The Mathematics Department holds regular seminars on a variety of topics. Please see below for further details.
Seminars
Seminar  Meeting Details  Title & Abstract 

Analysis Seminar  Wavelet representation and Sobolev regularity of quasiregular maps Extending the Sobolev theory of quasiconformal and quasiregular maps to subdomains of the complex plane motivates our investigation of Sobolev regularity of singular integral operators on domains. We introduce new paraproducts which lead to higher order T1type testing conditions. A special case provides weighted Sobolev estimates for the compressed Beurling transform which imply quantitative Sobolev estimates for the Beltrami resolvent. This is joint work with Francesco Di Plinio and Brett D. Wick. Speaker: Walton Green (Washington University in St Louis) 

Differential Equations Seminar  On some maximum principles for PFunctions and their applications In this talk we will survey some old and new results on maximum principles for Pfunctions and their applications to the study of partial differential equations. More precisely, we will show how one can employ the maximum principle in problems of physical or geometrical interest, in order to get the shape of some free boundaries, isoperimetric inequalities, symmetry results, convexity results and Liouville type results. In the first part of the talk we'll be mainly focused on some overdetermined problems, while in the second part of the talk we'll present our contributions to some MongeAmpere type problems and eventually discuss some open problems. Speaker: Cristian Enache (American University of Sharjah) 

Algebra Seminar  The Picard group of the stack of pointed hyperelliptic curves The problem of computing invariants of natural stacks of curves has a long history, starting from Mumford's seminal paper on the Picard group of the stack of 1pointed elliptic curves. The Picard group of the stack \(\mathcal{M}_{g,n}\) of \(n\)pointed smooth curves of genus \(g\geq3\) was later computed over \(\mathbb{C}\) by Harer. We study the closed substack \(\mathcal{H}_{g,n}\) in \(\mathcal{M}_{g,n}\) of \(n\)pointed smooth hyperelliptic curves of genus \(g\), and compute its Picard group. As a corollary, taking \(g=2\) and recalling that \(\mathcal{H}_{2,n}=\mathcal{M}_{2,n}\), we obtain \(\mathrm{Pic}(\mathcal{M}_{2,n})\) for all \(n\). Moreover, we give a very explicit description of the generators of the Picard group, which have evident geometric meaning. Speaker: Alberto Landi, Scuola Normale Superiore 

Analysis Seminar  Boundedness of the bilinear fractional integral operators on multiMorrey spaces Speaker: Naoya Hatano (Chuo University, Japan) 

Analysis Seminar  Update on singular integrals and entangled dilations We discuss various results on singular integrals adapted to entangled dilations from the past two years. The existing results are mostly on the socalled Zygmund dilations that constitute the simplest intermediate dilation structure lying in between the classical oneparameter setting and the multiparameter setting. We start with an overview of the subtle optimal weighted theory in the Zygmund case, the techniques behind that, and the implications these have for further results, such as, commutator estimates. We then discuss the more recent multilinear versions of this theory, the current limitations and, time permitting, some possible further directions and challenges in the area. Speaker: Henri Martikainen (Washington University in St Louis) 

Analysis Seminar  The compactness of multilinear Calder\’{o}nZygmund operators.
Speaker: Anastasios Frangos (Washington University St Louis) 

Algebra Seminar  On containment of trace ideals in ideals of finite projective or injective dimension Motivated by recent result of F. Perez and R.R.G. on equality of test ideal of module closure operation and trace ideal, and the wellknown result by K.E. Smith that parameter test ideal can never be contained in parameter ideals, we study the obstruction of containment of trace ideals in ideals of finite projective (or injective) dimension. As consequences of our results , we give upper bounds on madic order of trace ideals of certain modules. We also prove analogous results for ideal of entries of maps in a free resolution of certain modules. This is joint work with Souvik Dey. Speaker: Monalisa Dutta (University of Kansas) 

Differential Equations Seminar  Spectral analysis of the traveling waves of the CHKP equation under transverse perturbation The CamassaHolmKadomtsevPetviashvili equation (CHKP) is a two dimensional generalization of the CamassaHolm equation which has been recently derived in the context of shallow water waves and nonlinear elasticity. In this talk we will discuss the stability of the onedimensional traveling waves, solitary or periodic, with respect to two dimensional perturbations which are periodic in the transverse direction. We show that the stability or instability depends on a sign parameter of the transverse dispersion term. In particular, a nonlinear instability of the onedimensional solitary waves of any size can be proved for the socalled CHKPI model, while for onedimensional periodic waves we are able to obtain spectral instability for small amplitude CHKPI waves. This is a joint work with Lili Fan, Jie Jin, Xingchang Wang and Runzhang Xu. Speaker: Ming Chen (University of Pittsburgh) 

Differential Equations Seminar  Stability of the compacton waves for the degenerate KDV and NLS models This talk is based on the degenerate semilinear Schrödinger and Kortewegde Vries equations in one spatial dimension. We construct variationally special solutions of the two models, that is, standing wave solutions of NLS and traveling waves for KDV, which turn out to have compact support, hence compactons. We show that the compactons are unique bellshaped solutions of the corresponding PDE's and for appropriate variational problems as well. We also provide a complete spectral characterization of such waves, for all values of \(p\). Namely, we show that all waves are spectrally stable for \(2<p\leq 8\), while a single mode instability occurs for \(p>8\). This extends the previous work of Germain, HarropGriffiths and Marzuola, who have previously established orbital stability for some specific waves, in the range \(p<8\). This is a joint work with Atanas stefanov and Sevdhan Hakkaev. Speaker: Abba Ramadan (University of Alabama) 

Analysis Seminar  Estimates for elliptic and parabolic measures for operators satisfying an oscillation condition It has been known for quite some time that various conditions on the oscillation of the matrix $A$ in elliptic/parabolic operators of the form $L = div A \nabla$ or $L = \partial_t  \div A \nabla$ are sufficient to guarantee either the $L^p$ solvability of the Dirichlet problem or the logarithm of the density of the elliptic/parabolic measure, $k$, is in BMO or (locally) Hölder continuous. Conditions such as the global Hölder continuity of $A$ are classical, whereas other conditions quantifying the oscillation of $A$ in terms of Carleson measures are more recent. Only very recently in joint works with Toro and Zhao, and later with Egert and Saari, was it shown that the BMO norm of $\log k$ could be controlled by these Carleson conditions. These new results are elliptic and heavily relied on work of David, Li and Mayboroda.
In forthcoming work with Egert and Saari, we adapt these new results (B., Toro, Zhao and B. Egert, Saari) to the parabolic setting. Therefore we needed to also adapt the work of David, Li and Mayboroda to the parabolic setting and, in doing so, sharpened their results. It seems very likely that these sharper estimates will allow one to treat the classical results (Hölder continuous coefficients) and the modern results (Carleson conditions) with a unified method.
I will delve into the ideas underpinning these results. Speaker: Simon Bortz (University of Alabama) 