Past Seminars
Past Seminars
History: This seminar was initiated by Azita Mayeli in 2012 and has run continuously since then. It primarily hosts speakers in the fields of harmonic analysis and analysis of partial differential equations, as well as their applications. Topics of interest include, but are not limited to, sampling and frames theory, time-frequency analysis, Riesz bases and signal processing, Fuglede Conjecture, singular integral theory, oscillatory integral operators, restriction and Kakeya-type estimates, decoupling, analysis in data science, dispersive equations, hydrodynamic equations, the mathematics of turbulence, regularity theory, stochastic PDEs, ergodic theory, long-time behavior of dynamical systems. If you would like to present a talk or added to our mailing list, please contact one of the corresponding co-organizers.
Abstract: Steady states of the two-dimensional Euler equations generally come in infinite-dimensional families and play an important role in the long-time dynamics of generic initial data. In this talk we will introduce several classes of steady states, we will discuss recent results on their structures and we will provide a geometric characterization of them in the periodic channel and annulus.
Abstract: Despite the small scales involved, the compressible Euler equations seem to be a good model even in the presence of shocks. Introducing viscosity is one way to resolve some of these small scale effects. In this talk, we examine the vanishing viscosity limit near the formation of a generic shock in one spatial dimension for a class of viscous conservation laws which includes compressible Navier Stokes. In a nutshell, we recover the inviscid (singular) solution in the limit, and we uncover universal structure in the viscous correctors. The proof involves matching approximate solutions constructed in regions where the viscosity is perturbative and where it is dominant. This is joint work in progress with Sanchit Chaturvedi and Cole Graham.
Abstract: I will introduce a new flow for Liquid Crystals enjoying a geometric free boundary condition. This natural (simplified) model enjoys many properties similar to the system of PDEs introduced by Lin in the 80’s, some of them more rigid than the classical one. I will explain the existence and smoothness of weak solutions versus the appearance of blow-ups. The structure of the PDEs are shared with other types of models appearing in geometry and mathematical physics, like the Harmonic-Ricci flow or the Teichmuller flow for instance. I will try to explain a general strategy to approach regularity/singularity in such equations.
Abstract: I will discuss how the existence of functions satisfying certain PDE inequalities constrains the geometry of complete non-compact Riemannian manifolds. In particular I will discuss the following result. Let (Mⁿ,g) be a complete non-compact Riemannian manifold with two ends, and with n≥2. Denote Ric the (pointwise) lowest eigenvalue of the Ricci tensor, and assume u is a positive function on M satisfying −γΔu+Ric*u≥0 and γ<4/(n-1). Then, Mⁿ is isometric to the product of IR x Nⁿ⁻¹. This extends a result of Cheeger--Gromoll from 1971. It is a joint work with M. Pozzetta and K. Xu.
Abstract: In this talk I will discuss a finite-time vorticity blowup result for the 2D compressible Euler equations, with smooth, localized and non-vacuous initial data. The vorticity blowup occurs at the time of the first singularity, and is accompanied by an axisymmetric implosion in which the swirl velocity enjoys full stability, as opposed to finite co-dimension stability. This is a joint work with Jiajie Chen, Steve Shkoller and Vlad Vicol.
Abstract: Despite decades of significant progress in numerical algorithms, real-time solvers of partial differential equations (PDEs) describing nonlinear wave phenomena generally remain out of reach. The key obstacle is the apparent lack of low-dimensional structure in wave phenomena. For example, it is known that the Kolmogorov width decays slowly for solution manifolds of nonlinear conservation laws, prohibiting construction of efficient linear approximations. In this talk, I will discuss theoretical and computational results that point towards a certain compositional low-rankness in entropy solutions of scalar conservation laws, as described by a neural network architecture we call low rank neural representation (LRNR). These results lead to new approaches in dimensionality reduction as well as to new real-time solvers applicable to various PDEs. In particular, I will discuss how (1) a theoretical LRNR construction with small parameter dimension can efficiently approximate solutions to scalar nonlinear conservation laws involving arbitrarily complicated shock interactions, (2) LRNRs can be used within the physics informed neural networks (PINNs) to efficiently compute PDE solutions. This talk is based on joint works with Woojin Cho (Yonsei U.), KookjinLee (Arizona State U.), Noseong Park (KAIST), and Gerrit Welper (U. of Central Florida).
Abstract: We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our analysis is motivated by Landau's analysis of spherically-symmetric shock waves, who predicted that at large times, not just one, but two shocks emerge. These shocks are logarithmically-separated from the Minkowskian light cone and the fluid velocity decays at the non-time-integrable rate 1/(t(\log t)^{1/2}). We show that for initial data, which need not be spherically-symmetric, with two shocks in it and which is sufficiently close, in appropriately weighted Sobolev norms, to an N-wave profile, the solution to the shock-front initial value problem can be continued for all time and does not develop any further singularities. In particular this is the first proof of global existence for solutions (which are necessarily singular) of a quasilinear wave equation in three space dimensions which does not verify the null condition. The proof requires carefully-constructed multiplier estimates and analysis of the geometry of the shock surfaces.
