## Bruce Donald : Some mathematical and computational challenges arising in structural molecular biology

- Applied Math and Analysis ( 304 Views )Computational protein design is a transformative field with exciting prospects for advancing both basic science and translational medical research. New algorithms blend discrete and continuous mathematics to address the challenges of creating designer proteins. I will discuss recent progress in this area and some interesting open problems. I will motivate this talk by discussing how, by using continuous geometric representations within a discrete optimization framework, broadly-neutralizing anti-HIV-1 antibodies were computationally designed that are now being tested in humans - the designed antibodies are currently in eight clinical trials (See https://clinicaltrials.gov/ct2/results?cond=&term=VRC07&cntry=&state=&city=&dist= ), one of which is Phase 2a (NCT03721510). These continuous representations model the flexibility and dynamics of biological macromolecules, which are an important structural determinant of function. However, reconstruction of biomolecular dynamics from experimental observables requires the determination of a conformational probability distribution. These distributions are not fully constrained by the limited information from experiments, making the problem ill-posed in the sense of Hadamard. The ill-posed nature of the problem comes from the fact that it has no unique solution. Multiple or even an infinite number of solutions may exist. To avoid the ill-posed nature, the problem must be regularized by making (hopefully reasonable) assumptions. I will present new ways to both represent and visualize correlated inter-domain protein motions (See Figure). We use Bingham distributions, based on a quaternion fit to circular moments of a physics-based quadratic form. To find the optimal solution for the distribution, we designed an efficient, provable branch-and-bound algorithm that exploits the structure of analytical solutions to the trigonometric moment problem. Hence, continuous conformational PDFs can be determined directly from NMR measurements. The representation works especially well for multi-domain systems with broad conformational distributions. Ultimately, this method has parallels to other branches of applied mathematics that balance discrete and continuous representations, including physical geometric algorithms, robotics, computer vision, and robust optimization. I will advocate for using continuous distributions for protein modeling, and describe future work and open problems.

## Min Kang : Tropically Linear Interface Growth Models

- Applied Math and Analysis ( 249 Views )We first discuss a general method to derive macroscopic laws from certain microscopic interactions that can be applied to a large class of particle systems. In particular we consider a broad class of systems that are linear in a special algebra, so-called tropical algebra. Some natural connections among the scaling limits of these random systems, the solutions to specific partial differential equations (Hamilton-Jacobi type) and combinatorial optimization problems have been noticed. If time allows, we further discuss a useful application of the variational formula (microscopic version of Hopf-Lax formula) to a well-known interacting particle system, totally asymmetric simple exclusion process.

## Yian Ma : Bridging MCMC and Optimization

- Applied Math and Analysis ( 223 Views )In this talk, I will discuss three ingredients of optimization theory in the context of MCMC: Non-convexity, Acceleration, and Stochasticity.

I will focus on a class of non-convex objective functions arising from mixture models. For that class of objective functions, I will demonstrate that the computational complexity of a simple MCMC algorithm scales linearly with the model dimension, while optimization problems are NP-hard.

I will then study MCMC algorithms as optimization over the KL-divergence in the space of measures. By incorporating a momentum variable, I will discuss an algorithm which performs "accelerated gradient descent" over the KL-divergence. Using optimization-like ideas, a suitable Lyapunov function is constructed to prove that an accelerated convergence rate is obtained.

Finally, I will present a general recipe for constructing stochastic gradient MCMC algorithms that translates the task of finding a valid sampler into one of choosing two matrices. I will then describe how stochastic gradient MCMC algorithms can be applied to applications involving temporally dependent data, where the challenge arises from the need to break the dependencies when considering minibatches of observations.

