WVU Math Colloquia
Wednesdays at 4pm EDT
zoom link (pass mathAdi2)
The 2021-2022 Applied Mathematics Seminar is organized by Adrian Tudorascu and Casian Pantea.
The talks are on zoom until further notice. The regular time for the Seminar is on Wednesday at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).
If you'd like to suggest speakers for the fall semester please contact Adrian or Casian.
Schedule
date | speaker | institution | title | |
---|---|---|---|---|
September 28 | Lorand Parajdi | WVU | On the controllability of some systems modeling cell dynamics related to leukemia | |
September 15 | Adrian Tudorascu | WVU | On the convexity condition for the semi-geostrophic system | |
September 28 | Lorand Parajdi | WVU | On the controllability of some systems modeling cell dynamics related to leukemia | |
October 5, Online | Giuseppe Negro | University of Birmingham | Stability of sharp Fourier restriction to spheres | |
October 12, VV B139 | Rajula Srivastava | UW Madison | Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups | |
October 19, Online | Itamar Oliveira | Cornell University | A new approach to the Fourier extension problem for the paraboloid | |
October 26, VV B139 | Changkeun Oh | UW Madison | Decoupling inequalities for quadratic forms and beyond | |
October 29, Colloquium | Alexandru Ionescu | Princeton University | Polynomial averages and pointwise ergodic theorems on nilpotent groups | |
November 2, VV B139 | Liding Yao | UW Madison | An In-depth Look of Rychkov's Universal Extension Operators for Lipschitz Domains | |
November 9, VV B139 | Lingxiao Zhang | UW Madison | Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition | |
November 12, Colloquium | Kasso Okoudjou | Tufts University | An exploration in analysis on fractals | |
November 16, VV B139 | Rahul Parhi | UW Madison (EE) | On BV Spaces, Splines, and Neural Networks | |
November 30, VV B139 | Alexei Poltoratski | UW Madison | Title | |
December 7 | TBA | TBA | Title | |
December 14 | Tao Mei | Baylor University | Title | |
February 1 | Person | Institution | Title | |
February 8 | Person | Institution | Title | |
February 15 | Person | Institution | Title | |
February 22 | Person | Institution | Title | |
March 1 | Person | Institution | Title | |
March 8 | Brian Street | UW Madison | Title | |
March 15: No Seminar | Person | Institution | Title | |
March 23 | Person | Institution | Title | |
March 30 | Person | Institution | Title | |
April 5 | Person | Institution | Title | |
April 12 | Person | Institution | Title | |
April 19 | Person | Institution | Title | |
April 22, Colloquium | Detlef Müller | University of Kiel | Title | |
April 25, 4:00 p.m., Distinguished Lecture Series | Larry Guth | MIT | Title | |
April 26, 4:00 p.m., Distinguished Lecture Series | Larry Guth | MIT | Title | |
April 27, 4:00 p.m., Distinguished Lecture Series | Larry Guth | MIT | Title | |
May 3 | Person | Institution | Title | |
Date | Person | Institution | Title |
Abstracts
Lorand Parajdi
On the controllability of some systems modeling cell dynamics related to leukemia
In this talk, I will present two control problems for a model of cell dynamics related to leukemia. The first control problem is in connection with classical chemotherapy, which indicates that the evolution of the disease under treatment should follow a prescribed trajectory assuming that the drug works by increasing the cell death rates of both malignant and normal cells. In the second control problem, as for targeted therapies, the drug is assumed to work by decreasing the multiplication rate of leukemic cells only, and the control objective is that the disease state reaches a desired endpoint. The solvability of the two problems as well as their stability is proved by using a general method of analysis. Some numerical simulations are included to illustrate the theoretical results and prove their applicability. The results can possibly be used to design therapeutic scenarios such that an expected clinical evolution can be achieved.
Adrian Tudorascu
On the convexity condition for the semi-geostrophic system
We show that conservative distributional solutions to the Semi-Geostrophic system in a rigid domain are in some well-defined sense critical points of a time-shifted energy functional involving measure-preserving rearrangements of the absolute density and momentum, which arise as one-parameter flow maps of continuously differentiable, compactly supported divergence free vector fields. We also show directly, with no recourse to Monge- Kantorovich theory, that the convexity requirement on the modified pressure potentials arises naturally if these critical points are local minimizers of said energy functional for any admissible vector field.
Giuseppe Negro
Stability of sharp Fourier restriction to spheres
In dimension $d\in\{3, 4, 5, 6, 7\}$, we establish that the constant functions maximize the weighted $L^2(S^{d-1}) - L^4(R^d)$ Fourier extension estimate on the sphere, provided that the weight function is sufficiently regular and small, in a proper and effective sense which we will make precise. One of the main tools is an integration by parts identity, which generalizes the so-called "magic identity" of Foschi for the unweighted inequality with $d=3$, which is exactly the classical Stein-Tomas estimate.
Joint work with E.Carneiro and D.Oliveira e Silva.
Rajula Srivastava
Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups
We discuss $L^p\to L^q$ estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups, sharp up to endpoints. The proof shall be reduced to estimates for standard oscillatory integrals of Carleson-Sj\"olin-H\"ormander type, relying on the maximal possible number of nonvanishing curvatures for a cone in the fibers of the associated canonical relation. We shall also discuss a new counterexample which shows the sharpness of one of the edges in the region of boundedness. Based on joint work with Joris Roos and Andreas Seeger.