Abstract: Morrey's inequality measures the Hölder continuity of a function whose gradient belongs to an appropriate Lebesgue space. There has been recent interest in understanding the extremals of Morrey's inequality, which are the functions which saturate the inequality. We present a natural variant of Morrey's inequality on a domain which involves the distance to the boundary and discuss the question of whether or not an extremal exists. This is joint work with Simon Larson (Chalmers) and Erik Lindgren (KTH).
Abstract: One of the most common procedures in modern data analytics is filling in missing values in times series. For a variety of reasons, the data provided by clients to obtain a forecast, or other forms of data analysis, may have missing values, and those values need to be filled in before the data set can be properly analyzed. Many freely available forecasting software packages, such as the Facebook Prophet or the sktime library, have built-in mechanisms for filling in missing values. Here, we develop a simple procedure, based on discrete Fourier analysis and pruning, that works hand-in-hand with the existing methods to produce a more accurate method for filling in missing values in time series. A theoretical justification of our method is given. In addition, we adapt the classical $L^1$ minimization method for signal recovery to the filling of missing values in times, both by itself and in conjunction with the Fourier pruning procedure above. The theoretical justifications of these methods leverage results by Bourgain.
Abstract: Spectral estimation is a fundamental problem of inverting a Fourier sum from noisy samples. Major difficulties arise if the frequencies are unknown and non-harmonic, which is usually the case in many imaging and signal processing problems. The stability of this nonlinear inverse problem behaves differently depending on the relationship between the number of samples 2m-1 versus the separation between the frequencies $\Delta$. This talk will mainly focus on the favorable well-separated case where $m \Delta \gg 2 \pi$. We develop a novel algorithm called Gradient-MUSIC, which is a non-convex implementation of the classical MUSIC algorithm. We provide a geometric analysis of its associated objective function and prove that Gradient-MUSIC estimates the frequencies and amplitudes at minimax optimal rates, for a variety of deterministic and stochastic noise. Time permitting, we discuss some work on the super-resolution regime where $m \Delta \ll 2 \pi$. This is joint work with Albert Fannjiang and Wenjing Liao. This talk is accessible to graduate students.
Abstract: In this talk, we will discuss recent developments in constructing (nearly) self-similar singularities in the incompressible Euler, compressible Euler, and related equations. Our approach combines computer-assisted construction, weighted energy estimates, compact perturbation methods, and soft functional analysis arguments.
September 13
Arash Farashahi (University of Delaware)
Title: Covariant Functions of Characters of Compact Subgroups
Abstract: In this talk, we will study abstract harmonic analysis on classical Banach spaces of covariant functions of characters of compact subgroups. Let $G$ be a locally compact group and $H$ be a compact subgroup of $G$. Suppose $\xi:H\to\mathbb{T}$ is a continuous character, $1\le p<\infty$ and $L_\xi^p(G,H)$ is the set of all covariant functions of $\xi$ in $L^p(G)$. We prove that $L^p_\xi(G,H)$ is isometrically isomorphic to a quotient space of $L^p(G)$. Then we apply our results to show that $L^q_\xi(G,H)$ is isometrically isomorphic to the dual space $L^p_\xi(G,H)^*$, where $q$ is the conjugate exponent of $p$. We also discuss some results for the case that $G$ is compact.
September 20
Nestor Guillen (Texas State University)
Title: The Landau equation does not blow up
Abstract: In a recent preprint with Luis Silvestre we show the monotonicity of the Fisher information for smooth solutions of the homogeneous Landau equation. From this fact one concludes that initially smooth solutions do not blow up and remain smooth for all times. This is proved for a broad range of interaction potentials — including the important case of the Coulomb potential. This result is made possible by three ingredients: 1) a “lifting” of the Landau equation to a linear degenerate parabolic PDE in double the number of variables, 2) a decomposition of this linear PDE in terms of rotations and 3) a functional inequality on the sphere that is closely related to the log-Sobolev inequality on $\mathbb{S}^2$. In this talk I will describe these ingredients starting by motivating some of them for the simpler case of the heat equation. Further consequences and extensions of the result --including recently for the Boltzmann equation in a new preprint of Imbert, Silvestre, and Villani— will be discussed time permitting.
