CUNY Geometric Analysis Seminar
In Fall of 2018, we will meet on Tuesdays, at 3pm,
room 6496. The
organizers of this seminar are Renato Bettiol,
Zeno Huang,
Neil Katz
and Bianca Santoro. Please
email Bianca at bsantoro(NoSpamPlease)ccny.cuny.edu to schedule a guest
speaker.
The CUNY Graduate Center is located at 365 Fifth Avenue at 34th Street, diagonally across the street
from the Empire State Building, just two blocks from Penn Station (NYC).
Fall 2018:
- Thursday, September 13: Markus Upmeier (University of Oxford). Please note the different day: the talk will be held at 3pm, room 6496.
Orientations for Moduli Spaces in Higher-Dimensional Gauge Theory
The Donaldson-Segal programme proposes to extend familiar techniques for
anti-self-dual connections on 4-manifolds to special, higher-dimensional
geometries. This includes the study of Calabi-Yau 3-folds, G2-manifolds, and
Spin(7)-holonomy manifolds. The fundamental difficulties are compactness,
deformation invariance, and orientations of the moduli space of connections.
This questions are, at present, largely open. After introducing some background
material, I shall discuss in my talk the orientation problem for SU(2)-bundles over
G2-manifolds.
- Tuesday, September 18: No Seminar
Holiday
- Tuesday, September 25: Antonio de Rosa (NYU)
Anisotropic counterpart of Allard's rectifiability theorem and applications.
Abstract: We present our extension of Allard's celebrated rectifiability theorem to the setting of varifolds with locally bounded first variation with respect to an anisotropic integrand. In particular, we identify a necessary and sufficient condition on the integrand to obtain the rectifiability of every d-dimensional varifold with locally bounded first variation and positive d-dimensional density.
We can apply this result to the minimization of anisotropic energies among families of d-rectifiable closed subsets of $\mathbb{R}^n$. Applications of this compactness result are the solutions to three formulations of the Plateau problem: one introduced by Reifenberg, one proposed by Harrison and Pugh and another one studied by Guy David.
Moreover, we apply the rectifiability theorem to prove an anisotropic counterpart of Allard's compactness result for integral varifolds.
To conclude, we give some ideas of an ongoing project, which relies on the presented rectifiability theorem.
The main result is a joint work with G. De Philippis and F. Ghiraldin.
- Tuesday, October 2: Rafael Montezuma (Princeton University)
Extremal metrics for the min-max width and the 2-systole in conformal classes
Abstract: In this talk, we will present a study on the min-max width of three-spheres, and a 2-dimensional systole of real projective three-spaces in conformal classes of Riemannian metrics. These are natural geometric invariants that have been studied from various different points of view. The min-max width, for instance, can be thought as the first eigenvalue of a non-linear spectrum of a Riemannian metric, as suggested by Gromov. The 2-systole that we consider is closely related with those considered by Berger, and later by Gromov and many others, generalizing the well studied quantity given by the length of the shortest non-contractible loop. We will focus on optimal bounds for the above invariants in conformal classes involving their volumes. If time permits we will discuss on some general properties of extremal metrics for the min-max width. Our methods involve an integral-geometric formula on the space of minimal two-spheres in homogeneous three-spheres, as well as the classification of such two-spheres by Meeks, Mira, Perez, and Ros. In the case of the min-max width, we apply the foundational convergence result of Chow for the Yamabe flow on conformally flat metrics with positive Ricci curvature. The flow is combined with old and new tools from the theory of minimal surfaces and min-max methods. All these results are part of a joint work with Lucas Ambrozio.
- Tuesday, October 9: Bernhard Hanke (University of
Augsburg)
Gromov's local flexibility and counter-intuitive applications
Abstract: In his famous book on partial differential relations, Gromov formulates an exercise concerning local deformations of solutions to open partial differential relations. We will explain the content of this fundamental and useful assertion, and sketch a proof. In the sequel we will apply this:
- to increase the excitement of Oktoberfest roller coasters without violating safety restrictions.
- to construct C^{1,1}-Riemannian metrics which are positively curved "almost everywhere" on arbitrary manifolds.
This is joint work with Christian Bar (Potsdam).