## Casey Rodriguez : The Radiative Uniqueness Conjecture for Bubbling Wave Maps

- Applied Math and Analysis ( 191 Views )One of the most fundamental questions in partial differential equations is that of regularity and the possible breakdown of solutions. We will discuss this question for solutions to a canonical example of a geometric wave equation; energy critical wave maps. Break-through works of Krieger-Schlag-Tataru, Rodnianski-Sterbenz and Rapha Ì?el-Rodnianski produced examples of wave maps that develop singularities in finite time. These solutions break down by concentrating energy at a point in space (via bubbling a harmonic map) but have a regular limit, away from the singular point, as time approaches the final time of existence. The regular limit is referred to as the radiation. This mechanism of breakdown occurs in many other PDE including energy critical wave equations, Schro Ì?dinger maps and Yang-Mills equations. A basic question is the following: â?¢ Can we give a precise description of all bubbling singularities for wave maps with the goal of finding the natural unique continuation of such solutions past the singularity? In this talk, we will discuss recent work (joint with J. Jendrej and A. Lawrie) which is the first to directly and explicitly connect the radiative component to the bubbling dynamics by constructing and classifying bubbling solutions with a simple form of prescribed radiation. Our results serve as an important first step in formulating and proving the following Radiative Uniqueness Conjecture for a large class of wave maps: every bubbling solution is uniquely characterized by itâ??s radiation, and thus, every bubbling solution can be uniquely continued past blow-up time while conserving energy.

## Wenjun Ying : Recent developments of the kernel-free boundary integral method

- Applied Math and Analysis ( 188 Views )The kernel-free boundary integral method is a Cartesian grid based method for solving elliptic partial differential equations (PDEs). It solves elliptic PDEs in the framework of boundary integral equations (BIEs). The method evaluates boundary and volume integrals by solving equivalent simple interface problems on Cartesian grids. It takes advantages of the well-conditioning properties of the BIE formulation, the convenience of grid generation with Cartesian grids and the availability of fast and efficient elliptic solvers for the simple interface problems. In this talk, I will present recent developments of the method for the reaction-diffusion equations in computational cardiology, the nonlinear Poisson-Boltzmann equation in biophysics, the Stokes equation in fluid dynamics as well as some free boundary and moving interface problems.

## Jim Nolen : Asymptotic Spreading of Reaction-Diffusion Fronts in Random Media

- Applied Math and Analysis ( 169 Views )Some reaction-advection-diffusion equations admit traveling wave solutions; these are simple models of a combustion reaction spreading with constant speed. Even in a random medium, solutions to the initial value problem may develop fronts propagating with a well-defined asymptotic speed. First, I will describe this behavior when the nonlinearity is the Kolmogorov-Petrovsky-Piskunov (KPP) type nonlinearity and the randomness comes from a prescribed random drift (a simple model of turbulent combustion). Next, I will describe propagation of fronts when the nonlinearity is a random ignition-type nonlinearity. In the latter case, there exist special solutions that generalize the notion of a traveling wave in the random setting.

## Boris Malomed : Spatiotemporal optical solitons: an overview

- Applied Math and Analysis ( 157 Views )An introduction to the topic of multi-dimensional optical solitons ("light bullets"), localized simultaneously in the direction of propagation (as temporal solitons) and in one or two transverse directions (as spatial solitons) will be given, including a review of basic theoretical and experimental results. Also considered will be connection of this topic to the problem of the creation of multidimensional solitons in Bose-Einstein condensates. In both settings (optical and BEC), the main problem is stabilization of the multidimensional solitons against the spatiotemporal collapse. The stabilization may be provided in various ways (in particular, by means of an optical lattice in BEC). The talk will partly based on a review article: B.A. Malomed, D. Mihalache, F. Wise, and L. Torner, "Spatiotemporal optical solitons", J. Optics B: Quant. Semics. Opt. 7, R53-R72 (2005).

## Catalin Turc : Domain Decomposition Methods for the solution of Helmholtz transmission problems

- Applied Math and Analysis ( 143 Views )We present several versions of non-overlapping Domain Decomposition Methods (DDM) for the solution of Helmholtz transmission problems for (a) multiple scattering configurations, (b) bounded composite scatterers with piecewise constant material properties, and (c) layered media. We show that DDM solvers give rise to important computational savings over other existing solvers, especially in the challenging high-frequency regime.

## Vladimir Sverak : On long-time behavior of 2d flows

- Applied Math and Analysis ( 134 Views )Our knowledge of the long-time behavior of 2d inviscid flows is quite limited. There are some appealing conjectures based on ideas in Statistical Mechanics, but they appear to be beyond reach of the current methods. We will discuss some partial results concerning the dynamics, as well as some results for variational problems to which the Statistical Mechanics methods lead.

