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Algebra Seminar

Title: The supersingular locus of some unitary Shimura varieties

Speaker: Sungyoon Cho (Northwestern University)

Faculty Sponsor: Liang Xiao

Time: Wednesday, September 26, 2018 at 11:15 am

Place: MONT 313Abstract: A version of the Arithmetic Gan-Gross-Prasad conjecture predicts a relation between the intersection number of a certain arithmetic cycle in unitary Shimura variety to the non-vanishing of the central derivative of a certain L-function. Here, the supersingular locus plays an important role. In this talk, I will explain the geometric structure of the supersingular locus of the relevant unitary Shimura variety at a place with bad reduction.

Title: The eigenvalue problem of the Dirichlet to Neumann operator 4

Speaker: Gunhee Cho (University of Connecticut)

Time: Wednesday, September 26, 2018 at 2:30 pm

Place: MONT 214Abstract: We will try to compare the variational characterization of the Dirichlet Laplacian and the eigenvalue of the Dirichlet to Neumann map. In particular, we will see how people obtained the gap estimate of the eigenvalue in the case of the Dirichlet Laplacian, hoping to get the similar result in the case of the D-N map, which is unknown. This is still an open question, and I will show the computations why I was stuck in this problem last summer which I want to overcome eventually.Comments: This talk is about current research.
Our seminar is informal and we invite the audience to ask our speakers questions. Emphasis is placed on understanding topics in areas of mathematical physics (broadly defined).
If anyone is interested in giving a talk on something they would like to learn better, please contact Arthur Parzygnat at arthur.parzygnat@uconn.edu. To give you an idea of some of the topics that will be covered, we will have one set of talks on the decomposition of the spectrum (point, continuous, residual) and another set of talks on direct integrals to name a few.

Title: Incomplete Equilibrium with a Stochastic Annuity

Speaker: Kim Weston (Rutgers University)

Time: Wednesday, September 26, 2018 at 4:00 pm

Place: MONT 313Abstract: In this talk, I will present an incomplete equilibrium model to determine the price of an annuity. A finite number of agents receive stochastic income streams and choose between consumption and investment in the traded annuity. The novelty of this model is its ability to handle running consumption and general income streams. In particular, the model incorporates mean reverting income, which is empirically relevant but historically too intractable in equilibrium. The model is set in a Brownian framework, and equilibrium is characterized and proven to exist using a system of fully coupled quadratic BSDEs. This work is joint with Gordan Zitkovic of UT Austin.

Title: What are Manifolds?

Speaker: Ovidiu Munteanu (University of Connecticut)

Time: Wednesday, September 26, 2018 at 5:45 pm

Place: MONT 226Abstract: Manifolds locally resemble Euclidean space (**R**^{n} for some *n*), meaning that every point on a manifold belongs to a region that can be parametrized by Euclidean coordinates. Globally, manifolds may have complicated shapes and intriguing properties. Remarkably, in differential geometry we are able to understand such complicated structures in terms of certain simpler local properties of Euclidean space. The simplest examples of manifolds are curves and surfaces in Euclidean space. These will serve as our primary source of examples in this talk, and will motivate the concepts of Riemannian metric and curvature.Comments: Free pizza and drinks!

Title: Convergence of spectral measures and eigenvalue rigidity in random matrices

Speaker: Elizabeth Meckes (Case Western Reserve University)

Time: Thursday, September 27, 2018 at 4:00 pm

Place: MONT 214Abstract: The behavior of the eigenvalues of large random matrices is generally very predictable, on multiple scales. Macroscopically, results like the semi-circle law describe the overall shape of the eigenvalue distributions, and it is often the case that spectral measures are approximated asymptotically almost surely, and with known estimates on distances, by deterministic limiting measures. On a microscopic scale, we may see the phenomenon of eigenvalue rigidity, in which individual eigenvalues concentrate strongly at predicted locations. I will describe some general approaches to these phenomena, with many examples: Wigner matrices, Wishart matrices, random unitary matrices, truncations of random unitary matrices, Brownian motion on the unitary group, and others.

Title: Symmetric and skew symmetric degeneracy loci and constructions of hyperkahler manifolds

Speaker: Kristian Ranestad (University of Oslo)

Faculty Sponsor: Jerzy Weyman

Time: Friday, September 28, 2018 at 10:00 am

Place: MONT 214Abstract: Minors of matrices are defining equations for many classical varieties, which are therefore called degeneracy loci. Starting with Kummer surfaces, I shall explain a relation between symmetric and skew symmetric degeneracy loci. I shall go on to show how hyperkahler manifolds can be constructed from such degeneracy loci, in reporting on work with A. Iliev, G. Kapustka and M. Kapustka.

Title: Wrecked-ifiability: How bad could it be?

Speaker: Sean McCurdy (University of Washington)

Time: Friday, September 28, 2018 at 12:20 pm

Place: MONT 214Abstract: This talk will introduce several important topics in Geometric Measure Theory (GMT). In particular, we will introduce rectifiable sets, which are a natural (and non-smooth) generalization of surfaces. We will also discuss a quantified version of rectifiability called uniform rectifiability and its connection to an important tool in GMT, the Jones beta-numbers. We will end with a sketch of the construction of a family of examples which answer the question: How 'far' from uniformly rectifiable can a rectifiable set be? This is joint work with Max Goering (University of Washington, Seattle).

Title: A new proof of the concentration-compactness principle of Trudinger-Moser inequality

Speaker: Jungang Li (University of Connecticut)

Time: Friday, September 28, 2018 at 1:30 pm

Place: MONT 313Abstract: The concentration-compactness principle plays a key role in the study of PDEs with critical growth. The classical proof of the concentration-compactness principle of Trudinger-Moser inequality depends on a rearrangement argument. In this talk I will give a rearrangement-free proof of such principle. This proof enables us to extend the related results to non-Euclidean spaces, e.g. Heisenberg Groups and Riemannian manifolds. This is a joint work with Prof. Guozhen Lu and Prof. Maochun Zhu.

Title: no seminar

Speaker:

Time: Friday, September 28, 2018 at 3:00 pm

Place: MONT 313

Title: Strichartz estimates on compact symmetric spaces II

Speaker: Yunfeng Zhang (University of Connecticut)

Time: Friday, September 28, 2018 at 3:30 pm

Place: MONT 414Abstract: In the first talk, I will introduce the problems of Fourier restriction and Strichartz estimates. In the second talk, I will review the fundamental tools of harmonic analysis on compact symmetric spaces. In the final talk, I will sketch a proof of some Strichartz estimates on compact Lie groups.