical Hamiltonian perturbation theory. Somesimple examples will be discussed.If time allows I will present a few open questions that are currentlybeing investigated with Ram Band and August Krueger here at the Technion.

*Abstract:*

After reviewing the theory of singular limits of smooth solutions of evolutionary partial differential equations both for the standard case in which the large terms have constant-coefficients and for some equations having variable-coefficient large terms, an analysis of certain numerical schemes for singular limits will be presented that is analogous to the corresponding PDE theory. The analysis has so far be done for certain finite-difference schemes but some preliminary results are available for finite-volume schemes.

*Abstract:*

We study a compactification of certain graphs that goes back toideas of Royden. Given the boundary that arises from thiscompactification, we first study the Dirichlet problem. Secondly, inthe case of finite measure the associated Laplacians have purelydiscrete spectrum and one can give estimates on the eigenvalueasymptotics. Finally, the Markov extensions of the Laplacian can becharacterized by boundary conditions given by Dirichlet forms on theboundary.

(This comprises joint work with Agelos Georgakopoulos, SebastianHaeseler, Daniel Lenz, Marcel Schmidt, Michael Schwarz, RadoslawWojciechowski)

*Abstract:*

*Abstract:*

Compressible fluids are modeled through Navier Stokes equations for density and velocity.In this talk I consider the model in a bounded interval and discuss null controllability(steer the system to zero state in finite time) and stabilization (steer the system to a steady state as time goes to infinity). The control acts only on the velocity.

*Abstract:*

Based on the Weierstrass representation of second variation, we develop a non-spectral theory of stability for isoperimetric problem with minimizedand constrained two-dimensional functionals of general type and free endpoints allowed to move along two given planar curves.We establish the stability criterion and apply this theory to the axisymmetric liquid bridge between two axisymmetric solid bodies without gravity to determine the stability of menisci with free contactlines.For catenoid and cylinder menisci and different solid shapes, we determinethe stability domain. The other menisci (unduloid, nodoid and sphere) are considered in a simple setup between two plates or two spheres.

Joint work with B. Rubinstein.

*Abstract:*

We shall discuss non-self-adjoint Kac operators, and in particular the asymptotics of their largest eigenvalues in the semi-classical regime. Such operators appear in particular as transfer matrices of supersymmetric models which encode the spectral properties of one-dimensional random operators. [Joint work with Margherita Disertori]

*Abstract:*

Broadly speaking, an eigenvalue appearing in the boundary conditions of an elliptic operator is an eigenvalue of Steklov-type.

In this talk we shall discuss a few variants of the classical second order Steklov problem. In particular, we shall formulate the naturalfourth order Steklov problem which involves the biharmonic operator, providing a physical justification.

Shape optimization problems will be addressed and an isoperimetric inequality for the first eigenvalue of the above mentionedbiharmonic Steklov problem will be presented.We shall also point out that a class of Steklov-type problems could be viewed as a class of critical Neumann-type problems arisingin boundary mass concentration phenomena.

This talk is based on joint works with Davide Buoso and Luigi Provenzano.

*Abstract:*

Mean curvature flow is a geometric heat equation for hyper-surfaces, which is the gradient flow of the surface area functional. The flow typically becomes singular at finite time, after which it can be extended by an object called the "level set flow". In general, the level set flow is not that well behaved, but in the important mean convex case, where the initial hypersurface is a boundary of a domain which starts moving inward, a beautiful regularity and structure theory was developed in the last 20 years by Brian White. While parts of this theory work in full generality, parts were only known to hold in either the Euclidean setting or in low dimensions.

