The classical Hardy-Littlewood maximal operators (averaging over families of Euclidean balls and cubes) are known to satisfy L^p bounds that are independent of dimension. This talk will extend these results to spherical maximal functions acting on Cartesian products of cyclic groups equipped with the Hamming metric.

We consider composite media with a broad range of scales, whose effective properties are important in materials science, biophysics, and climate modeling. Examples include random resistor networks, polycrystalline media, porous bone, the brine microstructure of sea ice, ocean eddies, melt ponds on the surface of Arctic sea ice, and the polar ice packs themselves. The analytic continuation method provides Stieltjes integral representations for the bulk transport coefficients of such systems, involving spectral measures of self-adjoint random operators which depend only on the composite geometry. On finite bond lattices or discretizations of continuum systems, these random operators are represented by random matrices and the spectral measures are given explicitly in terms of their eigenvalues and eigenvectors. In this lecture we will discuss various implications and applications of these integral representations. We will also discuss computations of the spectral measures of the operators, as well as statistical measures of their eigenvalues. For example, the effective behavior of composite materials often exhibits large changes associated with transitions in the connectedness or percolation properties of a particular phase. We demonstrate that an onset of connectedness gives rise to striking transitional behavior in the short and long range correlations in the eigenvalues of the associated random matrix. This, in turn, gives rise to transitional behavior in the spectral measures, leading to observed critical behavior in the effective transport properties of the media.

In this talk, we shall look at discrepancy theory through the prism of harmonic and functional analysis. Discrepancy theory deals with finding optimal approximations of continuous objects by discrete sets of points and quantifying the inevitably arising errors (irregularities of distribution). This field lies at the interface of several areas of mathematics: approximation, probability, discrete geometry, number theory. Historically, methods of analysis (Fourier techniques, Riesz product, wavelet expansions etc) played a pivotal role in the development of the subject.

A number of exciting new connections of discrepancy theory to other fields were discovered recently and are not yet fully understood. These include approximation theory (metric entropy of spaces with mixed smoothness, hyperbolic approximations), probability (small deviations of Gaussian processes, empirical processes), harmonic analysis (small ball inequality, Sidon theorem), compressed sensing etc.

We shall describe some of the recent results in the field, the main ideas and methods, and numerous relations to other areas of mathematics.

Motion by mean curvature for a network of surfaces arises in many applications. An important example is the evolution of microstructure in a polycrystalline material under heat treatment. Most metals and ceramics are of this type: They consist of many small single-crystal pieces of differing orientation, called grains, that are stuck together. A famous model proposed by Mullins in the 60s describes the dynamics of the network of surfaces that separate neighboring grains from one another in such a material as gradient descent for a weighted sum of the (possibly anisotropic) areas of the surfaces. The resulting dynamics is motion by weighted mean curvature for the surfaces in the network, together with certain conditions that need to be satisfied at junctions along which three or more surfaces may intersect. Typically, many topological changes occur during the evolution, as grains shrink and disappear, pinch off, or junctions collide. A very elegant algorithm -- known as threshold dynamics -- for the motion by mean curvature of a surface was given by Merriman, Bence, and Osher: It generates the whole evolution simply by alternating two very simple operations: convolution with a Gaussian kernel, and thresholding. It also works for networks, provided that all surfaces in the network have isotropic surface energies with equal weights. Its correct extension to the more general setting of unequal weights and possibly anisotropic (normal dependent) surface energies remained elusive, despite keen interest in this setting from materials scientists. In joint work with Felix Otto, we give a variational formulation of the original threshold dynamics algorithm by identifying a Lyapunov functional for it. In turn, the variational formulation shows how to extend the algorithm correctly to the more general settings that are of interest for materials scientists (joint work with Felix Otto and Matt Elsey). Examples of how to use the new algorithms to investigate unsettled questions about grain size distribution and its evolution will also be given.

Density Functional Theory (DFT) and methods based on it are the primary production methods for electronic-structure based "first principles" simulations in materials science today. This talk focuses on the anatomy of the FHI-aims code: an all-electron implementation of DFT that makes no a priori shape approximations to the potential or solutions (orbitals), yet implements the necessary algorithms in a way that scales up to thousands of atoms and on massively parallel computers with (ten)thousands of cores for routine simulations. Particularly important developments include a scalable, massively parallel dense eigenvalue solver "ELPA" and a framework to expand the (expensive) two-electron Coulomb operator in a linear-scaling localized resolution of identity framework for large-scale calculations.