September 27
Daniil Glukhovskiy (Stony Brook University)
Title: Point vortices and “pensive billiards”
Abstract: Motivated by the motion of point vortices in two-dimensional ideal hydrodynamics in bounded domains, we define a new class of plane billiards - the `pensive billiard' - in which the billiard ball travels along the boundary for some distance depending on the incidence angle before reflecting. In addition to vortex motion, this system turns out to generalize so called `puck billiards' proposed by M.Bialy. We shall discuss dynamical properties of pensive billiards, such as variational origin, invariance of a symplectic structure, conditions for a twist map, the existence of periodic orbits, etc. We also demonstrate the appearance of both the golden and silver ratios in the corresponding hydrodynamical vortex setting. Finally, we introduce and describe basic properties of pensive outer billiards. This is joint work with Theo Drivas and Boris Khesin.
October 4
(Rosh Hashanah)
October 11
(Yom Kippur)
October 18
Igor Kukavica (University of Southern California)
Title: On the inviscid limit for the Navier-Stokes equations
Abstract: The question of whether the solution of the Navier-Stokes equation converges to the solution of the Euler equation as the viscosity vanishes is an important one in fluid dynamics. In the talk, we will review current results on this problem. We will also present a result, joint with V. Vicol and F. Wang, which shows that the inviscid limit holds for the initial data that is analytic only close to the boundary of the domain, and has Sobolev regularity in the interior. We will also discuss the Prandtl expansions of solutions of the Navier-Stokes equations.
October 25
Yupei Huang (Duke)
Title: Classification of the analytic steady states of 2D Euler equation
Abstract: Classification of the steady states for 2D Euler equation is a classical topic in fluid mechanics. In this talk, we consider the rigidity of the analytic steady states in bounded simply-connected domains. By studying an over-determined elliptic problem in Serrin type, we show the stream functions for the steady state are either radial functions or solutions to semi-linear elliptic equations. This is the joint work with Tarek Elgindi, Ayman Said and Chunjing Xie.
November 1
Maja Taskovic (Emory University) THIS TALK WILL START AT 3PM
Title: On the inhomogeneous wave kinetic equation and its associated hierarchy
Abstract: The wave kinetic equation is one of the fundamental models in the theory of wave turbulence, and provides a statistical description of weakly nonlinear interacting waves. This talk will address the global in time well-posedness of the spatially inhomogeneous wave kinetic equation by applying techniques inspired by the analysis of the Boltzmann equation – another model of statistical physics that describes evolution of rarefied gases in which particles undergo predominantly binary interactions. We will also discuss the well-posedness of the wave kinetic hierarchy – an infinite system of coupled equations closely related to the wave kinetic equation. Existence of global in time solutions of this hierarchy is obtained by combining the corresponding result for the wave kinetic equation together with a result from the probability theory known as the Hewitt-Savage theorem. Uniqueness of solutions to the wave kinetic hierarchy is proved with the help of a combinatorial technique, known as the Klainerman-Machedon board game argument, typically used for dispersive PDE, which allows us to control the factorial growth of the Dyson series. This is a joint work with Ioakeim Ampatzoglou, Joseph K. Miller and Natasa Pavlovic.
November 8
James Murphy (Tufts)
Title: Intrinsic Models in Wasserstein Space with Applications to Signal and Image Processing
Abstract: We study the problems of efficient modeling and representation learning for probability distributions in Wasserstein space. We consider a general barycentric coding model in which data are represented as 2-Wasserstein (W2) barycenters of a set of fixed reference measures. Leveraging the geometry of W2-space, we develop a tractable optimization program to learn the barycentric coordinates and provide a consistent statistical procedure for learning these coordinates when the measures are accessed only by i.i.d. samples. Our consistency results and algorithms exploit entropic regularization of the optimal transport problem, and the statistical convergence of entropic optimal transport maps will be discussed. We also consider the problem of learning reference measures given observed data. Our regularized approach to dictionary learning in W2-space addresses core problems of ill-posedness and in practice learns interpretable dictionary elements and coefficients useful for downstream tasks. We will also consider the question of representational capacity of barycentric models in W2-space and contrast with linear approximation methods. Applications of optimal transport to image inpainting, reduced order modeling of molecular dynamics simulations, and nonlinear hyperspectral unmixing will be considered.
November 15
Jackie Lok (Princeton)
Title: A subspace constrained randomized Kaczmarz method
Abstract: The Kaczmarz method is a classical iterative algorithm for solving large-scale, highly overdetermined systems of linear equations that has found applications in areas such as image and signal processing. In this talk, we will discuss a randomized Kaczmarz algorithm where the iterates are confined to the solution space of a selected subsystem. We will show that the subspace constraint leads to an accelerated convergence rate, especially when the system is coherent or has approximately low rank structure. Furthermore, for the problem of solving linear systems with sparsely corrupted measurements, we will discuss how this idea can be used in a Kaczmarz-based algorithm to incorporate external knowledge about corruption-free equations to help achieve convergence. This is joint work with Elizaveta Rebrova.
