Department of


Seminar Calendar
for Harmonic Analysis and Differential Equations events the year of Saturday, November 9, 2019.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, January 22, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, January 22, 2019

Singular limits of sign-changing weighted eigenproblems

Derek Kielty   [email] (Illinois Math)

Abstract: Eigenvalue problems with positive weights are related to heat flow and wave propagation in inhomogeneous media. Sign-changing weights have ecological interpretations, and generate spectra that accumulate at both positive and negative infinity. This talk will discuss recent results on limits of such eigenvalue problems when a negative portion of the weight is made arbitrarily large.

Tuesday, January 29, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, January 29, 2019

Traveling waves in an inclined channel and their stability

Zhao Yang (Indiana University Bloomington)

Abstract: The inviscid Saint-Venant equations are commonly used to model fluid flow in a dam or spillway. To classify known traveling wave solutions to the St. Venant equations, the condition of hydrodynamic stability introduces a dichotomy on the parameter F (Froude number): Namely, the constant flow solution is stable for F < 2 where one expect persistent asymptotically-constant traveling wave solutions and unstable for F > 2 where one expect rather complex pattern formation. We will discuss for F>2 Dressler's construction of the inviscid roll wave solution and for F<2 Yang-Zumbrun's construction of the smooth/discontinuous hydraulic shock profiles. We will then present recent stability results of these traveling waves. That is a complete spectral stability diagram for F>2 roll wave case obtained in [JNRYZ18] and spectral, linear orbital, and nonlinear orbital stability of all the hydraulic shock profiles obtained in [YZ18] and [SYZ18].

Tuesday, March 5, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, March 5, 2019

Some recent progress on the Falconer distance conjecture and applications

Alex Iosevich (U. Rochester)

Abstract: We are going to discuss some recent results related to the Falconer distance conjecture and applications of some of these methods to the theory of exponential bases and frames.

Tuesday, March 12, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, March 12, 2019

The lattice bump multiplier problem

Loukas Grafakos (University of Missouri-Columbia)

Abstract: Given a smooth bump supported in a ball centered at the origin in $R^n$, we consider the multiplier formed by adding the translations of this bump by $N$ distinct lattice points. We investigate the behavior as $N$ tends to infinity of the $L^p$ norm of the multiplier operators associated with this finite sum of $N$ bumps.

Tuesday, March 19, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, March 19, 2019

The Steklov and Laplacian spectra of Riemannian manifolds with boundary

Alexandre Girouard   [email] (Université Laval)

Abstract: The Dirichlet-to-Neumann map is a first order pseudodifferential operator acting on the smooth functions of the boundary of a compact Riemannian manifold M. Its spectrum is known as the Steklov spectrum of M. The asymptotic behaviour (as j tends to infinity) of the Steklov eigenvalues s_j is determined by the geometry of the boundary of M. Neverthless, each individual eigenvalue can become arbitrarily big if the Riemannian metric is perturbed adequately. This can be achieved while keeping the geometry of the boundary unchanged, but it requires wild perturbations in arbitrarily small neighborhoods of the boundary. In recent work with Bruno Colbois and Asma Hassannezhad, we impose constraints on the geometry of M on and near its boundary. This allows the comparison of each Steklov eigenvalue s_j with the corresponding eigenvalues l_j of the Laplace operator acting on the boundary. This control is uniform in the index j. The proof is based on a generalized Pohozaev identity and on comparison results for the principal curvatures of hypersurfaces that are parallel to the boundary.

Tuesday, April 2, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, April 2, 2019

Direct Scattering and Small Dispersion for the Benjamin-Ono Equation with Rational Initial Data

Alfredo Wetzel (Wisconsin-Madison)

Abstract: The Benjamin-Ono (BO) equation describes the weakly nonlinear evolution of one-dimensional interface waves in a dispersive medium. It is an integrable equation, with a known Lax pair and inverse scattering transform, that may be viewed as a prototypical problem for the study of multi-dimensional integrable equations and Riemann-Hilbert problems with a non-local jump condition. In this talk, we propose explicit formulas for the scattering data of the BO equation with a rational initial condition. For this class of initial conditions, the recovery of the scattering data can be done directly by exploiting the analyticity properties of the Lax pair solutions. Our procedure validates previous well-known formal results and provides new details concerning the leading order behavior of the scattering data in the small dispersion limit. In the small dispersion limit, we are able to derive formulas for the location and density of the eigenvalues, magnitude and phase of the reflection coefficient, and density of the phase constants.

Tuesday, April 9, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, April 9, 2019

Convexity of Whitham's wave of extreme form

Bruno Vergara (ICMAT, Spain)

Abstract: In this talk I will discuss a conjecture of Ehrnström and Wahlén concerning travelling wave solutions of greatest height to Whitham's non-local model of water waves. We will see that there exists a cusped periodic solution whose profile is convex between consecutive peaks of $C^{1/2}$-regularity. The talk is based on joint work with A. Enciso and J. Gómez-Serrano.

