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.

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.

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.

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.

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.

Stein's method for distributional approximation has become a valuable tool in probability and statistics by providing finite sample distributional bounds for a wide class of target distributions in a number of metrics. A key step in popular versions of the method involves making couplings constructions, and a family of couplings of Chen and Roellin vastly expanded the range of applications for which Stein's method for normal approximation could be applied. This family subsumes both Stein's classical exchangeable pair, and the size bias coupling. A further simple generalization includes zero bias couplings, and also allows for situations where the coupling is not exact. The zero bias versions result in bounds for which often tedious computations of a variance of a conditional expectation is not required. An example to the Lightbulb process shows that even though the method may be simple to apply, it may yield improvements over previous results that had achieved bounds with optimal rates and small, explicit constants.

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.

Let $G$ be a $p$-adic reductive group with $\mathfrak{g} = \mathrm{Lie}(G)$. I will summarize work with Matthias Strauch in which we construct an exact functor from category $\mathcal{O}^\infty$, the extension closure of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$ inside the category of $U(\mathfrak{g})$-modules, into the category of admissible locally analytic representations of $G$. This expands on an earlier construction by Sascha Orlik and Matthias Strauch. A key role in our new construction is played by $p$-adic logarithms on tori, and representations in the image of this functor are related to some that are known to arise in the context of the $p$-adic Langlands program.

[pre-talk at 1:20PM]

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.

A variety $X$ over an algebraically closed field $k$ of characteristic $p>0$ is Witt-liftable if it is the closed fiber of a flat morphism $\mathcal{X}\to\mathrm{Spec}W(k)$, where $W(k)$ denotes the ring of Witt vectors of $k$. The existence of such a lift allows us to study $X$ using techniques from complex geometry. Although it is well-known that such a lift does not always exist, it is conjectured that every globally F-split variety is Witt-liftable. We show a stronger result in dimension two, and apply this to the study of singularities of globally F-split del Pezzo and Calabi-Yau surfaces. This is a joint work with F. Bernasconi, T. Kawakami, and J. Witaszek.

Pre-talk: 3:30-4:00pm