Abstract: Arithmetic theta series are incarnations of theta functions in arithmetic algebraic geometry. The first examples were constructed by Kudla as generating series of special cycles on Shimura varieties. Their conjectural key features are (1) modularity of the generating series, and (2) the arithmetic Siegel-Weil formula, relating their enumerative geometry to the first derivative of Eisenstein series at special values. In joint work with Zhiwei Yun and Wei Zhang, we construct "higher" arithmetic theta series on moduli spaces of shtukas, which we conjecture to also enjoy (1) modularity and (2) a higher arithmetic Siegel-Weil formula relating their enumerative geometry to all derivatives of Eisenstein series at special values. We prove several results towards these conjectures, drawing upon ideas from Ngo's proof of the Fundamental Lemma in addition to new ingredients from Springer theory and derived algebraic geometry.
Abstract: A fundamental result in 3-manifold topology due to Lickorish and Wallace says that every closed oriented connected 3-manifold can be realized as surgery on a link in the 3-sphere. One may therefore ask: which 3-manifolds can be obtained by surgery on a link with a single component, i.e. a knot, in the 3-sphere? More specifically, one can ask: which 3-manifolds are obtained by zero surgery on a knot in the 3-sphere? In this talk, we give a brief outline of some known results to this question in the context of small Seifert fibered spaces. We then sketch a new method, using involutive Heegaard Floer homology, to show that certain 3-manifolds cannot be obtained by zero surgery on a knot in the three sphere. In particular, we produce a new infinite family of weight 1 irreducible small Seifert fibered spaces with first homology Z which cannot be obtained by zero surgery on a knot in the 3-sphere, extending a result of Hedden, Kim, Mark and Park.
Abstract: We examine three products on classes of structures, the semi-direct product, the direct product, and the free superposition. We investigate what properties of classes of structures, such as indivisibility and age indivisibility, are preserved under each product. This work is joint with Miriam Parnes and Lynn Scow.
Abstract: Inverse problems arise in a variety of applications: image processing, finance, mathematical biology, and more. Mathematical models for these applications may involve integral equations, partial differential equations, and dynamical systems, and solution schemes are formulated by applying algorithms that incorporate regularization techniques and/or statistical approaches. In most cases these solutions schemes involve the need to solve a large-scale ill-conditioned linear system that is corrupted by noise and other errors. In this talk we describe Krylov subspace-based regularization approaches to solve these linear systems that combine direct matrix factorization methods on small subproblems with iterative solvers. The methods are very efficient for large scale inverse problems, they have the advantage that various regularization approaches can be used, and they can also incorporate methods to automatically estimate regularization parameters.
Abstract: We will finish our presentation on the necessary background on probability theory that is perhaps not included in a course on measure theory. This includes material on independence, convergence in law, and characteristic functions. Then, we will start the material on infinitely divisible distributions. The reading material can be found in Sections 1.1 and 1.2 of David Applebaum's text, Levy Processes and Stochastic Calculus, 2ed.
Abstract: In 1983 Jim Arthur conjectured the existence of certain sets of representations of a reductive group G over a local field F, associated to homomorphisms from the Weil-Deligne group x SL(2) into the L-group of G. In the case of split classical groups he defined these "Arthur" packets in his 2010 book. For F=R Adams, Barbasch and Vogan gave a definition in their 1992 book, commonly referred to as ABV. Nicolas Arancibia and Paul Mezo, with some help from the speaker, have proved these two definitions agree when they are both defined.
The talk will be available online at https://umd.zoom.us/j/96890967721
Abstract: Let $M$ be a Riemannian, boundaryless, and compact manifold with $\dim M\geq 2$, and let $f$ be a $C^{1+\beta}$ ($\beta>0$) diffeomorphism of $M$. Let $\varphi$ be a H\"older continuous potential on $M$. We construct an invariant and absolutely continuous family of measures (with transformation relations defined by $\varphi$), which sit on local unstable leaves. We present two main applications. First, given an ergodic homoclinic class $H_\chi(p)$, we prove that $\varphi$ admits a local equilibrium state on $H_\chi(p)$ if and only if $\varphi$ is ``recurrent on $H_\chi(p)$" (a condition tested by counting periodic points), and one of the leaf measures gives a positive measure to a set of positively recurrent hyperbolic points; and if an equilibrium measure exists, the said invariant and absolutely continuous family of measures constitutes as its conditional measures. Second, we prove a Ledrappier-Young property for hyperbolic equilibrium states- if $\varphi$ admits a conformal family of leaf measures, and a hyperbolic local equilibrium state, then the leaf measures of the invariant family (respective to $\varphi$) are equivalent to the conformal measures (on a full measure set). This extends the celebrated result by Ledrappier and Young for hyperbolic SRB measures, which states that a hyperbolic equilibrium state of the geometric potential (with pressure 0) has conditional measures on local unstable leaves which are absolutely continuous w.r.t the Riemannian leaf volume.
Abstract: In this talk, we consider nematic liquid crystal flow (NLCF) and Landau-Lifshitz-Gilbert equation (LLG) in \R^2. (NLCF) is a system strongly coupling the nonhomogeneous incompressible Navier-Stokes equation and the transported harmonic map heat flow, while (LLG) serves as the basic evolution equation for the spin fields in the continuum theory of ferromagnetism. We will investigate how parabolic gluing method can be developed and applied to construct finite time blow-up solutions. This talk is based on joint works with Chen-Chih Lai, Fanghua Lin, Changyou Wang, Juncheng Wei and Qidi Zhang.
Abstract: This will be an introduction to the SYK (Sachdev-Ye-Kitaev) model. Recommended reference: the first section or two of Juan Maldacena and Douglas Stanford, Comments on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94, 106002 (2016), https://arxiv.org/abs/1604.07818 .
Abstract: There is a growing interest in using multiple-frame surveys in recent years in order to save survey costs and reduce different types of nonsampling errors. Following the pioneering work by Hartley, methods and theories have been developed. A key underlying assumption of current papers on multiple-frame surveys is known domain membership of each unit of the finite population. But this assumption is hardly met in practice. The effect of violation of this critical assumption on finite population inference is not fully understood. We first investigate the effect of misspecification of the domain membership on estimation and variance estimation. We then exploit the recent development of probabilistic record linkage techniques in adjusting for biases due to domain membership misspecification in the finite population inference. We study the properties of the proposed estimators and the associated variance estimators analytically and through Monte Carlo simulations.