Michael Novack - UT Austin
Math 258: Seminar of Differential Geometry
We introduce a "mesoscale flatness criterion" for hypersurfaces with bounded mean curvature, discussing its relation to and differences with classical blow-up and blow-down theorems, and then we exploit this tool for a complete resolution of relative isoperimetric sets with large volume in the exterior of a compact obstacle. This is joint work with Francesco Maggi (UT Austin).
Christian Klevdal - UCSD
Math 209 - Number Theory Seminar
APM 6402 and ZoomAbstract
(Joint with Stefan Patrikis.) In this talk, we discuss a strong form of independence of $\ell$ for canonical $\ell$-adic local systems on Shimura varieties, and sketch a proof of this for Shimura varieties arising from adjoint groups whose simple factors have real rank $\geq 2$. Notably, this includes all adjoint Shimura varieties which are not of abelian type. The key tools used are the existence of companions for $\ell$-adic local systems and the superrigidity theorem of Margulis for lattices in Lie groups of real rank $\geq 2$.
The independence of $\ell$ is motivated by a conjectural description of Shimura varieties as moduli spaces of motives. For certain Shimura varieties that arise as a moduli space of abelian varieties, the strong independence of $\ell$ is proven (at the level of Galois representations) by recent work of Kisin and Zhou, refining the independence of $\ell$ on the Tate module given by Deligne's work on the Weil conjectures.
Yinbang Lin - Tongji University
Math 208: Algebraic Geometry Seminar
We will discuss about resolutions of coherent sheaves by line bundles from strong full exceptional sequences over rational surfaces. We call them Gaeta resolutions. We then apply the results towards the study of the moduli space of sheaves, in particular Le Potier's strange duality conjecture. We will show that the strange morphism is injective in some new cases. One of the key steps is to show that certain Quot schemes are finite and reduced. The next key step is to enumerate the length of the finite Quot scheme, by identifying the Quot scheme as the moduli space of limit stable pairs, where we are able to calculate the (virtual) fundamental class. This is based on joint work with Thomas Goller.
Pre-talk for graduate students: 3:30pm - 4:00pm
Brian Tran - UCSD
Math 278A: Center for Computational Mathematics Seminar
APM 7218 and Zoom
Zoom ID 986 1678 1113Abstract
Adjoint systems are widely used to inform control, optimization, and design in systems described by ordinary differential equations and differential-algebraic equations. In this talk, we begin by exploring the geometric properties of adjoint systems associated to ordinary differential equations by investigating their symplectic and Hamiltonian structures. We then extend this to adjoint systems associated to differential-algebraic equations and develop geometric methods for such systems by utilizing presymplectic geometry to characterize the fundamental properties of such systems, such as the adjoint variational quadratic conservation laws admitted by these systems, which are key to adjoint sensitivity analysis. We develop structure-preserving numerical methods for such systems by extending the Galerkin Hamiltonian variational integrator construction of Leok and Zhang to the presymplectic setting. Such methods are natural, in the sense that reduction, forming the adjoint system, and discretization commute for suitable choices of these processes. We conclude with a numerical example. This is joint work with Prof. Melvin Leok.
Yuchen Wu - UCSD
Math 292: Topology Seminar (student seminar on equivariant homotopy theory)
Mona Merling - University of Pennsylvania
Math 292: Topology Seminar
Waldhausen's algebraic K-theory of manifolds satisfies a homotopical lift of the classical h-cobordism theorem and provides a critical link in the chain of homotopy theoretic constructions that show up in the classification of manifolds and their diffeomorphisms. I will give an overview of joint work with Goodwillie, Igusa and Malkiewich about the equivariant homotopical lift of the h-cobordism theorem.
Papri Dey - Georgia Tech
Math 278C: Optimization and Data Science
Meeting ID: 941 9922 3268
Abstract: In this talk, I shall introduce the notion of polynomials with Lorentzian signature. This class is a generalization to the remarkable class of Lorentzian polynomials. The hyperbolic polynomials and conic polynomials are shown to be polynomials with Lorentzian signature. Using the notion of polynomials with Lorentzian signature I shall describe how to compute the permanents of a special class of nonsingular matrices via hyperbolic programming. The nonsingular $k$ locally singular matrices are contained in the special class of nonsingular matrices for which computing the permanents can be done via hyperbolic programming.
Jacob Keller - UCSD
Food for Thought
Geometric invariant theory (GIT) is the main tool for taking quotients by group actions in algebraic geometry. In this talk I will try to show how GIT actually works by showing lots of examples.
Konrad Wrobel - McGill University
Math 211B - Group Actions Seminar
Zoom ID 967 4109 3409
(email an organizer for the password)Abstract
We prove various antirigidity and rigidity results around the orbit equivalence of wreath product actions. Let $F$ be a nonabelian free group. In particular, we show that the wreath products $A \wr F$ and $B \wr F$ are orbit equivalent for any pair of nontrivial amenable groups $A$, $B$. This is joint work with Robin Tucker-Drob.