November 22
Jincheng Yang (Institute for Advanced Study)
Title: Energy dissipation near the outflow boundary in the vanishing viscosity limit
Abstract: We consider the incompressible Navier-Stokes and Euler equations in a bounded domain with either characteristic or non-characteristic boundary condition, and study the energy dissipation in the zero-viscosity limit. For the characteristic case, we show energy dissipation rate is bounded from above by $\bar V ^3$, where $\bar V$ is the strength of Euler on the boundary. For the non-characteristic case, the energy dissipation rate is proportional to $\bar U \bar V ^2$, where $\bar U$ is the strength of the suction and $\bar V$ is the tangential component of the difference between Euler and Navier-Stokes on the outflow boundary. Moreover, we show that the rate of enstrophy production near the outflow boundary is inversely proportional to the viscosity. This talk will be based on joint work with Vincent R. Martinez, Anna L. Mazzucato, and Alexis F. Vasseur.
November 29
(Thanksgiving Break)
December 6
Liding Yao (Ohio State University)
Title: The Cauchy-Riemann problem via extension operators
Abstract: The Cauchy-Riemann problem, also known as the $\overline\partial$-problem, is a central problem in several complex variables. It concerns the regularity estimates to the equation $\overline\partial u=f$ on forms in a bounded domain $\Omega\subset\mathbb C^n$. We will talk about the background of the $\overline\partial$-regularity theory, its obstructions, and our recent works using new technique from extension operator. We use the called the Rychkov's extension operator, which extends functions on a bounded Lipschitz domain and has boundedness on all Besov spaces and Triebel-Lizorkin spaces. This is partly joint with Ziming Shi.
December 13
Zhiyuan Zhang (Northeastern University) THIS TALK WILL START AT 3PM
Title: Scattering of Schrödinger Operators with Step Potentials
Abstract: We consider the Schrödinger operator with a step potential, which has a non-zero asymptote at infinity. We build up a scattering and distorted Fourier transform theory for this operator, and use it to study the long time behavior of small data solutions of the 1D cubic NLS equation with the step potential. We also discuss some potential application of this theory, that is, the nonlinear stability problem of dark solitons for the 1D cubic NLS equation. This is ongoing joint work with J. Holmer (Brown University).
February 16
Sarah Strikwerda (UPenn)
Title: Analysis of a multiscale interface problem
Abstract: In biomechanics, the local phenomena such as tissue perfusion are impacted by global features like blood flow. We will discuss a model with a 3D description of fluid flow through biological systems coupled with a 0D model accounting for the effects of global circulation. The coupling leads to interface conditions enforcing the continuity of mass and the balance of stresses across models at different scales. We will discuss strategies to show the well-posedness of this system.
February 23
Haoya Li (Stanford)
Title: Towards practical and efficient quantum phase estimation
Abstract: Quantum phase estimation (QPE) is of essential importance in the field of quantum computing, serving as a foundational component of many quantum algorithms. This presentation will delve into the latest advancements that have been made to enhance the efficiency and practicality of QPE. We will begin with a succinct historical overview of the development of QPE algorithms. Following this, I will present a selection of near-optimal algorithms specifically designed for early fault-tolerant quantum computers, which are often constrained by limited quantum resources. We will examine both single-mode and multi-mode QPE algorithms under various model assumptions. To conclude, we will discuss a recent refinement that significantly increases the practicality of the QPE algorithms.
March 1
Patrick Phelps (Temple University)
Title: Asymptotic properties and separation rates for local energy solutions to the Navier-Stokes equations
Abstract: We present recent results on spatial decay and properties of non-uniqueness for the 3D Navier-Stokes equations. We show asymptotics for the ‘non-linear’ part of scaling invariant flows with data in subcritical classes. Motivated by recent work on non-uniqueness, we investigate how non-uniqueness of the velocity field would evolve in time in the local energy class. Specifically, by extending our subcritical asymptotics to approximations by Picard iterates, we may bound the rate at which two solutions, evolving from the same data, may separate pointwise. We conclude by extending this separation rate to solutions with no scaling assumption. Joint work with Zachary Bradshaw.
March 8
Brian Choi (USMA Westpoint)
Title: Nonlocal dispersive lattice dynamics and continuum limit
Abstract: Nonlocal phenomena rise naturally from the continuum limit of lattice systems with long range interactions. Mathematically they are described by partial differential equations with fractional derivatives. The talk will focus on how nonlocality and discreteness as physical parameters influence the dynamics of nonlinear dispersive phenomena in the context of discrete nonlinear Schrodinger equation. One important difference between dynamics on the discrete and continuum domain is the lack of translational invariance, which inhibits the existence of traveling waves. This topic is revisited in the context of nonlocal lattice interaction.