## Cyrill Muratov : On shape of charged drops: an isoperimetric problem with a competing non-local term

- Applied Math and Analysis ( 130 Views )In this talk I will give an overview of my recent work with H. Knuepfer on the analysis of a class of geometric problems in the calculus of variations. I will discuss the basic questions of existence and non-existence of energy minimizers for the isoperimetric problem with a competing non-local term. A complete answer will be given for the case of slowly decaying kernels in two space dimensions, and qualitative properties of the minimizers will be established for general Riesz kernels.

## Ruiwen Shu : Flocking hydrodynamics with external potentials

- Applied Math and Analysis ( 128 Views )We study the large-time behavior of hydrodynamic model which describes the collective behavior of continuum of agents, driven by pairwise alignment interactions with additional external potential forcing. The external force tends to compete with alignment which makes the large time behavior very different from the original Cucker-Smale (CS) alignment model, and far more interesting. Here we focus on uniformly convex potentials. In the particular case of \emph{quadratic} potentials, we are able to treat a large class of admissible interaction kernels, $\phi(r) \gtrsim (1+r^2)^{-\beta}$ with `thin' tails $\beta \leq 1$ --- thinner than the usual `fat-tail' kernels encountered in CS flocking $\beta\leq\nicefrac{1}{2}$: we discover unconditional flocking with exponential convergence of velocities \emph{and} positions towards a Dirac mass traveling as harmonic oscillator. For general convex potentials, we impose a necessary stability condition, requiring large enough alignment kernel to avoid crowd dispersion. We prove, by hypocoercivity arguments, that both the velocities \emph{and} positions of smooth solution must flock. We also prove the existence of global smooth solutions for one and two space dimensions, subject to critical thresholds in initial configuration space. It is interesting to observe that global smoothness can be guaranteed for sub-critical initial data, independently of the apriori knowledge of large time flocking behavior.

## Badal Joshi : A coupled Poisson process model for sleep-wake cycling

- Applied Math and Analysis ( 127 Views )Sleep-wake cycling is an example of switching between discrete states in mammalian brain. Based on the experimental data on the activity of populations of neurons, we develop a mathematical model. The model incorporates several different time scales: firing of action potentials (milliseconds), sleep and wake bout times (seconds), developmental time (days). Bifurcation diagrams in a deterministic dynamical system gives the occupancy time distributions in the corresponding stochastic system. The model correctly predicts that forebrain regions help to stabilize wake state and thus modifies the wake bout distribution.

## Mark Hoefer : Eulerian Dispersive Shock Waves and Instabilities

- Applied Math and Analysis ( 125 Views )Recent experimental and theoretical research in Bose-Einstein condensation and nonlinear optics have demonstrated novel supersonic, fluid-like phenomena. Shock waves in these and other systems are modeled by a dispersive regularization of Euler's equations, implemented by use of the Whitham averaging technique. Normal and oblique dispersive shock waves (DSWs) connecting two constant states are constructed. Numerical computations of supersonic, dispersive flow over a corner in the special case of systems modeled by the Nonlinear Schrodinger equation (NLS) exhibit stable pattern formation (oblique DSWs) or instability (turbulent-like behavior) depending on the flow parameters. A combination of analytical and computational approaches are used to demonstrate that this change in behavior can be identified with the transition from convective to absolute instability of dark solitons. The linearized NLS behavior about the dark soliton DSW trailing edge is studied in detail to identify the separatrix between convective and absolute instabilities.

## Pete Casazza : Applications of Hilbert space frames

- Applied Math and Analysis ( 124 Views )Hilbert space frames have traditionally been used in signal/image processing. Recently, there have arisen a variety of new applications to speeding up the internet, producing cell phones which won't fade, quantum information theory, distributed processing and more. We will review the fundamentals of frame theory and then look at the myriad of applications of frames.

## Xin Yang Lu : EVOLUTION EQUATIONS FROM EPITAXIAL GROWTH

- Applied Math and Analysis ( 122 Views )Epitaxial growth is a process in which a thin film is grown above a much thicker substrate. In the simplest case, no deposition is considered, and all the interactions are assumed to be purely elastic. However, since the film may potentially have different rigidity constant from the substate, such growth leads to a nonuniform film thickness. The equations governing epitaxial growth are high order (generally fourth order), nonlocal, and highly nonlinear. In this talk I will present some recent results about the regularity of solutions to several equations arising from epitaxial growth. Joint work with I. Fonseca and G.Leoni.

