Schedule

Speaker: Claude LeBrun (Stony Brook)

Title: Gravitational Instantons, Weyl Curvature, and Conformally Kaehler Geometry

Abstract: This talk will be based on  my recent joint paper with Olivier Biquard and Paul Gauduchon on ALF Ricci-flat Riemannian 4-manifolds. My collaborators had previously classified all such spaces that are toric and Hermitian, but not Kaehler. Our main result uses an open curvature condition to prove a rigidity result of the following type: any Ricci-flat metric that is sufficiently close to a non-Kaehler, toric, Hermitian ALF solution (with respect to a norm that imposes reasonable fall-off at infinity) is actually  one of the  known Hermitian toric  solutions. 

Speaker: Luca Di Cerbo (U Florida) 

Title: Curvature, Macroscopic Dimensions, and Symmetric Products of Curves

Abstract: In this talk, I will present a detailed study of the curvature and symplectic asphericity properties of symmetric products of surfaces. I show that these spaces can be used to answer nuanced questions arising in the study of closed Riemannian manifolds with positive scalar curvature. For example, symmetric products of surfaces sharply distinguish between two distinct notions of macroscopic dimension introduced by Gromov and Dranishnikov. As a natural generalization of this circle of ideas, I will also address the Gromov–Lawson and Gromov conjectures in the Kaehler projective setting and draw new connections between the theories of the minimal model, positivity in algebraic geometry, and macroscopic dimensions. This is joint work with Alexander Dranishnikov and Ekansh Jauhari.

Lunch Break

Lunch Break

Speaker: Sean Paul (U Wisconsin-Madison)

Title: The (spherical) Mahler measure of the $X-$discriminant

Abstract: Let $P$ be a homogeneous polynomial of degree $d$ in $N+1$ complex variables. The logarithmic Mahler measure of $P$  (denoted by $m(P) $) is the integral of $\log|P|$ over the sphere in  $C^{N+1}$ with respect to the usual Hermitian metric and measure on the sphere. Now let $X$ be a smooth variety embedded in $CP^{N} $ by a high power of an ample line bundle and let  $Delta$ denote a generalized discriminant of $X$ with respect to the given embedding, then $Delta$ is an irreducible homogeneous polynomial in the appropriate space of variables. In this talk I will discuss joint work in progress (with Song Sun and Junsheng Zhang) whose aim is to find an asymptotic expansion of $m(\Delta)$ in terms of the degree of the embedding. The technical machinery for this task was developed by Jean-Michel Bismut in several articles in the 1990's. 

Speaker: Pierre Albin (UIUC)

Title: Analytic torsion and the sub-Riemannian limit of a contact manifold

Abstract: Contact manifolds, which arise naturally in mechanics, dynamics, and geometry, carry natural Riemannian and sub-Riemannian structures and it was shown by Gromov that the latter can be obtained as a limit of the former. Subsequently, Rumin found a complex of differential forms reflecting the contact structure that computes the singular cohomology of the manifold. He used this complex to describe the behavior of the spectra of the Riemannian Hodge Lapacians in the sub-Riemannian limit. As many of the eigenvalues diverge, a refined analysis is necessary to determine the behavior of global spectral invariants. I will report on joint work with Hadrian Quan in which we describe some of the global behavior of the spectrum, such as that of analytic torsion, by explaining the structure of the heat kernel along this limit in a uniform way.

End of the day

Speaker:  Lei Ni (UCSD & ZJNU)

Note: This talk will be delivered on Zoom; Meeting ID: 975 7589 5137; Password: 142791

Title: Compact complex homogeneous manifolds without derivatives 

Abstract: In a joint work with Nolan Wallach we give a Lie algebraic treatment of classical works of H.-C. Wang, Tits on compact complex homogeneous manifolds. The method has the advantage of associating the integrable complex structure on a Lie algebra with a Cartan subalgebra canonically. 

