CUNY Graduate Center Harmonic Analysis & PDE Seminar

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:

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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.

May 17

Speaker (Institution)

Title: 

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