Xiaolong Li - Wichita
Math 258: Seminar in Differential Geometry
Zoom ID: 953 0943 3365Abstract
I will first give an introduction to the notion of the curvature operator of the second kind and review some known results, including the proof of Nishikawa's conjecture stating that a closed Riemannian manifold with positive (resp. nonnegative) curvature operator of the second is diffeomorphic to a spherical space form (resp. a Riemannian locally symmetric space). Then I will talk about my recent works on the curvature operator of the second kind on Kahler manifolds and product manifolds. Along the way, I will mention some interesting questions and conjectures.
Jeff Viaclovsky - UCI
Geometers are interested in the problem of finding a "best"metric on a manifold. In dimension 2, the best metric is usually one which possesses the most symmetries, such as the round metric on a sphere, or a flat metric on a torus. In higher dimensions, there are many more classes of geometrically interesting metrics. I will give a general overview of a certain class of Einstein metrics in dimension 4 which have special holonomy, and which are known as "gravitational instantons." I will then discuss certain aspects of their classification and connections with algebraic surfaces.
Brian Tran - UCSD
Computational Geometric Mechanics Research Seminar
Adjoint systems are widely used to inform control, optimization, and design in systems described by ordinary differential equations or differential-algebraic equations. In this session, we explore the geometric properties and develop methods for such adjoint systems. In particular, we utilize symplectic and presymplectic geometry to investigate the properties of adjoint systems associated with ordinary differential equations and differential-algebraic equations, respectively. We show that the adjoint variational quadratic conservation laws, which are key to adjoint sensitivity analysis, arise from (pre)symplecticity of such adjoint systems. We discuss various additional geometric properties of adjoint systems, such as symmetries and variational characterizations. For adjoint systems associated with a differential-algebraic equation, we relate the index of the differential-algebraic equation to the presymplectic constraint algorithm of Gotay and Nester. As an application of this geometric framework, we discuss how the adjoint variational quadratic conservation laws can be used to compute sensitivities of terminal or running cost functions. Furthermore, we develop structure-preserving numerical methods for such systems using Galerkin Hamiltonian variational integrators which admit discrete analogues of these quadratic conservation laws. We additionally show that such methods are natural, in the sense that reduction, forming the adjoint system, and discretization all commute, for suitable choices of these processes. We utilize this naturality to derive a variational error analysis result for the presymplectic variational integrator that we use to discretize the adjoint DAE system. This is joint work with Prof. Melvin Leok.
In the post-talk discussion session, we plan to discuss future directions; in particular, exploring the geometry of adjoint systems for infinite-dimensional spaces with the application of PDE-constrained optimization in mind.
Maryam Yashtini - Georgetown University
Math 278C: Optimization and Data Science
Meeting ID: 941 9922 3268
Counting objects is a fundamental but challenging problem. In this paper, we propose diffusion-based, geometry-free, and learning-free methodologies to count the number of objects in images. The main idea is to represent each object by a unique index value regardless of its intensity or size, and to simply count the number of index values. First, we place different vectors, referred to as seed vectors, uniformly throughout the mask image. The mask image has boundary information of the objects to be counted. Secondly, the seeds are diffused using an edge-weighted harmonic variational optimization model within each object. We propose an efficient algorithm based on an operator splitting approach and alternating direction minimization method, and theoretical analysis of this algorithm is given. An optimal solution of the model is obtained when the distributed seeds are completely diffused such that there is a unique intensity within each object, which we refer to as an index. For computational efficiency, we stop the diffusion process before a full convergence, and propose to cluster these diffused index values. We refer to this approach as Counting Objects by Diffused Index (CODI). We explore scalar and multi-dimensional seed vectors. For Scalar seeds, we use Gaussian fitting in histogram to count, while for vector seeds, we exploit a high-dimensional clustering method for the final step of counting via clustering. The proposed method is flexible even if the boundary of the object is not clear nor fully enclosed. We present counting results in various applications such as biological cells, agriculture, concert crowd, and transportation. Some comparisons with existing methods are presented.
Li Gao - University of Houston
Math 243: Functional Analysis Seminar
Email email@example.com for Zoom infoAbstract
Logarithmic Sobolev inequalities, first introduced by Gross in 70s, have rich connections to probability, geometry, as well as information theory. In recent years, logarithmic Sobolev inequalities for quantum Markov semigroups attracted a lot of attentions for its applications in quantum information theory and quantum many-body systems. In this talk, I'll present a simple, information-theoretic approach to modified logarithmic Sobolev inequalities for both quantum Markov semigroup on matrices, and classical Markov semigroup on matrix-valued functions. In the classical setting, our results implies every sub-Laplacian of a Hörmander system admits a uniform modified logarithmic Sobolev constant for all its matrix valued functions. For quantum Markov semigroups, we improve a previous result of Gao and Rouzé by replacing the dimension constant by its logarithm. This talk is based on a joint work with Marius Junge, Nicholas, LaRacunte, and Haojian Li.