Tuesday, April 23, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, April 23, 2019

On Hardy-Rellich-type inequalities

Fritz Gesztesy (Baylor University)

Abstract: We will illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisely, using this factorization method, we will derive a general inequality and demonstrate how particular choices of the parameters contained in this inequality yield well-known inequalities, such as the classical Hardy and Rellich inequalities, as special cases. Actually, other special cases yield additional and apparently less well-known inequalities. We will indicate that our method is quite flexible when it comes to a variety of generalized situations involving the inclusion of remainder terms and higher-order operators. If time permits, we might illustrate a very recent new and most elementary proof in the one-dimensional context. This talk will be accessible to students. This is based on joint work with Lance Littlejohn, Isaac Michael, and Michael Pang.

Tuesday, April 30, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, April 30, 2019

Blow-up and Soliton Stability in KdV-type equations

Svetlana Roudenko (Florida International University)

Abstract: While the KdV equation and its generalizations with higher power nonlinearities (gKdV) have been long studied, a question about existence of blow-up solutions for higher power nonlinearities has posed lots of challenges and far from being answered. One of the main obstacles is that unlike other dispersive models such as the nonlinear Schrodinger or wave equations, the gKdV equation does not have a suitable virial quantity which is the key to prove the finite time blow-up. Partially, the question of existence and formation of singularities intertwines with the soliton stability or actually the instability, which may lead to a blowup. Only at the dawn of this century the groundbreaking works of Martel and Merle showed the existence of finite-time blow-up solutions for the quintic (critical) gKdV equation, as well as the asymptotic stability of solitons in the subcritical gKdV equation. We consider a higher dimensional extension of the gKdV equation, called generalized Zakharov-Kuznetsov (gZK) equation (the gKdV is limited as a spatially one-dimensional model), and investigate stability of solitons and the existence of blow-up solutions. We positively answer the question of existence of blowup in the two dimensional version of critical Zakharov-Kuznetsov equation and also obtain the asymptotic stability in the subcritical setting. We will discuss some of the important ingredients to obtain these results, including the Liouville-type theorem, which uses time-decay estimates, a la virial type quantity and spectral properties associated to it (this is a joint work with Luiz Farah, Justin Holmer and Kai Yang).

Tuesday, August 27, 2019

1:00 pm in Altgeld Hall,Tuesday, August 27, 2019

Deflated Continuation: A bifurcation analysis tool for Nonlinear Schrodinger (NLS) Systems

Stathis Charalampidis (Mathematics Department, California Polytechnic State University)

Abstract: Continuation methods are numerical algorithmic procedures for tracing out branches of fixed points/roots to nonlinear equations as one (or more) of the free parameters of the underlying system is varied. On top of standard continuation techniques such as the sequential and pseudo-arclength continuation, we will present a new and powerful continuation technique called the deflated continuation method which tries to find/construct undiscovered/disconnected branches of solutions by eliminating known branches. In this talk we will employ this method and apply it to the one-component Nonlinear Schrodinger (NLS) equation in two spatial dimensions. We will present novel nonlinear steady states that have not been reported before and discuss bifurcations involving such states. Next, we will focus on a two-component NLS system and discuss about recent developments by using the deflated continuation method where the landscape of solutions of such a system is far richer. A discussion about the challenges in the two-component setting will be offered and a summary of open problems will be emphasized.

Tuesday, September 3, 2019

12:50 pm in 347 Altgeld Hall,Tuesday, September 3, 2019

Multi-frequency class averaging for three-dimensional cryo-electron microscopy

Zhizhen Jane Zhao (Illinois ECE)

Abstract: We introduce a novel intrinsic classification algorithm--multi-frequency class averaging (MFCA)--for clustering noisy projection images obtained from three-dimensional cryo-electron microscopy (cryo-EM) by the similarity among their viewing directions. This new algorithm leverages multiple irreducible representations of the unitary group to introduce additional redundancy into the representation of the transport data, extending and outperforming the previous class averaging algorithm that uses only a single representation. We will discuss the formal algebraic model and representation theoretic patterns of the proposed MFCA algorithm. We conceptually establish the consistency and stability of MFCA by inspecting the spectral properties of a generalized localized parallel transport operator on the two-dimensional unit sphere through the lens of Wigner D-matrices. We will also show how this algorithm can be applied to directly denoise the real data.

Tuesday, September 17, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, September 17, 2019

Finite Elements for Curvature

Kaibo Hu   [email] (University of Minnesota)

Abstract: We review the elasticity (linearized Calabi) complex, its cohomology and potential applications in differential geometry and continuum defect theory. We construct discrete finite element complexes. In particular, this leads to new finite element discretization for the 2D linearized curvature operator. Compared with classical discrete geometric approaches, e.g., the Regge calculus, the new finite elements are conforming. The construction is based on a Bernstein-Gelfand-Gelfand type diagram chase with various finite element de Rham complexes. This is a joint work with Snorre H. Christiansen.