- Tuesday, October 16: Ruobing Zhang (Princeton University)
Nilpotent structure and collapsing Ricci-flat metrics on K3 surfaces
Abstract: We exhibit families of Ricci-flat KŠhler metrics on K3 surfaces which collapse to an interval, with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a corresponding continuous surjective map from the K3 surface to the interval, with regular fibers diffeomorphic to either 3-tori or Heisenberg nilmanifolds. This is joint work with H. Hein, J. Viaclovsky and S. Song.
- Tuesday, October 23: TBA
TBA
Abstract: TBA
- Tuesday, October 30: Xumin Jiang (Rutgers University)
Minimal graphs in the hyperbolic space
Abstract: We talk about the asymptotic behavior of the minimal surfaces in the hyperbolic space. To this end, we regard the minimal surface as a graph of certain function, of which we establish the boundary expansion with logarithmic terms. We also discuss the convergence theorem of the expansion under reasonable assumptions.
- Tuesday, November 6: TBA
TBA
Abstract: TBA
- Tuesday, November 13: Siyi Zhang(Princeton University)
A conformally invariant gap theorem characterizing complex projective space via the Ricci flow
Abstract: In this talk, we extend a sphere theorem proved by A. Chang, M. Gursky, and P. Yang to give a conformally invariant characterization of complex projective space.
In particular, we introduce a conformal invariant defined on closed four-manifolds. We shall show the manifold is diffeomorphic to the complex projective space if the conformal invariant is pinched sufficiently closed to that of the Fubini-Study metric. This is a joint work with Alice Chang and Matthew Gursky.
- Tuesday, November 20: Stephen Kleene (University of Rochester)
TBA
Abstract: TBA
- Tuesday, November 27: Aleksander Doan (Stony Brook University)
Harmonic Z/2 spinors and wall-crossing in Seiberg-Witten theory
Abstract: The notion of a harmonic Z/2 spinor was introduced by Taubes as an abstraction of various limiting objects appearing in compactifications of gauge-theoretic moduli spaces. I will explain this notion and discuss an existence result for harmonic Z/2 spinors on three-manifolds. The proof uses a wall-crossing formula for solutions of generalized Seiberg-Witten equations in dimension three, a result itself motivated by Yang-Mills theory on Riemannian manifolds with special holonomy G_2. The talk is based on joint work with Thomas Walpuski.
- Tuesday, December 4: Lu Wang (University of Wisconsin)
TBA
Abstract: TBA
- Tuesday, December 11: Ronan Conlon (Florida International University)
TBA
Abstract: TBA
Spring 2018:
- Thursday, February 15: Mark Stern (Duke University).
Monotonicity and Betti Number Bounds
Abstract: In this talk I will discuss the application of techniques initially developed to study singularities of Yang Mill's fields and harmonic maps to obtain Betti number bounds, especially for negatively curved manifolds.
- Thursday, February 22: Baris Coskunuzer (Boston College)
Asymptotic H-Plateau Problem in H^3
Abstract: In this talk, we show that for any C^0 Jordan curve C in the sphere at infinity of H^3, there exists an embedded constant mean curvature H-plane P_H in H^3 with asymptotic boundary C for any H in (-1,1). As a corollary, we will show that any quasi-Fuchsian hyperbolic 3-manifold M=SxR contains an H-surface S_H in the homotopy class of the core surface S for any H in (-1,1).
- Thursday, March 1: Antoine Song (Princeton University)
Local min-max surfaces and existence of minimal Heegaard splittings
Abstract: Let M be a closed oriented 3-manifold not diffeomorphic to the 3-sphere, and suppose that there is a strongly irreducible Heegaard splitting H. Previously, Rubinstein announced that either there is a minimal surface of index at most one isotopic to H or there is a non-orientable minimal surface such that the double cover with a vertical handle attached is isotopic to H. He sketched a natural outline of a proof using min-max, which is however incomplete. We will explain how to justify it. The key point is a version of min-max theory producing interior minimal surfaces when the ambient manifold has minimal boundary. Some corollaries of the theorem include the existence in any RP^3 of either a minimal torus or a minimal projective plane with stable universal cover. Several consequences for metric with positive scalar curvature are also derived. The first part is joint work with Dan Ketover and Yevgeny Liokumovich.