March 15
Gavin Stewart (Rutgers University)
Title: A wave packet method for Nonlinear Schrödinger Equations with potential
Abstract: In this talk, I'll discuss the asymptotics of the cubic nonlinear Schrödinger equation with potential in dimension 1 for small, localized initial data. In the case when the potential V = 0, it has been known for some time that solutions exhibit modified scattering. Due to additional complications introduced by the potential, the case with V nonzero has not been addressed until recently. Here, we present a method to obtain asymptotics for this problem based on the method of testing with wave packets introduced by Ifrim and Tataru. Compared to previous results, this method can handle potentials with significantly slower decay at infinity.
March 22
Federico Pasqualotto (UC Berkeley)
Title: From instability to singularity formation in incompressible fluids
Note: This talk will take place at 4pm
Abstract: In this talk, I will first review the singularity formation problem in incompressible fluid dynamics, describing how particle transport poses the main challenge in constructing blow-up solutions for the 3d incompressible Euler equations. I will then outline a new mechanism that allows us to overcome the effects of particle transport, leveraging the instability seen in the classical Taylor-Couette experiment. Using this mechanism, we construct the first swirl-driven singularity for the incompressible Euler equations in R^3. This is joint work with Tarek Elgindi (Duke University).
March 29
(Break)
April 5
Giulia Carigi (University of L'Aquila)
Title: Long-time behaviour of stochastic geophysical fluid dynamics models
Abstract: The introduction of random perturbations by noise in partial differential equations has proven extremely useful to understand more about long-time behaviour in complex systems like atmosphere and ocean dynamics or global temperature. Considering additional transport by noise in fluid models has been shown to induce convergence to stationary solutions with enhanced dissipation, under specific conditions. On the other hand, the presence of simple additive forcing by noise helps to find a stationary distribution (invariant measure) for the system and understand how this distribution changes with respect to changes in model parameters (response theory). I will discuss these approaches with a multi-layer quasi-geostrophic model as example.
April 12
Note: Double talk today. 2-3 and 3-4.
William Verreault (University of Toronto), 2-3 pm
Title: Nonlinear expansions in reproducing kernel Hilbert spaces
Abstract: Over the last few years, many mathematicians became interested in a nonlinear analogue of Fourier series that allows them to approximate a signal by a sum of terms whose components represent frequency and amplitude. It is the Blaschke unwinding series introduced by Coifman, or adaptive Fourier decomposition. Because it has many advantages over the classical Fourier series, this series expansion has been used in several other problems since. Yet, the question of convergence of the series has remained a major problem for a few decades. We only know that it converges in certain weighted subspaces of H^2 and, by recent work, in Hardy spaces.
I will introduce an expansion scheme in reproducing kernel Hilbert spaces which is motivated by operator theory and de Branges–Rovnyak spaces, and which as a special case covers the Blaschke unwinding series. The expansion scheme can also be generalized to cover certain reproducing kernel Banach spaces. I will discuss convergence results for this series expansion and present a few applications and examples.
This is based on joint work with Javad Mashreghi.
Stefano De Marchi (Tullio Levi-Civita, Italy), 3-4 pm
Title: Radial Basis Functions and Variably Scaled (Discontinuous) Kernels
April 19
Sanchit Chaturvedi (NYU)
Title: Phase mixing in astrophysical plasmas with an external Kepler potential
Abstract: In Newtonian gravity, a self-gravitating gas around a massive object such as a star or a planet is modeled via Vlasov Poisson equation with an external Kepler potential. The presence of this attractive potential allows for bounded trajectories along which the gas neither falls in towards the object or escape to infinity. We focus on this regime and prove first a linear phase mixing result in 3D outside symmetry with exact Kepler potential. Then we also prove a long-time nonlinear phase mixing result in spherical symmetry. The mechanism is phenomenologically similar to Landau damping on a torus but mathematically the situation is quite a lot more complex.
This is based on an upcoming joint work with Jonathan Luk at Stanford.
April 26
Maria Soria Carro (Rutgers University)
Title:
Abstract:
May 3
Dima Batenkov (NYU and Tel Aviv University)
Title: Stability of exponential fitting, with applications to inverse problems
Abstract: The problem of recovering parameters in exponential sums from inaccurate samples has a long history and appears in many applications, including spectral estimation, sampling theory, super-resolution of sparse measures, to name a few. It is of great interest to quantify the stability of this inverse problem, and develop optimal reconstruction algorithms. I will present some recent results in these directions. For the super-resolution problem in the regime of near-colliding point sources and in the presence of noise, we derive a novel recovery algorithm based on the well-known Prony's method, and show that it provably attains the previously established min-max bounds for this problem. Time permitting, we also discuss the case of so-called ``structured noise'', which appears in the problem of recovering reaction-diffusion dynamics, and show that the exponential fitting approach provides surprisingly stable solution in this case as well.
