The Geometry and Geometric Analysis Seminar at Purdue in Fall 2024 is usually on Mondays 10:30am-11:20am Eastern Time in MATH 731 if we meet in person. Some talks will be on zoom and the link will be included in the email announcement.
Lvzhou Chen and Nicholas McCleerey are organizing this seminar in Fall 2024. If you have any questions or would like to suggest speakers, please contact one of us. If you would like to be on the mailing list, please email Lvzhou.
Title: Rigidity and Finiteness of Totally Geodesic Hypersurfaces in Negative Curvature
Abstract: This talk will discuss progress towards answering the following question: if a closed negatively curved manifold M has infinitely many totally geodesic hypersurfaces, then must it have constant curvature (and thus, by work of Bader-Fisher-Stover-Miller and Margulis-Mohamaddi, be homothetic to an arithmetic hyperbolic manifold)? First, I will talk about work that gives a partial answer to this question under the assumption that M is hyperbolizable. This part uses Ratner’s theorems in an essential way. I will also talk about work that gives an affirmative answer to the question if the metric on M is analytic. In this case the analyticity condition together with properties of the geodesic flow in negative curvature allow us to conclude what in constant curvature would follow from Ratner’s theorems. This talk is based on joint work with Fernando Al Assal, and Simion Filip and David Fisher.
Title: (Co)homology and Abelian Covers of Integrally Equivalent Manifolds
Abstract: Integral equivalence of manifolds implies a sort of refined iso-spectrality, first studied by Arapura et. al. They proved that integrally equivalent manifolds admit isomorphisms in cohomology compatible with restriction and corestriction. We extend this theorem to obtain similar isomorphisms in homology and use this to define a correspondence of abelian covers. We conclude by proving that corresponding abelian covers have the same normal closure over a common base, under mild hypotheses.
Title: Higher rank lattice actions with positive entropy
Abstract: In this talk, I will discuss about smooth higher rank lattice actions on manifolds with positive entropy. Especially, we see how one can construct a “homogeneous structure” on the manifold from dynamics. Then, for instance, when lattices in SL(n,R) (with n>2) act on an n-dimensional manifold with positive entropy, we can see that the lattice is commensurable with SL(n,Z) and some information about manifolds. This is joint work with Aaron Brown.
Title: The virtual Rokhlin property
Abstract: A topological group has the Rokhlin property (RP) if it contains an element whose conjugacy class Is dense; it has the virtual Rokhlin property (vRP) if it contains a closed finite-index subgroup that has the RP. The goal of the talk is to introduce and motivate the vRP. Despite its topological nature, vRP has both algebraic and geometric consequences. In joint work with Justin Lanier, we classified the 2-manifolds whose homeomorphism groups have the vRP: I will discuss this result and the strategy for establishing the vRP as well as to how to obstruct it.
Title: Hyperbolicity of Subvarieties in Non-Compact Ball Quotients
Abstract: Let X be an n-dimensional complex ball quotient by a non-uniform neat lattice, and Y be its unique toroidal compactification. We establish positivity properties of the cotangent bundle of Y depending intrinsically on X. Specifically, we show that all subvarieties of X are of general type when the cusps of X have sufficiently large depth. Furthermore, we derive a uniform lower bound for the volume of the canonical bundle of these subvarieties.
Title: Zeros of polynomials with integer coefficients
Abstract: A classical result of Fekete gives necessary conditions on a compact set in the complex plane so that it contains infinitely many sets of conjugate algebraic integers. In this talk, we discuss an effective version of Fekete’s theorem in terms of a height function. As an application, we give a lower bound on the growth of the leading coefficient of certain polynomial sequences, generalizing a result by Schur. Lastly, if time permits, we discuss distribution of zeros of polynomial mappings in C^2 with integer coefficients. This talk is based on a joint work with Norm Levenberg.
Title: More Complete Calabi-Yau Metrics of Calabi Type
Abstract: We construct more complete Calabi-Yau metrics asymptotic to Calabi ansatz. They are the higher-dimensional analogues of two dimensional ALH* gravitational instantons. Our work builds on and generalizes the results of Tian-Yau and Hein-Sun-Viaclovski-Zhang, creating Calabi-Yau metrics that are only polynomially close to the model space. We also prove the uniqueness of such metrics in a given cohomology class with fixed asymptotic behavior.
Title: Regularity of capillary minimal hypersurfaces
Abstract: A capillary surface is a hypersurface meeting some container at a prescribed angle, like the surface of water in a test tube. We describe two recent results concerning the boundary regularity of capillary surfaces which either minimize or are critical for their relevant energy. The first (joint with O. Chodosh and C. Li) is an improved dimension bound for energy minimizers: we show the boundary singular set has codimension at least 4 in the surface, and even higher codimension when the contact angle is close to 0, \pi/2, or \pi. The second result (joint with L. de Masi, C. Gasparetto, and C. Li) is a regularity theorem for energy-critical capillary surfaces, which implies regularity at generic boundary points of density < 1.
Title: Finite conjugacy classes and split exact cochain complexes
Abstract: We will present the theory behind and new results on the cohomology of super-reflexive Banach G-modules X, where G is a countable discrete group. In particular, we shall show how the cohomology is controlled by the FC-centre of G, that is, the subgroup of elements having finite conjugacy classes. For example, using purely cohomological tools, we show that when X is an isometric super-reflexive Banach G-module so that X has no almost invariant unit vectors under the action of the FC-centre, then the associated cochain complex is split exact. Further connections to work of Bader-Furman-Gelander-Monod, Nowak, and Bader-Rosendal-Sauer may also be presented.
Title: From curve graphs to fine curve graphs, and back
Abstract: In this talk I will introduce curve graphs and fine curve graphs of surfaces and discuss how they interact. In particular, I will talk about how, and to which extent, curve graphs can be used to approximate fine curve graphs, and how fine curve graphs results can be used to show the existence of a parabolic isometry of a graph of curves of certain infinite-type surfaces. This is joint work with Sebastian Hensel.
Title: Complex Monge-Ampère equation for positive (p,p) forms on compact Kähler manifolds
Abstract: A complex Monge-Ampère equation for differential (p,p) forms is introduced on compact Kähler manifolds. For any 1≤ p <n, we show the existence of smooth solutions unique up to adding constants. For p=1, this corresponds to the Calabi-Yau theorem proved by S. T. Yau, and for p=n−1, this gives the Monge-Ampère equation for (n−1) plurisubharmonic functions solved by Tosatti-Weinkove. For other p values, this defines a non-linear PDE that falls outside of the general framework of Caffarelli-Nirenberg-Spruck. Further, we define a geometric flow for higher-order forms that preserves their cohomology classes, and extends the Kähler-Ricci flow naturally to (p,p) forms. As a consequence of our main theorem, we show that this flow exists in a maximal time interval and can be shown to converge under some assumptions. The convergence of the associated normalized flow is shown and some potential applications are discussed.