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Abstract: In this talk, we define the Zak transform and examine some of its properties. In particular, the Zak transform may be used to analyze Gabor systems. We will show how it may be used to determine whether a critical density Gabor system is complete, a frame, a Riesz basis, an orthonormal basis or a Bessel sequence.

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Abstract: Kottwitz introduced certain compact ''fake'' unitary group Shimura varieties and determined their local Hasse-Weil zeta functions at primes of good reduction. For primes where the level is arbitrarily deep, the local Hasse-Weil zeta functions were further studied in the case of signature $(1,n-1)$ (by Xu Shen), and in the case of arbitrary signature but under the assumption that the group at $p$ is a product of Weil restrictions of general linear groups, by Scholze and Shin. In this talk, I will explain joint work with Jingren Chi in which we generalize the work of Scholze-Shin by allowing the group at $p$ to be any inner form of a product of Weil restrictions of general linear groups. New phenomena arise when the group at $p$ is not quasi-split, for example a crucial vanishing property of twisted orbital integrals of the test functions at $p$.

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Abstract: The work presented in this talk uses ideas from semidefinite programming and information geometry to efficiently simulate gas dynamics in the presence of shock waves. The latter cause severe numerical challenges for classical and learning-based solvers.The talk begins by observing that shock formation arises from the deformation map reaching the boundary of the manifold of diffeomorphisms. This motivates using the log-determinant barrier function of semidefinite programming to modify the geometry of the manifold such that the deformation map approaches but never reaches its boundary. This information geometric regularization (IGR) preserves the original long-time behavior without forming singular shocks, greatly simplifying numerical simulation. The modified geometry on the diffeomorphism manifold is also the information geometry of the mass density. I will show how this observation motivates information geometric mechanics that views the solutions of continuum mechanical equations as parameters of probability distributions to be evolved on a suitable information geometry, promising far-reaching extensions of IGR. 

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Abstract: The space of quasi-Fuchsian surface group representations into PSL(2,C) has been largely studied in recent years, also due to the direct relation with hyperbolic geometry. In this framework, Bers' simultaneous uniformization theorem shows that it is parameterized by two copies of Teichmüller space of the closed surface. In this seminar, we are interested in surface group representations into SL(3,C) acting naturally on a homogeneous space called the bi-complex hyperbolic space. After briefly describing its main features, we will define minimal Lagrangian surface immersions and their structural equations. If they are equivariant under the action of a representation into SL(3,C), we will see that their embedding data provide a parameterization of the space of SL(3,C)-quasi-Fuchsian representations by two copies of the bundle of holomorphic cubic differentials over Teichmüller space. This is a joint work with Andrea Tamburelli.

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Abstract: The five human acrocentric chromosomes (13, 14, 15, 21, and 22) are hypothesized to recombine in a non-homologous way resulting in distal regions that are not chromosome specific. Here, we extract segmental duplications from the distal regions of human acrocentric chromosomes, build multiple sequence alignments, and infer phylogenetic trees. The phylogenetic trees show evidence of recombination between heterologous chromosomes. This work began as a summer internship project with Dr. Adam Phillippy at the National Human Genome Research Institute at the NIH.


Speaker #2: Soyoung Park (PhD student, UMD Mathematics Department)


Title: Mathematical assessment of the roles of vaccination and Pap  screening on the incidence of HPV and related cancers in the Republic of  Korea


Abstract: Human Papillomavirus (HPV) is a major sexually-transmitted infection that causes various cancers and genital warts in humans. In addition to accounting for about 99% of cervical cancer cases, it significantly contributes to anal, penile, vaginal, and head and neck cancers. Although HPV is vaccine-preventable (and highly efficacious vaccines exist for preventing infection with some of the most oncogenic HPV subtypes in the targeted population), the disease continues to cause major public health burden globally (largely due to inequity in access to the main control resources (i.e., access to Pap smear and vaccination) and low vaccination coverage in most communities that implement routine HPV vaccination). This talk is based on the use of a new mathematical model (for the natural history of HPV, together with the associated neoplasia) for assessing the combined population-level impacts of Pap cytology screening and vaccination against the spread of HPV in a heterogeneous (heterosexual and homosexual) population. The model, which takes the form of a deterministic system of nonlinear differential equations, will be calibrated and validated using HPV-related cancer data from the Republic of Korea. Theoretical and numerical simulation results will be presented. Conditions for achieving vaccine-derived herd-immunity threshold (for achieving HPV elimination in Korea) will be derived.