Based on joint works with N.Diab, R.Katz and G.Giordano.
May 10
Penghang Yin (SUNY Albany)
Title: Quantization and Compression of Neural Networks
Abstract: Quantized neural networks offer compelling advantages in terms of memory and power efficiency upon deployment. In this talk, I will first discuss the challenges inherent in training quantized neural networks using gradient-based algorithms. I will focus on the theoretical analysis of a heuristic trick known as straight-through estimator, which is employed to overcome the issues of discontinuity and discreteness during minimization. In addition, I will present an innovative backpropagation-free (or gradient-free) algorithm for quantizing and compressing large-scale neural networks, especially beneficial for models such as large language models. Experimental results demonstrate that our proposed method achieves state-of-the-art performance in tasks such as image classification and natural language processing.
October 27
Tristan Leger (Princeton University)
Title: L^p bounds for spectral projectors on hyperbolic surfaces
Abstract: In this talk I will present L^p boundedness results for spectral projectors on hyperbolic surfaces. I will focus on the case where the spectral window has small width. Indeed the negative curvature assumption leads to improvements over the universal bounds of C.Sogge, thus illustrating how these objects are sensitive to the global geometry of the underlying manifold. The proof relies on new Strichartz and smoothing estimates for the Schrödinger semi-group, thus illustrating how dispersive PDE techniques can lead to new results in classical harmonic analysis.
November 3
Joonhyun La (Princeton University/KIAS)
Title: Local well-posedness and smoothing of MMT kinetic wave equation
Abstract: In this talk, I will prove local well-posedness of kinetic wave equation arising from MMT equation, which is introduced by Majda, Mclaughlin, and Tabak and is one of the standard toy models to study wave turbulence. Surprisingly, our result reveals a regularization effect of the collision operator, which resembles the situation of non-cutoff Boltzmann. This talk is based on a joint work with Pierre Germain (Imperial College London) and Katherine Zhiyuan Zhang (Northeastern).
November 10
Rajendra Beekie (Duke University)
Title: Uniform vorticity depletion and inviscid damping for periodic shear flows in the high Reynolds number regime
Abstract: We study the dynamics of the two-dimensional Navier Stokes equations linearized around a shear flow on a (non-square) torus which possesses exactly two non-degenerate critical points. The main task is to understand the associated Rayleigh and Orr-Sommerfeld equations, under the natural assumption that the linearized operator around the shear flow in the inviscid case has no discrete eigenvalues. We obtain linear inviscid damping and vorticity depletion estimates for the linearized flow that are uniform with respect to the viscosity, and enhanced dissipation type decay estimates. The key difficulty is to understand the behavior of the solution to Orr-Sommerfeld equations in three distinct regimes depending on the spectral parameter: the non-degenerate case when the spectral parameter is away from the critical values, the intermediate case when the spectral parameter is close but still separated from the critical values, and the most singular case when the spectral parameter is inside the viscous layer. This is based on joint work with Shan Chen (UMN) and Hao Jia (UMN)
November 17 (Zoom Only)
Jacob Carruth (Princeton University)
Title: The Whitney Extension Problem for the Sobolev Space L^{2,p}(R^2) when p<2
Abstract: Let L^{2,p}(R^2) be the Sobolev space of real-valued functions on R^2 whose derivatives up to order 2 belong to L^p. For a subset E of R^2, let L^{2,p}(E) denote the space of restrictions to E of functions in L^{2,p}(R^2). Does there exist a bounded linear operator from L^{2,p}(E) to L^{2,p}(R^2) (an "extension operator")? When p > 2, a bounded linear extension operator exists (thanks to Fefferman, Israel, Luli). When p < 2, it is unknown whether a bounded linear extension operator exists in general. We will show how to construct such an operator for certain sets E. In a particularly interesting case, the problem reduces to an extension problem on a tree.
December 1
THIS TALK WILL TAKE PLACE AT 4PM IN ROOM 4213.03
Joseph Miller (UT Austin)
Title: On the effective dynamics of Bose-Fermi Mixtures
Abstract: In this talk, I will be discussing recent work with Esteban Cardenas and my advisor Natasa Pavlovic: https://arxiv.org/abs/2309.04638. In this work, we describe the dynamics of a Bose-Einstein condensate interacting with a degenerate Fermi gas, at zero temperature. First, we analyze the mean-field approximation of the many-body Schrödinger dynamics and prove emergence of a coupled Hartree-type system of equations. We obtain rigorous error control that yields a non-trivial scaling window in which the approximation is meaningful. Second, starting from this Hartree system, we identify a novel scaling regime in which the fermion distribution behaves semi-clasically, but the boson field remains quantum-mechanical; this is one of the main contributions of the present article. In this regime, the bosons are much lighter and more numerous than the fermions. We then prove convergence to a coupled Vlasov-Hartee system of equations with an explicit convergence rate.