- Thursday, March 8: Shengwen Wang (Johns Hopkins University)
Hausdorff stability of round spheres under small-entropy perturbation
Abstract: Colding-Minicozzi introduced the entropy functional on the space of all hypersurfaces in the Euclidean space when studying generic singularities of mean curvature flow. It is a measure of complexity of hypersurfaces. Bernstein-Wang proved that round spheres $\mathbb S^n$ minimize entropy among all closed hypersurfaces for $n\leq6$, and the result is generalized to all dimensions by Zhu. Bernstein-Wang later also proved that the round 2-sphere is actually Hausdorff stable under small-entropy perturbations. I will present in this talk the generalization of the Hausdorff stability to round hyper-spheres in all dimensions.
- Thursday, March 15: Alex Waldron (SCGP, Stony Brook)
Yang-Mills flow in dimension four
Abstract: Among the classical geometric evolution equations, YM flow is the least nonlinear and best behaved. Nevertheless, curvature concentration is a subtle problem when the base manifold has dimension four. I'll discuss my proof that finite-time singularities do not occur, and briefly describe the infinite-time picture.
- Thursday, March 22: Daniel Stern (Princeton University)
Level Set PDE Approaches to the Construction of Codimension-Two Minimal Submanifolds
Abstract: We will present a new approach to the construction of generalized minimal submanifolds of codimension 2 on a given compact manifold, via variational methods for the complex Ginzburg-Landau functionals. In the course of our discussion, we will also highlight some new estimates for the Ginzburg-Landau functionals on manifolds with nonzero first Betti number, which may be of independent interest.
- Thursday, March 29: No Seminar
Spring Break
- Thursday, April 5: No Seminar
Spring Break
- Thursday, April 12: Yueh-Ju Lin (Princeton University)
Deformations of Q-curvature
Abstract: Stability (local surjectivity) and rigidity of the scalar curvature have been studied in an early work of Fischer-Marsden on "vacuum static spaces". Inspired by this
line of research, we seek similar properties for Q-curvature by studying Q-singular
spaces", which were introduced by Chang-Gursky-Yang.
In this talk, we investigate deformation problems of Q-curvature on closed Riemannian manifolds with dimensions n 3. In particular, we prove local surjectivity for non-Q-singular spaces and local rigidity of flat manifolds.
For global results, we show that any smooth functions can be realized as a Q-curvature on generic Q-flat manifolds. However, a locally conformally flat metric
on n-tori with nonnegative Q-curvature has to be flat.
This is joint work with Wei Yuan.
- Thursday, April 19: Sophia Jahns (TŸbingen University)
Trapped Light in Stationary Spacetimes
Abstract: Light can circle a massive object (like a black hole or a neutron star) at a ''fixed distance'', or, more generally, circle the object without falling in or escaping to infinity. This phenomenon is called trapping of light and well understood in static, asymptotically flat (AF) spacetimes. If we drop the requirement of staticity, similar behavior of light is known, but there is no definition of trapping available.
We present some known results about trapping of light in static AF spacetimes. Using the Kerr spacetime as a model, we then show how trapping can be better understood in the framework of phase space. We prove that the Kerr photon region is a submanifold of the phase space and characterize its topology.
This is joint work with Carla Cederbaum.
- Wednesday, April 25 to Friday, April 27: Nonlinear Days in New York
3-day workshop organized by Z. Huang, M. Lucia (CUNY) and M. Squassina (Universita Cattolica del Sacro Cuore, Italy)
For the list of speakers, please click here.
- Thursday, May 3: Yu Li (Stony Brook University)
Ricci flow on asymptotically Euclidean manifolds
Abstract: In this talk, we prove that if an asymptotically Euclidean (AE) manifold with nonnegative scalar curvature has long time existence of Ricci flow, it converges to the Euclidean space in the strong sense. By convergence, the mass will drop to zero as time tends to infinity. Moreover, in three-dimensional case, we use Ricci flow with surgery to give an independent proof of the positive mass theorem.