## Qin Li : Low rankness in forward and inverse kinetic theory

- Applied Math and Analysis ( 120 Views )Multi-scale kinetic equations can be compressed: in certain regimes, the Boltzmann equation is asymptotically equivalent to the Euler equations, and the radiative transfer equation is asymptotically equivalent to the diffusion equation. A lot of detailed information is lost when the system passes to the limit. In linear algebra, it is equivalent to being of low rank. I will discuss such transition and how it affects the computation: mainly, in the forward regime, inserting low-rankness could greatly advances the computation, while in the inverse regime, the system being of low rank typically makes the problems significantly harder.

## Lek-Heng Lim : Multilinear Algebra and Its Applications

- Applied Math and Analysis ( 119 Views )In mathematics, the study of multilinear algebra is largely limited to properties of a whole space of tensors --- tensor products of k vector spaces, modules, vector bundles, Hilbert spaces, operator algebras, etc. There is also a tendency to take an abstract coordinate-free approach. In most applications, instead of a whole space of tensors, we are often given just a single tensor from that space; and it usually takes the form of a hypermatrix, i.e.\ a k-dimensional array of numerical values that represents the tensor with respect to some coordinates/bases determined by the units and nature of measurements. How could one analyze this one single tensor then? If the order of the tensor k = 2, then the hypermatrix is just a matrix and we have access to a rich collection of tools: rank, determinant, norms, singular values, eigenvalues, condition number, etc. This talk is about the case when k > 2. We will see that one may often define higher-order analogues of common matrix notions rather naturally: tensor ranks, hyperdeterminants, tensor norms (Hilbert-Schmidt, spectral, Schatten, Ky Fan, etc), tensor eigenvalues and singular values, etc. We will discuss the utility as well as difficulties of various tensorial analogues of matrix problems. In particular we shall look at how tensors arise in a variety of applications including: computational complexity, control engineering, mathematical biology, neuroimaging, quantum computing, signal processing, spectroscopy, and statistics.

## Wencai Liu : Spectral transitions for Schr\odinger operators with decaying potentials and Laplacians on asymptotically flat (hyperbolic) manifolds

- Applied Math and Analysis ( 116 Views )We apply piecewise constructions and gluing technics to construct asymptotically flat (hyperbolic) manifolds such that associated Laplacians have dense embedded eigenvalues or singular continuous spectra. The method also allows us to provide various examples of operators with embedded singular spectra, including perturbed periodic operators, periodic Jacobi operators, and Stark operators. We establish the asymptotic behavior (WKB for example) of eigensolutions under small perturbations, which implies certain rules for the absence of singular spectra. As a result, several sharp spectral transitions (even criteria) for a single (finitely many or countably many) embedded eigenvalues, singular continuous spectra and essential supports of spectral measures are obtained. The talk is based on several papers, some joint with Jitomirskaya and Ong.

## Liliana Borcea : Pulse Reflection in a Random Waveguide with a Turning Point

- Applied Math and Analysis ( 116 Views )Guided waves arise in a variety of applications like underwater acoustics, optics, the design of musical instruments, and so on. We present an analysis of wave propagation and reflection in an acoustic waveguide with random sound soft boundary and a turning point. The waveguide has slowly bending axis and variable cross section. The variation consists of a slow and monotone change of the width of the waveguide and small and rapid fluctuations of the boundary, on the scale of the wavelength. These fluctuations are modeled as random. The turning point is many wavelengths away from the source, which emits a pulse that propagates toward the turning point, where it is reflected. We consider a regime where scattering at the random boundary has a significant effect on the reflected pulse. We determine from first principles when this effects amounts to a deterministic pulse deformation. This is known as a pulse stabilization result. The reflected pulse shape is not the same as the emitted one. It is damped, due to scattering at the boundary, and is deformed by dispersion in the waveguide. An example of an application of this result is in inverse problems, where the travel time of reflected pulses at the turning points can be used to determine the geometry of the waveguide.