In many physical and biological systems, there are some competing effects such as focus and de-focus, attraction and repulsion, spread and concentration. These competing effects usually are represented by terms with different signs in a free energy. The dynamics of the physical system sometimes can be described by a gradient flow driven by the free energy. Some functional inequalities can be used to determine the domination among these competing effects in the free energy, and provided sharp conditions on initial data or coefficients in the system for the global existence. In this talk, we will introduce some important relations between functional inequalities and sharp conditions for the global existence. For example, the Hardy-Littlewood-Sobolev inequality vs parabolic-elliptic Keller-Segel model, Onofri's inequality vs parabolic-parabolic Keller-Segel model, and Sz. Nagy inequality vs 1-D thin film equation, and provide the results on the global existence and blow-up for above models under sharp conditions. Moreover, we obtain the uniqueness of the weak solution for the linear diffusion Keller-Segel model using the refined hyper-contractivity of the $L^p$ of the solution under the sharp initial condition, and prove the $L^{\infty}$ estimate of the solution utilizing the bootstrap method. We also provide some results on existence of the global smooth solution.

Phase retrieval problems appear in many imaging applications in which only the magnitude of (e.g., Fourier) transform coefficients of a given signal can be measured. In such settings one desires to learn the original signal (up to a global phase factor) using only such magnitude information. In this talk we discuss methods to rapidly re-learn such lost phase information by using the magnitudes of well-designed combinations of the original transform coefficients. In particular, we develop a fast phase retrieval method which is near-linear time, making it computationally feasible for large dimensional signals. Both theoretical and experimental results demonstrate the method's speed, accuracy, and robustness. We then use this new phase retrieval method to help establish the first known sublinear-time compressive phase retrieval algorithm capable of recovering a given $s$-sparse signal ${\bf x} \in \mathbbm{C}^d$ (up to an unknown phase factor) in just $\mathcal{O}(s \log^5 s \cdot \log d)$-time using only $\mathcal{O}(s \log^4 s \cdot \log d)$ magnitude measurements. This is joint work with Aditya Viswanathan and Yang Wang.

I will present the proof of the random homogenization of general coercive Hamiltonian in 1d with the form as H(p,x,\omega)=H(p)+V(x, \omega). Some interesting and complex phenomena associated with non-convex Hamiltonian will also be discussed. This is a joint work with Scott Armstrong and Hung Tran.

Elliptic boundary value problems are well-understood in the case when the boundary, the data, and the coefficients exhibit smoothness. However, it has been long recognized in physics and engineering that irregularities (non-smooth boundary, abrupt change of media, noise or disorder) can decisively influence the properties of the solutions and give rise to completely new phenomena. 

The analysis of general non-smooth elliptic PDEs gives rise to decisively new challenges: possible failure of maximal principle and positivity, breakdown of boundary regularity, lack of the classical L^2 estimates, to mention just a few. Further progress builds on an involved blend of harmonic analysis, potential theory and geometric measure theory techniques. In this talk we are going to discuss some highlights of the history, conjectures, paradoxes, and recent discoveries such as the higher-order Wiener criterion and maximum principle for higher order PDEs, solvability of rough elliptic boundary problems, harmonic measure, as well as an intriguing phenomenon of localization of eigenfunctions -- within and beyond the limits of the famous Anderson localization.

Blood vessel systems and leaf venations are typical biological transport networks. The energy consumption for such a system to perform its biological functions is determined by the network structure. In the first part of this talk, I will discuss the optimized structure of vessel networks, and show how the blood vessel system adapts itself to an optimized structure. Mathematical models are used to predict pruning vessels in the experiments of zebra fish. In the second part, I will discuss our recent modeling work on the initiation process of transport networks. Simulation results are used to illustrate how a tree-like structure is obtained from a continuum adaptation equation system, and how loops can exist in our model. Possible further application of this model will also be discussed.