- Thursday, May 10: Shaosai Huang (Stony Brook University)
Epsilon-regularity of four dimensional gradient shrinking Ricci solitons
Abstract: A central issue in studying uniform behaviors of Riemannian manifolds is to obtain uniform local L^{\infty}-bounds of the curvature tensor. For manifolds whose Riemannian metric satisfying certain elliptic equations, e.g. Einstein manifolds and Ricci solitons, local curvature bound are expected when the local energy is sufficiently small. Such estimates, referred to as \epsilon-regularity, are usually obtained via Moser iteration arguments, which requires a uniform control of the Sobolev constant. This requirement may fail in many natural situations. In this talk, I will discuss an epsilon-regularity result for 4-dimensional shrinking Ricci solitons without a priori control of the Sobolev constant.
Fall 2017:
- Thursday, September 14: Yiming Zhao (NYU/St. Johns University)
Minkowski-type problems in convex geometry
Abstract: An overview of Minkowski-type problems will be presented. Minkowski-type problems are related to Monge-Ampere equations. Particular focus will be on the dual Minkowski problem, which is the characterization problem for a family of new geometric measures, called dual curvature measures, introduced in [Huang, Lutwak, Yang, Zhang; Acta 2016]. These are the "duals" of Federer's curvature measure. A solution to the dual Minkowski problem in the symmetric case will be demonstrated. The solution is strongly connected to measure concentration phenomenon.
- Thursday, September 21: No Seminar
Holiday
- Thursday, September 28: Klaus Kroencke (University of Hamburg)
Stability of ALE Ricci-flat manifolds under Ricci-flow
Abstract: We prove that if an ALE Ricci-flat manifold (M,g) is linearly stable and integrable, it is dynamically stable under Ricci flow, i.e. any Ricci flow starting close to g exists for all time and converges modulo diffeomorphism to an ALE Ricci-flat metric close to g. By adapting Tian's approach in the closed case, we show that integrability holds for ALE Calabi-Yau manifolds which implies that they are dynamically stable. This is joint work with Alix Deruelle
- Thursday, October 5: Xiaochun Rong (Rutgers University)
Collapsed manifolds with Ricci curvature and local rewinding volume bounded below
Abstract: I'll report some recent progress in the study of manifolds in the title. Given a number rho>0 and a point x in M, a complete n-manifold, the local rewinding volume of the rho-ball at x, B_rho(x), is the volume of the \rho-ball at \tilde xon the (incomplete) Riemannian universal cover of B_rho(x). Part of this work is joint with Hongzhi Huang, Zuohai Jiang and Shicheng Xu of Capital Normal University.
- Thursday, October 12: Hengyu Zhou (Sun Yat-Sen University)
Mean Curvature Flows in Warped Product Manifolds
Abstract: In this talk we discuss the mean curvature flow of starshaped hypersurface in warped product manifolds admitting a totally geodesic slice Sigma. With some natural conditions on the warping function and Ricci curvature, long time existences and convergences to Sigma are established. This is a joint work with Zhou Zhang (Sydney) and Zheng Huang (CUNY).
- Thursday, October 19: CUNY symposium on nonlinear problems in geometry
TBA
- Thursday, October 26: Valentino Tosatti (Northwestern University)
Collapsing hyperkahler manifolds
Abstract: Consider a projective hyperkahler manifolds with a surjective holomorphic map with connected fibers onto a lower-dimensional manifold. In the case the base must be half-dimensional projective space, and the generic fibers are holomorphic Lagrangian tori. I will explain how hyperkahler metrics on the total space with volume of the torus fibers shrinking to zero, collapse smoothly away from the singular fibers to a special Kahler metric on the base, whose metric completion equals the global collapsed Gromov-Hausdorff limit, which has a singular set of real Hausdorff codimension at least 2. The resulting picture is compatible with the Strominger-Yau-Zaslow mirror symmetry, and can be used to prove a conjecture of Kontsevich-Soibelman and Gross-Wilson for large complex structure limits which arise via hyperkahler rotation from this construction. This is joint work with Yuguang Zhang.