We prove two new estimates for the level set flow of mean convex domains in general Riemannian manifolds. Our estimates give control - exponential in time - for the infimum of the mean curvature, and the ratio between the norm of the second fundamental form and the mean curvature. In particular, the estimates remove the above mentioned stumbling block that has been left after the work of White and thus allow us to extend the structure theory for weak mean convex level set flow to general ambient manifolds of arbitrary dimension. While the setting and motivation of the work is geometric, almost the entire labor turned out to be analytic. For instance, what is readily seen to be the main obstacle in generalizing the result from the Euclidean to the non-Euclidean setting is, in fact, the lack of an a-priori C^0 estimate for solutions to a certain family of elliptic PDEs. Although completely decoupling the analysis from the geometry of the problem would be misleading, the talk will highlight the analytic aspects of the work, and should be accessible to everyone doing analysis. This is a joint work with Robert Haslhofer.

*Abstract:*

The Keller-Segel equations model chemotaxis of bio-organisms. In a reduced form, considered in this talk, they are related to Vlasov equation for self-gravitating systems and are used in social sciences in descriptions of crime patterns.It is relatively easy to show that in the critical dimension 2 and for the mass of initial conditions greater than 8 \pi, the solutions break down in finite time. Understanding the mechanism of this breakdown turned out to be a subtle problem defying solution for a long time.Preliminary results indicate that the solutions 'blowup'. This blowup is supposed to describe the chemotactic aggregation of the organisms and understanding its universal features would allow comparison of theoretical results with experimental observations.In this talk I discuss recent results on dynamics of solutions of the (reduced) Keller-Segel equations in the critical dimension 2, which include a formal derivation and partial rigorous results on the blowup dynamics of solutions. The talk is based on joint work with S. I. Dejak, D. Egli and P.M. Lushnikov.

*Abstract:*

Consider a compact domain with the smooth boundary in the Euclidean space. Fractional Laplacian is defined on functions supported in this domain as a (non-integer) power of the positive Laplacian on the whole space restricted then to this domain. Such operators appear in the theory of stochastic processes. It turns out that the standard results about distribution of eigenvalues (including two-term asymptotics) remain true for fractional Laplacians. There are however some unsolved problems.

*Abstract:*

We shall discuss the Chirikov standard map, an area-preservingmap of the torus to itself in which quasi-periodic and chaotic dynamicsare believed to coexist. We shall describe how the problem can be relatedto the spectral properties of a one-dimensional discrete Schrödinger operator, and present a recent result. Based on joint work with T. Spencer.

*Abstract:*

First-order systems of partial differential equations appear in many areas of physics, from the Maxwell equations to the Dirac operator. The aim of the talk is to describe a general method for the study ofthe spectral density of all such systems, connecting it to traces on the (geometric-optical) "slowness surfaces" . The Holder continuity of the spectral density leads to a derivationof the limiting absorption principle and global spacetime estimates. (based on joint work with Tomio Umeda).

*Abstract:*

We study harmonic functions defined in Z^d. We define their L^2-growth in terms of the random walk, and show that it satisfiesa strong convexity phenomenon. This is related to high powers of the Laplace operator and a discrete three circles theorem with an inherent error term. We discuss the optimality of the error term.This is joint work with Gabor Lippner.

*Abstract:*

Let \Omega be a simply connected domain in R^N. For p\in[1,2) there is a natural decomposition of the space W^{1,p}(\Omega;S^1) into distinct classes. In this talk, based on a joint work with Brezis and Mironescu, we will present estimates for both the usual distance and the Hausdorff distance between different classes.

*Abstract:*

We derive an adiabatic theory for a stochastic differential equation, $ \varepsilon\, \mathrm{d} X(s) = L_1(s) X(s)\, \mathrm{d} s + \sqrt{\varepsilon} L_2(s) X(s) \, \mathrm{d} B_s, $ under a condition that instantaneous stationary states of $L_1(s)$ are also stationary states of $L_2(s)$. We use our results to derive the full statistics of tunneling for a driven stochastic Schrödinger equation describing a dephasing process. The work is motivated by a recent interest in quantum trajectories. We explain the connection, and , in particular, we include a short discussion of the quantum stochastic calculus.

