Revision as of 17:11, 12 November 2021

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	notes
December 1	Veronica Ciocanel	Duke University	TBA
November 17	Radu Precup	Babes-Bolyai University	Harnack type inequalities and localization of solutions
November 10	Jue Wang	Union College	Locating Objects of Interest from Screening Ultrasound
November 3	Matthew Johnston	Lawrence Technological University	A Mathematical Model of COVID-19 Spread by Vaccination Status
October 20	Xiang Xu	Old Dominion University	Blowup rate estimates of a singular potential in the Landau-de Gennes theory for liquid crystals
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

Abstracts

Veronica Ciocanel

TBA

Radu Precup

Harnack type inequalities and localization of solutions

First it is explained the problem of localization of solutions for boundary value problems in general. Next, the focus will be on radial solutions for phi-Laplace equations. Numerical solutions are obtained for some concrete equations.

Jue Wang

Locating Objects of Interest from Screening Ultrasound

A fast Enclosure Transform is developed to localize complex objects of interest from medical ultrasound images. This approach explores spatial constraints on regional features from a sparse image feature representation. Unrelated, broken ridge features surrounding an object are organized collaboratively, giving rise to the enclosureness of the object. Three enclosure likelihood measures are constructed, consisting of the enclosure force, potential energy, and encloser count. In the transform domain, the local maxima manifest the locations of interest objects, for which only the intrinsic dimension is known a priori. I will demonstrate two medical applications in detecting (1) suspicious breast masses in screening breast ultrasound, and (2) the location of the prostate on trans-abdominal ultrasound for verification of patient positioning in radiotherapy treatment of prostate cancer.

Matthew Johnston

A Mathematical Model of COVID-19 Spread by Vaccination Status

The novel coronavirus SARS-CoV-2, and the corresponding illness COVID-19, has afflicted hundreds of millions of people worldwide, killed at least 5 million, and devastated the world economy. The rapid development of effective and safe vaccines, which were authorized in the United States under Emergency Use Authorization in December 2020, have offered some hope that the end of the pandemic may be sight; however, at the same time we have seen the rise of variants, such as Delta, which are much more transmissible than the original strains. In this talk, I will introduce an n-stage vaccination model and corresponding system of differential equations which can simulate a disease outbreak by breaking the population down according to their vaccination status. This allows the mitigation effects of vaccination and accelerating effects of variants such as Delta to be uncoupled from one another, and offers valuable insight for the future course of the COVID-19 pandemic. I fit the model to 2021 data from the Virginia Department of Health.

Xiang Xu

Blowup rate estimates of a singular potential in the Landau-de Gennes theory for liquid crystals

The Landau-de Gennes theory is a type of continuum theory that describes nematic liquid crystal configurations in the framework of the Q-tensor order parameter. In the free energy, there is a singular bulk potential which is considered as a natural enforcement of a physical constraint on the eigenvalues of symmetric, traceless Q-tensors. In this talk we shall discuss some analytic properties related to this singular potential. More specifically, we provide precise estimates of both this singular potential and its gradient as the Q-tensor approaches its physical boundary.

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.

@@ Line 108: / Line 108: @@
 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

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