December 8
Title:
Abstract:
December 15 : There are two presentations scheduled for this day.
Quyuan Lin (Clemson University), 2-3 pm
Title: Anisotropic Viscosities Estimation for the Stochastic Primitive Equations
Abstract: The primitive equations (PE) serve as a fundamental model for exploring large-scale oceanic and atmospheric dynamics. Introducing the element of randomness, we investigate the 3D PE perturbed by an additive noise. Our objective is to derive effective estimators for the viscosity within the model, grounded in observational data, and subsequently examine their asymptotic consistency and normality. In particular, our approach belongs to the category of the so-called spectral methods, where observations are conducted in Fourier space and continuously over a finite time interval. The novelty of our work lies in the simultaneous estimation of both horizontal and vertical viscosity, which are typically treated as distinct parameters.
Papageorgiou Effie (Tufts University and Paderborn University) 3-4 pm
Title: Asymptotic behavior of solutions to the heat equation on certain Riemannian manifolds
Abstract: Here
February 3 Zoom (Recording)
Kevin D. Stubbs (UC Berkeley)
A Mathematical Invitation to Wannier Functions
Wannier functions, first proposed in the 1930s, have had a long history in computational chemistry as a practical means to speed up calculations. Stated in a mathematical language, Wannier functions are an orthonormal basis for certain types of spectral subspaces which are generated by the action of a translation group. In the 1980s however, it was realized that there is an intimate connection between Wannier functions and topology. In particular, Wannier functions with fast spatial decay exist if and only if a certain vector bundle is topologically trivial. Materials with non-trivial topology host a number of remarkable properties which are robust to physical imperfections. In this talk, I will give a brief introduction to topological materials and Wannier functions in periodic systems. I will then discuss my work on extending these results to systems without any underlying periodicity.
February 10 (In-Person)
Philip Greengard (Columbia University, Department of Statistics)
Efficient Fourier representations for Gaussian process regression
Over the last couple of decades a large number of numerical methods have been introduced for efficiently performing Gaussian process regression. Most of these methods focus on fast inversion of the covariance matrix that appears in the Gaussian density. In this talk I describe a slightly different approach to Gaussian process regression that relies on efficient weight-space representations of Gaussian processes. These representations -- complex exponential expansions with Gaussian coefficients -- have several advantages in Gaussian process regression tasks including theoretical guarantees, computational efficiency, and model-interpretability benefits.
February 17 Zoom (Recording)
Raghav Venkatraman (Courant Institute of Mathematical Sciences, Department of Mathematics)
Homogenization questions inspired by machine learning and the semi-supervised learning problem
This talk comprises two parts. In the first part, we revisit the problem of pointwise semi-supervised learning (SSL). Working on random geometric graphs (a.k.a point clouds) with few labeled points, our task is to propagate these labels to the rest of the point cloud. Algorithms that are based on the graph Laplacian often perform poorly in such pointwise learning tasks since minimizers develop localized spikes near labeled data. We introduce a class of graph-based higher order fractional Sobolev spaces, Hs, and establish their consistency in the large data limit, along with applications to the SSL problem. A crucial tool is recent convergence results for the spectrum of the graph Laplacian to that of the continuum. Obtaining optimal convergence rates for such spectra is an open question in stochastic homogenization. In the rest of the talk, we'll discuss how to get state-of-the-art and optimal rates of convergence for the spectrum, using tools from stochastic homogenization. The first half is joint work with Dejan Slepcev (CMU), and the second half is joint work with Scott Armstrong (Courant).
March 10 Zoom
Fushuai Jiang (Univeristy of Maryland, Department of Mathematics)
Quasi-optimal C2(Rn) Interpolation with Range Restriction
Experimental data often have range or shape constraints imposed by nature. For example, probability density or chemical concentration are non-negative quantities, and the trajectory design through an obstacle course may need to avoid two boundaries. In this talk, we investigate the theory of multivariate smooth interpolation with range restriction from the perspective of Whitney Extension Problems. Given a function defined on a finite set with no underlying geometric assumption, I will describe an O(N(log N)-n) procedure to compute a twice continuously differentiable interpolant that preserves a prescribed shape (e.g. nonnegativity) and whose second derivatives are as small as possible up to a constant factor (i.e., quasi-optimal). I will also provide explicit numerical results in one dimension. This is based on the joint works with Charles Fefferman (Princeton), Chen Liang (UC Davis), Yutong Liang (former UC Davis), and Kevin Luli (UC Davis).