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Abstract: In this talk, we construct new examples of constant scalar curvature Kähler(cscK) metrics of Poincaré type from existing cscK ones. The construction is obtained via gluing a cscK metric on a compact Kähler manifold to a complete scalar-flat Kähler metric of Poincaré type on $\mathbb{C}^n$ removing the origin. Assuming the compact Kähler manifold has no non-trivial holomorphic vector field, we prove the existence of cscK metrics of Poincaré type on this compact manifold removing finitely many points.

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Abstract: The use of weights and weighted norm inequalities has a rich history in harmonic analysis and the study of regularity properties to solutions of partial differential equations (PDE). Starting from classical results, we will present an overview of the application of some of these ideas to the numerical analysis of PDEs. Our main attention will be on some recent results concerning the use of weights in fractional diffusion, problems with singular data and some degenerate/singular PDE problems. Although these seem as disparate and unrelated applications, it is remarkable that the only structural assumption on the weight is that it belongs to a so-called Muckenhoupt $A_p$ class, which has been thoroughly studied in harmonic analysis since the 1970's.

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Abstract: We consider the limit of a linear kinetic equation with a degenerate scattering kernel and a reflection-transmission-absorption condition at an interface. An equation of this type arises from the kinetic limit of a microscopic harmonic chain of oscillators whose dynamics is perturbed by a stochastic term, conserving energy and momentum. The chain is in contact, via one oscillator, with a heat bath, which, in the limit, generates the boundary condition at the
interface.

It is known that in the absence of the interface, the solution of the kinetic equation exhibits either superdiffusive, or diffusive behavior, in the proper long time - large scale limit, depending on the dispersion relation of the harmonic chain. We discuss how the presence of the interface influences the boundary condition for the limiting diffusion, or anomalous diffusion.

The presented results have been obtained in collaboration with G. Basile (Univ.
Roma I), A. Bobrowski (Lublin Univ. of Techn.), K. Bogdan (Wrocław Univ. of Sci. and
Techn.), L. Arino (Ensta, Paris), S. Olla (Univ. Paris-Dauphine and GSSI, L’Aquila),
L. Ryzhik (Stanford Univ.), H. Spohn (TU, Munich).

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Abstract: The $N$-branching Brownian motion with selection ($N$-BBM) is a particle system consisting of $N$ independent particles that diffuse as Brownian motions in $\mathbb{R}$, branch at rate one, and whose size is kept constant by removing the leftmost particle at each branching event. It is a simple toy model for the evolution of a population under selection that has generated some fascinating research since its introduction by Brunet and Derrida in the early 200ss.

If one recentre the positions by the left most particle, this system has a stationary distribution. I will show that, as $N\to \infty$ the stationary empirical measure of the $N$-particle system converges to the minimal travelling wave of an associated free boundary PDE. This resolves an open question going back at least to works of e.g. Maillard in 2012.
It follows a recent related result by Oliver Tough (with whom this is joint work) establishing a similar selection principle for the so-called Fleming-Viot particle system.

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Abstract: We study the boundary layer theory for slightly viscous stationary flows forced by an imposed slip velocity at the boundary. According to the theory of Prandtl (1904) and Batchelor (1956), any Euler solution arising in this limit and consisting of a single ’eddy’ must have constant vorticity. Feynman and Lagerstrom (1956) gave a procedure to select the value of this vorticity by demanding a necessary condition for the existence of a periodic Prandtl boundary layer description. In the case of the disc, the choice – known to Batchelor (1956) and Wood (1957) - is explicit in terms of the slip forcing. For domains with non-constant curvature, Feynman and Lagerstrom give an approximate formula for the choice which is in fact only implicitly defined and must be determined together with the boundary layer profile. We show that this condition is also sufficient for the existence of a periodic boundary layer described by the Prandtl equations. Due to the quasilinear coupling between the solution and the selected vorticity, we devise a delicate iteration scheme coupled with a high-order energy method that captures and controls the implicit selection mechanism.

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Abstract: For a finite group G, the vertices of the prime graph Γ(G) are the primes that divide |G|, and two vertices p and q are connected by an edge if and only if there is an element of order pq in G. Though this notion was used in the 1970s to study cohomological questions of group rings, prime graphs have been studied for their own sake ever since. In studying prime graphs of a specific class of groups, we encounter interesting number-theoretic questions, overcome computational obstacles, and devise heavy group-theoretic machinery that streamlines further work on the subject.