In this work we study the long time, inviscid limit of the 2D Navier-Stokes equations near the periodic Couette flow, and in particular, we confirm at the nonlinear level the qualitative behavior predicted by Kelvin's 1887 linear analysis. At high Reynolds number Re, we prove that the solution behaves qualitatively like 2D Euler for times $t << Re^(1/3)$, and in particular exhibits "inviscid damping" (vorticity mixes and weakly approaches a shear flow). For times $t >> Re^(1/3)$, which is sooner than the natural dissipative time scale $O(Re)$, the viscosity becomes dominant and the streamwise dependence of the vorticity is rapidly eliminated by a mixing-enhanced dissipation effect. Afterwards, the remaining shear flow decays on very long time scales $t >> Re$ back to the Couette flow. The class of initial data we study is the sum of a sufficiently smooth function and a small (with respect to $Re^(-1)$) $L^2$ function. Joint with Nader Masmoudi and Vlad Vicol.

We discuss some recent results on stability of channel flows and boundary layers expansions in the inviscid limit.

The use of PDE models to describe complex systems in the social sciences, such as socio-economic segregation and crime, has been popularized during the past decade. In this talk I will introduce some PDE models which can be seen as basic models for a variety of social phenomena. I will then discuss how these models can be used to explore and gain understanding of the real-world systems they describe. For example, we learn that a populations innate views toward criminal activity can play a significant role in the prevention of crime-wave propagation.

Mathematical models of complex physical processes often involve large number of degrees of freedom as well as events occurring on different time scales. Therefore, direct simulations based on these models face tremendous challenge. This focus of this talk is on the Mori-Zwanzig (MZ) projection formalism for reducing the dimension of a complex dynamical system. The goal is to mathematically derive a reduced model with much fewer variables, while still able to capture the essential properties of the system. In many cases, this formalism also eliminates fast modes and makes it possible to explore events over longer time scales. The models that are directly derived from the MZ projection are typically too abstract to be practically implemented. We will first discuss cases where the model can be simplified to generalized Langevin equations (GLE). Furthermore, we introduce systematic numerical approximations to the GLE, in which the fluctuation-dissipation theorem (FDT) is automatically satisfied. More importantly, these approximations lead to a hierarchy of reduced models with increasing accuracy, which would also be useful for an adaptive model refinement (AMR). Examples, including the NLS, atomistic models of materials defects, and molecular models of proteins, will be presented to illustrate the potential applications of the methods.

Approximate separable representations of Green’s functions for differential operators is a basic and important question in the analysis of differential equations, the development of efficient numerical algorithms and imaging. Being able to approximate a Green’s function as a sum with few separable terms is equivalent to low rank properties of corresponding numerical solution operators. This will allow for matrix compression and fast solution techniques. Green's functions for coercive elliptic differential operators have been shown to be highly separable and the resulting low rank property for discretized system was explored to develop efficient numerical algorithms. However, the case of Helmholtz equation in the high frequency limit is more challenging both mathematically and numerically. We introduce new tools based on the study of relation between two Green’s functions with different source points and a tight dimension estimate for the best linear subspace approximating a set of almost orthogonal vectors to prove new lower bounds for the number of terms in the representation for the Green's function for Helmholtz operator in the high frequency limit. Upper bounds are also derived. We give explicit sharp estimates for cases that are common in practice and present numerical examples. This is a joint work with Bjorn Engquist.

I shall present some recent work on phase-field model for multiphase incompressible flows. We shall pay particular attention to situations with large density ratios as they lead to formidable challenges in both analysis and simulation. I shall present efficient and accurate numerical schemes for solving this coupled nonlinear system, in many case prove that they are energy stable, and show ample numerical results which not only demonstrate the effectiveness of the numerical schemes, but also validate the flexibility and robustness of the phase-field model.

Chemotaxis is a movement of micro-organisms or cells towards the areas of high concentration of a certain chemical, which attracts the cells and may be either produced or consumed by them. In its simplest form, the chemotaxis model is described by a system of nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction-diffusion equation for the chemoattractant concentration. It is well-known that solutions of such systems may develop spiky structures or even blow up in finite time provided the total number of cells exceeds a certain threshold. This makes development of numerical methods for chemotaxissystems extremely delicate and challenging task. In this talk, I will present a new family of high-order finite-volume finite-difference methods for the Keller-Segel chemotaxis system and several related models. Applications of the proposed methods to the classical Patlak-Keller-Segel model, its extensions to the two-species case as well as to the coupled chemotaxis–fluid system will also be discussed.