Speaker: Brett Kotschwar (ASU)

Title: Asymptotically conical shrinking solitons

Abstract: Asymptotically conical shrinking Ricci solitons are an important class of singularity models for the Ricci flow in dimension four and higher, and give rise to solutions which, near infinity, flow smoothly into a cone at the singular time. We will present some uniqueness and nonexistence results which can be inferred from this latter fact using a unique continuation result valid at the singular time, and discuss their applications to the classification problem for asymptotically conical shrinkers.

Lunch Break

Speaker: Eric Chen (UIUC)

Title: Ricci flow and integral curvature pinching

Abstract: Early applications of the Ricci flow by Hamilton and others characterized Riemannian manifolds with certain pointwise curvature pinching conditions as spherical space forms.  In some cases, curvature pinching in averaged, integral senses can extend such results on topological restrictions.  I will describe some works on critical, scale-invariant integral curvature pinching and smoothing obtained using the Ricci flow and consequences of Perelman's W-entropy, joint with Guofang Wei and Rugang Ye.

Speaker: Yueh-Ju Lin (WSU)

Title: Renormalized Curvature Integrals on Poincare-Einstein manifolds

Abstract: Poincare–Einstein (PE) manifolds such as Poincare ball model are complete Einstein manifolds with a well-defined conformal boundary. A first step in studying the moduli space of PE manifolds is to develop a good understanding of its global invariants. In even dimensions, renormalized curvature integrals give many such invariants. In this talk, I will discuss a general procedure for computing renormalized curvature integrals on PE manifolds that is independent of Alexakis’ classification. In particular, this explains the connection between the Gauss-Bonnet-type formulas of Albin and Chang-Qing-Yang for the renormalized volume, and explicitly identify the scalar conformal invariant in the latter formula. Our procedure also produces similar formulas for compact Einstein manifolds. If time permits, I will mention more examples and applications. This talk is based on joint works with Jeffrey Case, Ayush Khaitan, Aaron Tyrrell, and Wei Yuan.

Speaker: Xiaodong Cao (Cornell University)

Title: Curvature of the Second Kind and a conjecture of Nishakawa

Abstract: In this talk, we will discuss the 1st , 2nd Riemanian curvature operators and an application to the Nishakawa Conjecture, based on the recent work of Cao-Gursky-Tran and Li.

Speaker: Yuanqi Wang (KU)

Note: This talk will be delivered on Zoom; Meeting ID: 975 7589 5137; Password: 142791

Title: Atiyah classes and the leading term related to singular G2 instantons

Abstract: $G_{2}-$manifolds via twisted connected sums are a potential carrier of critical connections and submanifolds with $1-$dimensional singularities, due to the outer circle of each asymptotic cylindrical $G_{2}$ building block. Following SáEarp-Walpuski gluing construction for smooth instantons,  Walpuski proposed constructing singular $G_{2} $- instantons on these manifolds. This requires a package of analysis adapted to the underlying geometry. In this talk, we discuss a partial analytic obstruction to such a deformation theory. It is the cohomology of a particular bundle on $CP^{2}$ and is related to a version of the Atiyah classes that govern the movement of singularities in this case. It vanishes if and only if the tangent connection is a particular one on the tangent bundle of $CP^{2}$, which is the only case we wish to carry out the analysis. Under model data, we identify a “leading term” in the solution of the linearized equation with the vector fields that “move” the singular locus. This is a joint project with Henrique Sá Earp and Grégoire Menet.  

Lunch Break

Speaker: Thalia Jeffres (WSU)

Title: Gradient bounds for Viscosity Solutions to Certain Elliptic Equations

Abstract: This will be a preliminary report on work still in progress with Xiaolong Li. Building upon Li's earlier work in the parabolic case, we show how to identify a one-dimensional operator for which the modulus of continuity of a viscosity solution to an elliptic equation is a subsolution. In favorable cases, this can be used to obtain a gradient bound for the solution to the original elliptic equation. 

End of Workshop