Thu, Oct 6 2022
##### A mesoscale flatness criterion and its application to exterior isoperimetry

Math 258: Seminar of Differential Geometry

APM 7321

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).

##### Strong independence of $\ell$ for Shimura varieties

Math 209 - Number Theory Seminar

APM 6402 and Zoom

(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.

Fri, Oct 7 2022
##### Gaeta resolutions and strange duality over rational surfaces

Math 208: Algebraic Geometry Seminar

(via zoom)

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

Tue, Oct 11 2022
##### Geometric Integration of Adjoint DAE Systems

Math 278A: Center for Computational Mathematics Seminar

APM 7218 and Zoom
Zoom ID 986 1678 1113

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.

##### Bredon homology

Math 292: Topology Seminar (student seminar on equivariant homotopy theory)

APM 7218

##### Equivariant A-theory and spaces of equivariant h-cobordisms

Math 292: Topology Seminar

APM 7218

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.

Wed, Oct 12 2022
##### Computing Permanents via Hyperbolic Programming

Math 278C: Optimization and Data Science

https://ucsd.zoom.us/j/94199223268

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.

##### GIT 101

Food for Thought

HSS 4025

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.

Thu, Oct 13 2022
##### Orbit equivalence and wreath products

Math 211B - Group Actions Seminar

Zoom ID 967 4109 3409
(email an organizer for the password)

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.

##### The Curvature Operator of the Second Kind

Math 258: Seminar in Differential Geometry

Zoom ID: 953 0943 3365

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.

##### Gravitational instantons and algebraic surfaces

Department Colloquium

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.

Tue, Oct 18 2022
##### Geometric Methods for Adjoint Systems

Computational Geometric Mechanics Research Seminar

APM 6402

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.

Wed, Oct 19 2022
##### Counting Objects by Diffused Index: geometry-free and training-free approach

Math 278C: Optimization and Data Science

https://ucsd.zoom.us/j/94199223268

Meeting ID: 941 9922 3268