*Abstract:*

In this talk we propose algebraic criteria that yield sharp Hölder and reverse Hölder types of inequalities for the product of functions on Gaussian random vectors with arbitrary covariance structure. While the lower inequality appears to be new, we prove that the upper inequality gives an equivalent formulation for the Brascamp-Lieb inequality. We will see that this result generalizes, Hölder's inequality, Nelson's hypercontractivity and the sharp Young inequality as well as their reverse forms. Moreover, we will give one more application: Barthe's and Prekopa-Leindler inequalities.

Based on a joint work with Wei-Kuo Chen and Grigoris Paouris

*Abstract:*

We study the distribution of eigenvalues of operators which are compact perturbations of bounded operators , obtaining bounds on thenumber of eigenvalues in regions of the complex plane which are separated from the essential spectrum. Our results can be understood as quantitative versions of some well-known qualitative results of spectral theory.Our methods include a finite-dimensional reduction procedure and employingcomplex analysis. The talk will present some background and related results,our main results and methods, and some questions that remain open.

Joint work with: Michael Demuth, Franz Hanauska and Marcel Hansmann.

*Abstract:*

To compute wave propagation in unbounded domains, the domain is often truncated to a finite size, by introducing an artificial boundary at some, ideally not too large, distance. Boundary conditions are then needed on the artificial boundary, that render the boundary invisible to outgoing waves. In this work, we describe absorbing boundary treatment for the linearized water wave equation, governing incompressible, irrotational free surface flow. We use Fourier analysis to identify the structure of outgoing water waves and derive a one-way version of the equation, which we implement as an absorbing layer near the artificial boundary. Additional wave damping may also be incorporated and will be discussed.The one-way equation involves a fractional derivative operator corresponding to a half-derivative in space. The equation is viewed as a conservation law with a linear nonlocal flux involving a convolution with a singular integrable kernel. We construct high order numerical methods, based on local polynomial approximation of the solution followed by exact integration of the singular convolution. In this talk, we will discuss the water wave equation, its one-way counterparts, the numerical method, and present numerical results.

*Abstract:*

We discuss properties of positive solutions of linear equations of the form $\Delta u +a\delta(x)^{-2}\,u=0$ and of a class of corresponding semilinear equations with absorption term.

*Abstract:*

I'll consider an optimal partition of resources (e.g. customers) between several agents (e.g. experts), given utility functions for the agents and their capacities. This problem is a variant of optimal transport (Monge-Kantorovich) between two measure spaces where one of the measures is discrete (capacities) and the cost of transport is the utilities of agents. I'll concentrate on the individual value for each agent under optimal partition and show that, counter-intuitively, this value may decrease if the agent's utility is increased. Sufficient and necessary conditions for increment of the individual value will be given, independently of the other agents. The sharpness of these conditions will be discussed, as well.If time permit I'll discuss some applications to cooperative games.

*Abstract:*

P. W. Anderson in 1958 argued that disorder can cause localization of electron states, which manifests itself in time evolution (non-spreading of wave packets) and vanishing of conductivity. Starting from the early 80's, the topic rapidly became one of the most intensively studied ones in mathematical physics community.

In this talk, we will introduce a new approach for proving localization for the Anderson model at high disorder. In contrast to the usual strategy, we do not study finite volume Green's functions. Instead, we perform a multiscale analysis based on finite volume eigensystems, establishing localization of finite volume eigenfunctions with high probability. (Joint work in progress with A. Klein.)

*Abstract:*

In this talk I will discuss a transient dynamics describedby the solutions of the reaction-diffusion equations in which thereactionterm consists of a combination of a superlinear power-law absorptionand a time-independent point source. In one space dimension,solutions of these problems with zero initial data are known toapproach the stationary solution in an asymptotically self-similarmanner. Here I will show that this conclusion remains true in twospace dimensions, while in three and higher dimensions the sameconclusion holds true for all powers of the nonlinearity not exceedingthe Serrin critical exponent. The analysis requires dealing withsolutions that contain a persistent singularity and involves avariational proof of existence of ultra-singular solutions, aspecial class of self-similar solutions in the considered problem.