Many experimental systems can spend extended periods of time relative to their natural time scale in localized regions of phase space, transiting infrequently between them. This display of metastability can arise in stochastically driven systems due to the presence of large energy barriers, or in deterministic systems due to the presence of narrow passages in phase space. To investigate metastability in these different cases, we take the Langevin equation and determine the effects of small damping, small noise, and dimensionality on the dynamics and mean transition time. In finite dimensions, we show the limit of small noise and small damping do not interchange. In the limit of infinite dimensions, we argue the equivalence of the finitely-damped system and the zero-damped infinite energy Hamiltonian system.

The equations governing the motion of a system consisting of a deformable body attached to a rigid body are the partial differential equations for the deformable body subject to boundary conditions that are the equations of motion for the rigid body. (For the ostensibly elementary problem of a mass point on a light spring, the dynamics of the spring itself is typically ignored: The spring is reckoned merely as a feedback device to transmit force to the mass point.) If the inertia of a deformable body is small with respect to that of a rigid body to which it is attached, then the governing equations admit an asymptotic expansion in a small inertia parameter. Even for the simple problem of the spring considered as a continuum, the asymptotics is tricky: The leading term of the regular expansion is not the usual equation for a mass on a massless spring, but is a curious evolution equation with memory. Under very special physical circumstances, an elementary but not obvious process shows that the solution of this equation has an attractor governed by a second-order ordinary differential equation. (This survey of background material is based upon joint work with Michael Wiegner, J. Patrick Wilber, and Shui Cheung Yip.) This lecture describes the rigorous asymptotics and the dimensions of attractors for the motion in space of light nonlinearly viscoelastic rods carrying heavy rigid bodies and subjected to interesting loads. (The motion of the rod is governed by an 18th-order quasilinear parabolic-hyperbolic system.) The justification of the full expansion and the determination of the dimensions of attractors, which gives meaning to these curious equations, employ some simple techniques, which are briefly described (together with some complicated techniques, which are not described). These results come from work with Suleyman Ulusoy.

I will describe a recently discovered equivalence between the first two objects mentioned in the title. The stationary Schr\"odinger's equation, a.k.a. Hill’s equation, is ubiquitous in mathematics, physics, engineering and chemistry. Just to mention one application, the main idea of the Paul trap (for which W. Paul earned the 1989 Nobel Prize in physics) amounts to a certain property of Hill's equation. I will show that Hill's equation is equivalent to a seemingly unrelated problem of “tire tracks”. There is, in addition, a yet another connection between the ``tire tracks” problem and the high frequency forced vibrations which I will also outline briefly.

We are interested in large systems of agents collectively looking for a consensus (about e.g. their direction of motion, like in bird flocks). In spite of the local character of the interactions (only a few neighbours are involved), these systems often exhibit large scale coordinated structures. The understanding of how this self-organization emerges at the large scale is still poorly understood and offer fascinating challenges to the modelling science. We will discuss a few of these issues among (time permitting) phase transitions, propagation of chaos and the derivation of macroscopic models.

In this talk I will discuss several issues concerning the motions of gases expanding into vacuum with or without self-gravitations which are governed by a free-boundary value problem for the 3-dimnesional compressible Euler system with/or without Poisson equation. A general uniqueness theorem for classical solutions to such a free boundary-value problem is presented for physical vacuums. A typical physical vacuum solution includes the famous Lane-Emdan solution in astrophysics. The uniqueness is proved by a relative entropy argument. Then a local well-posedness theory for spherically symmetric motions is established in a less regular space by a deliberate choice of weighted functional to overcome difficulties arising both at the free surface and the symmetry center. Finally, the uniqueness of the spherically symmetric motions is discussed for general equation of state without self-gravitations. This is a joint work with Professor Tao Luo and Professor Huihui Zeng.

Historically, catalytic research and many areas of surface science have used phenomenological rate equations to build kinetic models for surface processes. The models treat surfaces as a lattice of sites, track the probability of finding a site in a particular state, and use maximal-entropy/well-mixed assumption to reconstruct spatially correlated information. This well-mixed assumption, however, often fails. This talk will develop a hierarchy of models that are able to take into account short range spatial correlations. The hierarchy is developed in the context of averaging an underlying master equation. The talk will continue with some simple examples, an example in catalysis, and conclude with ideas on several other applications for this framework.