Tuesday, October 15, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, October 15, 2019

Angled crested type water waves

Siddhant Agrawal (U Mass Amherst)

Abstract: We consider the two-dimensional water wave equation which is a model of ocean waves. The water wave equation is a free boundary problem for the Euler equation where we assume that the fluid is inviscid, incompressible, irrotational and the air density is zero. In the case of zero surface tension, we show that the singular solutions recently constructed by Wu (19) are rigid. In the case of non-zero surface tension, we construct an energy functional and prove a local wellposedness result without assuming the Taylor sign condition. This energy reduces to the energy obtained by Kinsey and Wu (18) in the zero surface tension case and allows angled crest interfaces. For non zero surface tension, the energy does not allow singularities in the interface but allows interfaces with large curvature. We show that in an appropriate regime, the zero surface tension limit of our solutions is a solution of the gravity water wave equation which includes waves with angled crests.

Tuesday, October 22, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, October 22, 2019

The Dirichlet problem for elliptic operators having a BMO antisymmetric part

Linhan Li (UMN Math)

Abstract: In this talk, we are going to introduce our result on the absolute continuity of elliptic measure with respect to the Lebesgue measure for a divergence form elliptic operator with non-smooth coefficients that have a BMO antisymmetric part. In particular, the coefficients of the operator are not necessarily bounded. Our method relies on kernel estimates and off-diagonal estimates for the semigourp e^{-tL}, solution to the Kato problem, and various estimates for the Hardy norms of certain commutators. This is a joint work with S. Hofmann, S. Mayboroda, and J. Pipher.

Tuesday, October 29, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, October 29, 2019

The Abnormally Normal Behavior of the Nonlinear Schroedinger Equation

Katelyn Leisman (Illinois Math)

Abstract: The Nonlinear Schroedinger Equation (NLS) is an important partial differential equation that models many different physical applications, including super-fast lasers, Bose-Einstein condensates, and light traveling in optical fibers and wave guides. The goal in studying this equation is to know how its solutions (and thus the physical systems they model) behave over time. One way to do this for linear ("normal") equations is by finding a relationship between the wavelength and the frequency, called the dispersion relation. Unfortunately, the traditional dispersion relation approach does not work for nonlinear waves (like the NLS). However, I've found that some numerical solutions of the NLS have an effective dispersion relation. In this talk, I'll discuss this important equation and this apparent abnormally "normal" behavior. This talk will be accessible to a general audience.

Tuesday, November 5, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, November 5, 2019

A bilinear proof of decoupling for the quartic moment curve

Zane Li (Indiana Math)

Abstract: Using a bilinear method inspired from Wooley's nested efficient congruencing method, we prove a sharp $l^2 L^{20}$ decoupling inequality for the moment curve in $\mathbb{R}^4$. This is joint work with Shaoming Guo, Po-Lam Yung, and Pavel Zorin-Kranich.

Tuesday, November 12, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, November 12, 2019

Turing patterns in the Schnakenberg equations: From normal to anomalous diffusion

Yanzhi Zhang (Department of Mathematics and Statistics, Missouri University of Science and Technology)

Abstract: In recent years, anomalous diffusion has been observed in many biological experiments. Instead of classical Laplace operator, the anomalous diffusion is described by the fractional Laplacian. In this talk, we will discuss the Turing patterns of the Schnakenberg equation and compare the pattern selection under normal and anomalous diffusion. Our analysis shows that the wave number of the Turing instability increases with the exponent of the fractional Laplacian. The interplay of the nonlinearity and long-range diffusion are studied numerically, and especially comparisons are provided to understand the nonlocal effects of the fractional Laplacian.

Tuesday, November 19, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, November 19, 2019

Global Strichartz estimates for the semiperiodic Schrodinger equation.

Alex Barron (illinois Math)

Abstract: We will discuss some recent results related to space-time estimates for solutions to the linear Schrodinger equation on manifolds which are products of tori and Euclidean space (e.g. a cylinder embedded in R^3 ). On these manifolds it is possible to prove certain analogues of the classical Euclidean Strichartz estimates which are scale-invariant and global-in-time. These estimates are strong enough to prove small-data scattering for solutions to the critical quintic NLS on R × T and the critical cubic NLS on R^2 × T (where T is the one-dimensional torus).

Tuesday, December 3, 2019

1:00 pm in 347 Altgeld Hall,Tuesday, December 3, 2019

Steklov Spectral Asymptotics for Polygons

David Sher   [email] (DePaul Mathematics)

Abstract: We consider the Steklov eigenvalue problem on curvilinear polygons in the plane, with all interior angles measuring less than pi. In this setting, we formulate and prove precise spectral asymptotics, with error converging to zero as the spectral parameter increases. These asymptotics have a surprising dependence on arithmetic properties of the angles. Moreover, the problem turns out to have an interesting relationship to a scattering-type eigenvalue problem on the one-dimensional boundary of the polygon, viewed as a quantum graph.