*Abstract:*

Tangent vector fields are widely used in computer graphics, where they are applied to various tasks such as texture generation on surfaces, and physical simulation for animation.

In this talk I will describe a new discretization of tangent vector fields which is inspired by the classic point of view in differential geometry, namely that vector fields are simply linear operators on functions.Taking this approach allows us to numerically simulate various PDEs on discrete surfaces, such as function advection and fluid flow, in an efficient and stable manner.

I will show a few examples of graphics applications, including turbulent flows and viscous thin film flows on triangle meshes.

*Abstract:*

We consider maps $u:\Omega\to {\mathbb S}^1$ having some Sobolev regularity $u\in W^{s,p}$. In several interesting situations, either

a) such maps need not have a phase $\varphi$ with the same regularity as $u$, or

b) the phase $\varphi$ exists but is not controlled by the norm of $u$.

In case a), I obtained some time ago a structure result (« factorization ») allowing to write each such $u$ as $u=v\, w$, where $v$ lifts and $w$ is « smoother » than $u$. In the first part of the talk, I will present a very simple proof of this result, based on the theory of weighted Sobolev spaces, that I will revisit.

Case b) occurs e.g. in the problem of the existence of a « minimal » (in the sense of an appropriate Sobolev seminorm) map $u: {\mathbb S}^1\to {\mathbb S}^1$ winding once around the unit circle. In the second part of the talk, I will present a modest progress in this direction, based on a geometric approach to the maximum principle in two dimensions.The common theme of the two arguments is a geometric detection of the energy concentration of manifold-valued maps.

*Abstract:*

In his 1962 paper, F. Dyson introduced a then novel approach for studying random matrix ensembles in terms of Brownian dynamics in the space of matrices.

He then proposed a Fokker-Planck evolution for the spectral distribution function, whose stationary solution provides the spectral join probability distribution function, $P(\lambda_1, …,\lambda_N)$.

Here, we reformulate the approach for the traces, $t_n = \sum_{k=1}^{N} \lambda_{k}^{n}$ (= spectral moments), and derive the Fokker-Planck equations and their joint probability distribution $Q(t_1, …,t_N)$. The advantages of this version of Dyson’s theory will be discussed, and a few new identities between symmetric polynomials will be derived.

*Abstract:*

Spatially localized and stationary states have recently brought to the spotlight of mathematical analysis of nonlinear PDEs via analysis of models exhibiting free energy and/or conservation. However, many chemical and biological systems exhibit rather localized traveling pulses, such as action potentials in axons and cardiac muscles. We identify and describe a qualitative novel property of excitable media that enables us to generate a sequence of traveling pulses of any desired length, using a one-time initial stimulus. The existence of these states is related here to the presence of homoclinic snaking in the vicinity of a subcritical, finite wavenumber Hopf bifurcation. The pulses are organized in a slanted snaking structure resulting from the presence of a heteroclinic cycle between small and large amplitude traveling waves. Connections of this type require a multivalued dispersion relation. This dispersion relation is computed numerically and used to interpret the profile of the pulse group. The different spatially localized pulse trains can be accessed by appropriately customised initial stimuli thereby blurring the traditional distinction between oscillatory and excitable systems. The results reveal a new class of phenomena relevant to spatiotemporal dynamics of excitable media, particularly in chemical and biological systems with multiple activators and inhibitors.

*Abstract:*

I will talk about a certain topic related to the first eigenvalue of the Laplace-Beltrami operator on Riemannian manifolds. This topic was inspired by the beautiful theorem of Hersch about the first eigenvalue of the Laplacian on a Riemannian 2-sphere.