With Jon Weare (Chicago), we study the continuum limit of a family of kinetic Monte Carlo models of crystal surface relaxation that includes both the solid-on-solid and discrete Gaussian models. With computational experiments and theoretical arguments we are able to derive several partial differential equation limits identified (or nearly identified) in previous studies and to clarify the correct choice of surface tension appearing in the PDE and the correct scaling regime giving rise to each PDE. We also provide preliminary computational and analytic investigations of a number of interesting qualitative features of the large scale behavior of the models. The PDE models involved are fully non-linear Fourth order diffusion type equations with many interesting geometric features. We will given time discuss recent progress analyzing properties of solutions to such PDE.

$\ell_1$ regularization for sparsity has played important role in recent developments in many fields including signal processing, statistics, optimization. The concept of sparsity is usually for the coefficients (i.e., only a small set of coefficients are nonzero) in a well-chosen set of modes (e.g. a basis or dictionary) for representation of the corresponding vectors or functions. In this talk, I will discuss our recent work on a new use of sparsity-promoting techniques to produce “compressed modes/compress plane waves" - modes that are sparse and localized in space - for efficient solutions of constrained variational problems in mathematics and physics. In particularly, I will focus on L1 regularized variational Schrodinger equations for creating spatially localized modes and orthonormal basis, which can efficiently represent localized functions and has promising potential to a variety of applications in many fields such as signal processing, solid state physics, materials science, etc. (This is a joint work with Vidvuds Ozolins, Russel Caflisch and Stanley Osher)

I will describe a linear time algorithm for computing the Riemann map from the unit disk onto an n-gon. The method depends on results from computational geometry (fast computation of the medial axis) and hyperbolic geometry (a theorem of Dennis Sullivan about convex sets in hyperbolic 3-space), as well as classical conformal and quasiconformal theory. Conversely, the fast mapping algorithm implies new results in computational geometry, e.g., (1) quadrilateral meshing for polygons and PSLGs (planar straight line graphs) with optimal time and optimal angle bounds, (2) the first polynomial time algorithm for refining general planar triangulations into non-obtuse triangulations (no angles > 90 degrees; this is desirable for various applications and 90 is the best bound that can be achieved in polynomial time). The talk is intended to be a colloquium-style overview, but I would be happy to discuss more technical details, as requested.

In this talk, we settle a long-standing problem on the connectivity of spaces of finite unit norm tight frames (FUNTFs), essentially affirming a conjecture first appearing in Dykema and Strawn (2003). Our central technique involves continuous liftings of paths from the polytope of eigensteps (see Cahill et al. (2012)) to spaces of FUNTFs. After demonstrating this connectivity result, we refine our analysis to show that the set of nonsingular points on these spaces is also connected, and we use this result to show that spaces of FUNTFs are irreducible in the algebro-geometric sense, and that generic FUNTFs are full spark.

The talk is concerned with a matrix-valued Boltzmann equation derived from the Fermi-Hubbard or Bose-Hubbard model for weak interactions. The quantum analogue of the classical distribution function is the Wigner function, which is matrix-valued to accommodate the spin degree of freedom. Conservation laws and the H-theorem can be proven analytically, and numerical simulations illustrate the time dynamics.

Functionalized polymer membranes have a strong affinity for solvent, imbibing it to form charge-lined networks which serve as charge-selective ion conductions in a host of energy conversion applications. We present a continuum model, based upon a reformulation of the Cahn-Hilliard free energy, which incorporates solvation energy and counter-ion entropy to stabilize a host of network morphologies. We derive geometric evolution for co-dimension 1 bilayers and co-dimension two pore morphologies and show that the system possesses a simple mechanism for competitive evolution of co-existing networks through the common far-field chemical potential.

We study a two-parameter family of solutions of the surface Euler equations in which solutions return to a spatial translation of their initial condition at a later time. Pure standing waves and pure traveling waves emerge as special cases at fixed values of one of the parameters. We find many examples of wave crests that nearly sharpen to a corner, with corner angles close to 120 degrees near the traveling wave of greatest height, and close to 90 degrees for large-amplitude pure standing waves. However, aside from the traveling case, we do not believe any of these solutions approach a limiting extreme wave that forms a perfect corner. We also compute nonlinear wave packets, or breathers, which can take the form of NLS-type solitary waves or counterpropagating wave trains of nearly equal wavelength. In the latter case, an interesting phenomenon occurs in which the pressure develops a large DC component that varies in time but not space, or at least varies slowly in space compared to the wavelength of the surface waves. These large-scale pressure zones can move very rapidly since they travel at the envelope speed, and may be partially responsible for microseisms, the background noise observed in earthquake seismographs.