Itamar Oliveira
A new approach to the Fourier extension problem for the paraboloid
An equivalent formulation of the Fourier Extension (F.E.) conjecture for a compact piece of the paraboloid states that the F.E. operator maps $ L^{2+\frac{2}{d}}([0,1]^{d}) $ to $L^{2+\frac{2}{d}+\varepsilon}(\mathbb{R}^{d+1}) $ for every $\varepsilon>0 $. It has been fully solved only for $ d=1 $ and there are many partial results in higher dimensions regarding the range of $ (p,q) $ for which $L^{p}([0,1]^{d}) $ is mapped to $ L^{q}(\mathbb{R}^{d+1}) $. One can reduce matters to proving that a model operator satisfies the same mapping properties, and we will show that the conjecture holds in higher dimensions for tensor functions, meaning for all $ g $ of the form $ g(x_{1},\ldots,x_{d})=g_{1}(x_{1})\cdot\ldots\cdot g_{d}(x_{d}) $. We will present this theorem as a proof of concept of a more general framework and set of techniques that can also address multilinear versions of this problem and get similar results. This is joint work with Camil Muscalu.
Changkeun Oh
Decoupling inequalities for quadratic forms and beyond
In this talk, I will present some recent progress on decoupling inequalities for some translation- and dilation-invariant systems (TDI systems in short). In particular, I will emphasize decoupling inequalities for quadratic forms. If time permits, I will also discuss some interesting phenomenon related to Brascamp-Lieb inequalities that appears in the study of a cubic TDI system. Joint work with Shaoming Guo, Pavel Zorin-Kranich, and Ruixiang Zhang.
Alexandru Ionescu
Polynomial averages and pointwise ergodic theorems on nilpotent groups
I will talk about some recent work on pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two. Our proof is based on almost-orthogonality techniques that go far beyond Fourier transform tools, which are not available in the non-commutative nilpotent setting. In particular we develop what we call a nilpotent circle method}, which allows us to adapt some the ideas of the classical circle method to the setting of nilpotent groups.
Liding Yao
An In-depth Look of Rychkov's Universal Extension Operators for Lipschitz Domains
Given a bounded Lipschitz domain $\Omega\subset\mathbb{R}^n$, Rychkov showed that there is a linear extension operator $\mathcal E$ for $\Omega$ which is bounded in Besov and Triebel-Lizorkin spaces. We introduce a class of operators that generalize $\mathcal E$ which are more versatile for applications. We also derive some quantitative blow-up estimates of the extended function and all its derivatives in $\overline{\Omega}^c$ up to boundary. This is a joint work with Ziming Shi.
Lingxiao Zhang
Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition
We study operators of the form $Tf(x)= \psi(x) \int f(\gamma_t(x))K(t)\,dt$ where $\gamma_t(x)$ is a real analytic function of $(t,x)$ mapping from a neighborhood of $(0,0)$ in $\mathbb{R}^N \times \mathbb{R}^n$ into $\mathbb{R}^n$ satisfying $\gamma_0(x)\equiv x$, $\psi(x) \in C_c^\infty(\mathbb{R}^n)$, and $K(t)$ is a `multi-parameter singular kernel' with compact support in $\mathbb{R}^N$; for example when $K(t)$ is a product singular kernel. The celebrated work of Christ, Nagel, Stein, and Wainger studied such operators with smooth $\gamma_t(x)$, in the single-parameter case when $K(t)$ is a Calder\'on-Zygmund kernel. Street and Stein generalized their work to the multi-parameter case, and gave sufficient conditions for the $L^p$-boundedness of such operators. This paper shows that when $\gamma_t(x)$ is real analytic, the sufficient conditions of Street and Stein are also necessary for the $L^p$-boundedness of $T$, for all such kernels $K$.
Kasso Okoudjou
An exploration in analysis on fractals
Analysis on fractal sets such as the Sierpinski gasket is based on the spectral analysis of a corresponding Laplace operator. In the first part of the talk, I will describe a class of fractals and the analytical tools that they support. In the second part of the talk, I will consider fractal analogs of topics from classical analysis, including the Heisenberg uncertainty principle, the spectral theory of Schrödinger operators, and the theory of orthogonal polynomials.
Rahul Parhi
On BV Spaces, Splines, and Neural Networks
Many problems in science and engineering can be phrased as the problem of reconstructing a function from a finite number of possibly noisy measurements. The reconstruction problem is inherently ill-posed when the allowable functions belong to an infinite set. Classical techniques to solve this problem assume, a priori, that the underlying function has some kind of regularity, typically Sobolev, Besov, or BV regularity. The field of applied harmonic analysis is interested in studying efficient decompositions and representations for functions with certain regularity. Common representation systems are based on splines and wavelets. These are well understood mathematically and have been successfully applied in a variety of signal processing and statistical tasks. Neural networks are another type of representation system that is useful in practice, but poorly understood mathematically.
In this talk, I will discuss my research which aims to rectify this issue by understanding the regularity properties of neural networks in a similar vein to classical methods based on splines and wavelets. In particular, we will show that neural networks are optimal solutions to variational problems over BV-type function spaces defined via the Radon transform. These spaces are non-reflexive Banach spaces, generally distinct from classical spaces studied in analysis. However, in the univariate setting, neural networks reduce to splines and these function spaces reduce to classical univariate BV spaces. If time permits, I will also discuss approximation properties of these spaces, showing that they are, in some sense, "small" compared to classical multivariate spaces such as Sobolev or Besov spaces.
This is joint work with Robert Nowak.
Previous_Analysis_seminars
https://www.math.wisc.edu/wiki/index.php/Previous_Analysis_seminars
Extras
Blank Analysis Seminar Template
Graduate Student Seminar:
https://www.math.wisc.edu/~sguo223/2020Fall_graduate_seminar.html