## Christoph Ortner : Multi-scale simulation of crystal defects

- Applied Math and Analysis ( 114 Views )PART 1: I will construct a mathematical model of a defect embedded in an infinite homogeneous crystal. I will then establish a regularity result for minimisers, which given the crucial information on which approximation schemes are based. As an elementary application of this framework I will prove convergence rates for two computational schemes: (1) clamped far-field and (2) coupling to harmonic far-field model.

PART 2: The conditions under which the theory of Part 1 holds are separability and locality of the total energy. In Part 2 I will show how for a tight-binding model (a minimalistic electronic structure model) these two condition arise. This analysis raises some interesting (open) questions.

PART 3: Finally, I will use the theory developed in PART 1 and PART 2 to construct and analyse a new family of QM/MM embedding schemes with rigorous error estimates.

## Cynthia Rudin : 1) Regulating Greed Over Time: An Important Lesson For Practical Recommender Systems and 2) Prediction Uncertainty and Optimal Experimental Design for Learning Dynamical Systems

- Applied Math and Analysis ( 113 Views )I will present work from these two papers:
1) *Regulating Greed Over Time.* Stefano Traca and Cynthia Rudin. 2015
Finalist for 2015 IBM Service Science Best Student Paper Award
2) *Prediction Uncertainty and Optimal Experimental Design for Learning Dynamical Systems.* Chaos, 2016.
Benjamin Letham, Portia A. Letham, Cynthia Rudin, and Edward Browne.

There is an important aspect of practical recommender systems that we noticed while competing in the ICML Exploration-Exploitation 3 data mining competition. The goal of the competition was to build a better recommender system for Yahoo!'s Front Page, which provides personalized new article recommendations. The main strategy we used was to carefully control the balance between exploiting good articles and exploring new ones in the multi-armed bandit setting. This strategy was based on our observation that there were clear trends over time in the click-through-rates of the articles. At certain times, we should explore new articles more often, and at certain times, we should reduce exploration and just show the best articles available. This led to dramatic performance improvements.

As it turns out, the observation we made in the Yahoo! data is in fact pervasive in settings where recommender systems are currently used. This observation is simply that certain times are more important than others for correct recommendations to be made. This affects the way exploration and exploitation (greed) should change in our algorithms over time. We thus formalize a setting where regulating greed over time can be provably beneficial. This is captured through regret bounds and leads to principled algorithms. The end result is a framework for bandit-style recommender systems in which certain times are more important than others for making a correct decision.

If time permits I will discuss work on measuring uncertainty in parameter estimation for dynamical systems. I will present "prediction deviation," a new metric of uncertainty that determines the extent to which observed data have constrained the model's predictions. This is accomplished by solving an optimization problem that searches for a pair of models that each provide a good fit for the observed data, yet have maximally different predictions. We develop a method for estimating a priori the impact that additional experiments would have on the prediction deviation, allowing the experimenter to design a set of experiments that would most reduce uncertainty.

## Rayan Saab : Quantization of compressed sensing measurements and frame expansions

- Applied Math and Analysis ( 111 Views )Compressed sensing, as a signal acquisition technique, has been shown to be highly effective for dimensionality reduction. On the other hand, reconstruction from compressed sensing measurements is highly non-linear and requires digital computers. Thus, quantizing (i.e., digitizing) compressed sensing measurements is an important, albeit under-addressed topic. In this talk, we show that by using $\Sigma\Delta$ quantization instead of the most commonly assumed approach (uniform quantization), a significant reduction in the reconstruction error is possible. In particular, we prove error decay rates of $\lambda^{-c r}$ where $\lambda$ is the ratio of the number of measurements to the sparsity of the underlying signal, and $r$ is the order of the $\Sigma\Delta$ scheme. In addition to the compressed sensing scenario we also consider the quantization of frame expansions, where one collects more measurements than the ambient dimension. We show state of the art results for certain frames (including random frames) and $\Sigma\Delta$ schemes. In particular, we prove error rates of $e^{-c\sqrt{\lambda}}$, where $\lambda$ is the oversampling ratio.

## Andrea Bertozzi : Geometry based image processing - a survey of recent results

- Applied Math and Analysis ( 108 Views )I will present a survey of recent results on geometry-based image processing. The topics will include wavelet-based diffuse interface methods, pan sharpening and hyperspectral sharpening, and sparse image representation.