March 17 Zoom (Recording)
Arthur Danielyan (University of South Florida, Department of Mathematics)
On a converse of Fatou's theorem
Fatou's theorem states that a bounded analytic function in the unit disc has radial limits a.e. on the unit circle T. This talk presents the following new theorem in the converse direction.
Theorem 1. Let E be a subset on T. There exists a bounded analytic function in the open unit disc which has no radial limits on E but has unrestricted limits at each point of T \ E if and only if E is an Fσ set of measure zero.
The sufficiency part of this theorem immediately implies a well-known theorem of Lohwater and Piranian the proof of which is complicated enough. However, the proof of Theorem 1 only uses the Fatou's interpolation theorem, for which too the author has recently suggested a new simple proof.
It turns out that for the Blaschke products, a well-known subclass of bounded analytic functions, Theorem 1 takes the following form.
Theorem 2. Let E be a subset on the unit circle T. There exists a Blaschke product which has no radial limits on E but has unrestricted limits at each point of T \ E if and only if E is a closed set of measure zero.
The proof of the necessity part of Theorem 2 is completely elementary, but it still contains some methodological novelty. The proof of the sufficiency uses Theorem 1 as well as some known results on Blaschke products. (Theorem 2 is a joint result with Spyros Pasias.)
References.
1. A. A. Danielyan, On Fatou's theorem, Anal. Math. Phys. V. 10, Paper no. 28, 2020.
2. A. A. Danielyan, A proof of Fatou's interpolation theorem, J. Fourier Anal. Appl., V. 28, Paper no. 45, 2022.
3. A. A. Danielyan and S. Pasias, On a boundary property of Blaschke products, to appear in Anal. Mathematica
March 24 Zoom
Geet Varma (Royal Melbourne Institute of Technology, School of Science)
Weaving Frames Linked with Fractal Convolutions
Weaving frames have been introduced to deal with some problems in signal processing and wireless sensor networks. More recently, the notion of fractal operator and fractal convolutions have been linked with perturbation theory of Schauder bases and frames. However, the existing literature has established limited connections between the theory of fractals and frame expansions. In this paper we define Weaving frames generated via fractal operators combined with fractal convolutions. The aim is to demonstrate how partial fractal convolutions are associated to Riesz bases, frames and the concept of Weaving frames. This current view point deals with ones sided convolutions i.e both left and right partial fractal convolution operators on Lebesgue space Lp for 1 ≤ p < ∞ . Some applications via partial fractal convolutions with null function have been obtained for the perturbation theory of bases and weaving frames.
March 31 Zoom (Recording)
Michael Perlmutter (UCLA, Department of Mathematics)
Geometric Scattering on Measure Spaces
Geometric Deep Learning is an emerging field of research that aims to extend the success of convolutional neural networks (CNNs) to data with non-Euclidean geometric structure. Despite being in its relative infancy, this field has already found great success in many applications such as recommender systems, computer graphics, and traffic navigation. In order to improve our understanding of the networks used in this new field, several works have proposed novel versions of the scattering transform, a wavelet-based model of CNNs for graphs, manifolds, and more general measure spaces. In a similar spirit to the original Euclidean scattering transform, these geometric scattering transforms provide a mathematically rigorous framework for understanding the stability and invariance of the networks used in geometric deep learning. Additionally, they also have many interesting applications such as drug discovery, solving combinatorial optimization problems, and predicting patient outcomes from single-cell data. In particular, motivated by these applications to single-cell data, I will also discuss recent work proposing a diffusion maps style algorithm with quantitative convergence guarantees for implementing the manifold scattering transform from finitely many samples of an unknown manifold.
April 14 Zoom
Nabil Fadai (University of Nottingham, Department of Mathematics)
Semi-infinite travelling waves arising in moving-boundary reaction-diffusion equations
Travelling waves arise in a wide variety of biological applications, from the healing of wounds to the migration of populations. Such biological phenomena are often modelled mathematically via reaction-diffusion equations; however, the resulting travelling wave fronts often lack the key feature of a sharp edge. In this talk, we will examine how the incorporation of a moving boundary condition in reaction-diffusion models gives rise to a variety of sharp-fronted travelling waves for a range of wave speeds. In particular, we will consider common reaction-diffusion models arising in biology and explore the key qualitative features of the resulting travelling wave fronts.
April 21 Zoom
Lu Zhang (Columbia University, Department of Applied Physics and Applied Mathematics)
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April 28 Zoom
Chun-Kit Lai (San Francisco State University, Department of Mathematics)
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