## Yifei Lou : Nonconvex Approaches in Data Science

- Applied Math and Analysis ( 108 Views )Although Â?big dataÂ? is ubiquitous in data science, one often faces challenges of Â?small data,Â? as the amount of data that can be taken or transmitted is limited by technical or economic constraints. To retrieve useful information from the insufficient amount of data, additional assumptions on the signal of interest are required, e.g. sparsity (having only a few non-zero elements). Conventional methods favor incoherent systems, in which any two measurements are as little correlated as possible. In reality, however, many problems are coherent. I will present a nonconvex approach that works particularly well in the coherent regime. I will also address computational aspects in the nonconvex optimization. Various numerical experiments have demonstrated advantages of the proposed method over the state-of-the-art. Applications, ranging from super-resolution to low-rank approximation, will be discussed.

## Pierre Degond : Collective dynamics and self-organization

- Applied Math and Analysis ( 107 Views )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.

## Shi Jin : Asymptotic-preseving schemes for the Boltzmann equation and relative problems with multiple scales

- Applied Math and Analysis ( 106 Views )We propose a general framework to design asymptotic preserving schemes for the Boltzmann kinetic and related equations. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We propose to penalize the nonlinear collision term by a BGK-type relaxation term, which can be solved explicitly even if discretized implicitly in time. Moreover, the BGK-type relaxation operator helps to drive the density distribution toward the local Maxwellian, thus naturally imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver or the use of Wild Sum. It is uniformly stable in terms of the (possibly small) Knudsen number, and can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. We will show how this idea can be applied to other collision operators, such as the Landau-Fokker-Planck operator, Ullenbeck-Urling model, and in the kinetic-fluid model of disperse multiphase flows, and can be implemented in the Monte-Carlo framework in which is time step is not limited by the possibly small mean free time.

## Ravi Srinivasan : Kinetic theory for shock clustering and Burgers turbulence

- Applied Math and Analysis ( 105 Views )A remarkable model of stochastic coalescence arises from considering shock statistics in scalar conservation laws with random initial data. While originally rooted in the study of Burgers turbulence, the model has deep connections to statistics, kinetic theory, random matrices, and completely integrable systems. The evolution takes the form of a Lax pair which, in addition to yielding interesting conserved quantities, admits some rather intriguing exact solutions. We will describe several distinct derivations for the evolution equation and, time-permitting, discuss properties of the corresponding kinetic system. This talk consists of joint work with Govind Menon (Brown).

## Chris Bishop : Conformal maps and optimal meshes

- Applied Math and Analysis ( 105 Views )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.

## Yoshiaki Teramoto : Benard-Marangoni problem of heat convection with free surface

- Applied Math and Analysis ( 103 Views )When a fluid layer is heated from below with temperature larger than a certain critical value, the convective motion appears in the fluid. The convection caused by the thermocapillary effect is called Benard-Marangoni heat convection. The thermocapillary effect is the dependence of the surface tension on the temperature. Near a hot spot on a free surface of fluid a thermocapillary tangential stress generates a fluid motion. In this talk the mathematical model system for this convection is explained. The Oberbeck-Boussinesq approximation is used for the system and the upper boundary is a free surface with surface tension which depends on the temperature. After formulating the linearized problem around the conductive state, we derive the resolvent estimates which guarantee the sectorial property. Stationary and Hopf bifurcations (periodic solutions) are proved to exist depending on the parameters (Raylegh and Marangoni numbers).

## Jun Kitagawa : Free discontinuity regularity and stability in optimal transport

- Applied Math and Analysis ( 103 Views )Regularity of solutions in the optimal transport problem requires very rigid hypotheses (e.g., convexity of certain sets). When such conditions are not available, one can consider the question of partial regularity, in other words, the in-depth analysis of the structure of singular sets. In this talk, I will discuss the regularity of the set of ``free singularities`` which arise in an optimal transport problem with inner product cost, from a connected set to a disconnected set, along with the stability of such sets under suitable perturbations of the data involved. Some of these results are proven via a non-smooth implicit function theorem for convex functions, which is of independent interest. This talk is based on joint work with Robert McCann.