- Thursday, November 2: Renato Bettiol (University of Pennsylvania)
Sectional curvature and Weitzenbšck formulae
Abstract: We prove an algebraic characterization of lower and upper sectional
curvature bounds in terms of the curvature terms in the Weitzenbšck
formulae for symmetric p-tensors. By introducing a symmetric analogue
of the Kulkarni-Nomizu product, we also derive an explicit formula for
such curvature terms, as a function of scalar, traceless Ricci, and
Weyl curvatures. This is based on joint work with R. Mendes.
- Thursday, November 9: TBA
TBA
Abstract: TBA
- Thursday, November 16: Xinliang An (University of Toronto)
On Singularity Formation in General Relativity
Abstract: In the process of gravitational collapse, singularities may form, which are either covered by trapped surfaces (black holes) or visible to faraway observers (naked singularities). In this talk, with three different approaches coming from hyperbolic PDE, quasilinear elliptic PDE and dynamical system, I will provide answers for four physical questions: i) Can black holes form dynamically in vacuum? ii) To form a black hole, what is the least size of initial data? iii) Can we find the boundary of a black hole region? Can we show that a black hole region is emerging from a point? iv) For Einstein vacuum equations, could singularities other than black hole type form in gravitational collapse?
- Thursday, November 23: No seminar
Thanksgiving
- Thursday, November 30: Lizhi Chen (Lanzhou University)
Systoles of 3-manifolds
Abstract: In systolic geometry, we study infimum volume of cycles representing homotopy or homology classes. The homotopy 1-systole of a Riemannian manifold is defined to be the shortest length of a noncontractible loop. Gromov proved that the homotopy 1-systole can be upper bounded in terms of volume for a closed essential manifold. In the talk, I am going to discuss the relation between topological complexity and the optimal constant in a systolic inequality for certain essential 3-manifolds. Moreover, we define homology systole as the infimum volume of cycles representing nontrivial homology classes. A conjecture of Gromov indicates that the volume of a Riemannian manifold can be bounded below in terms of homology 2-systoles over mod two coefficients. However, there are some counterexamples in dimension three.
- Thursday, December 7: Double Header - Nan Li (New York City Tech). Usual time and place: room 6496, 3pm to 4pm.
Quantitative Estimates on the Singular Sets of Alexandrov Spaces
Abstract: The notion of quantitative singular sets for spaces with lower Ricci curvature bounds was initiated by Cheeger and Naber. Volume estimates were proved for these singular sets in a non-collapsing setting. For Alexandrov spaces, we obtain stronger and volume-free estimates. We also show that the (k,\epsilon)-singular sets are k-rectifiable and such structure is sharp in some sense. This is a joint work with Aaron Naber.
- Thursday, December 7: Double Header - Luca Di Cerbo (Stony Brook University). Please note the different time: 4-5pm, room 6496.
Price Inequality and Betti numbers of manifolds without conjugate points
Abstract: In this talk, I will present a Price type inequality for harmonic forms on manifolds without conjugate points and negative Ricci curvature. The techniques employed in the proof work particularly well for manifolds of non-positive sectional curvature, and in this case one can prove a strengthened result. Equipped with these Price type inequalities, I then study the asymptotic behavior of Betti numbers along infinite towers of regular coverings. If time permits, I will discuss the case of hyperbolic manifolds in some detail. This is joint work with M. Stern.
Spring 2017:
- Thursday, February 16: Yusheng Wang (Beijing Normal University and Rutgers University)
Curvature>=1, diameter>=\pi/2 and rigidity (in Alexandrov geometry)
Abstract: In this talk, we will review some classical and recent rigidity results in Alexandrov geometry with curvature >=1 (including Riemannian geometry with sectional curvature
>=1) brought by distance>=\pi/2 between points or convex subsets.
- Thursday, February 23: Jorge Basilio (Graduate Center, CUNY)
Sequences of three-dimensional manifolds with positive scalar curvature
Abstract: Here we explore to what extent one may hope to preserve geometric properties of three dimensional manifolds with lower scalar curvature bounds under Gromov-Hausdorff and Intrinsic Flat limits. We introduce a new construction of three dimensional manifolds with positive scalar curvature called sewing. We produce sequences of such manifolds which converge to spaces demonstrating that almost rigidity theorems for manifolds with positive or nonnegative scalar curvature fail to hold for these limits including the Scalar Torus Rigidity Theorem and the rigidity part of the Positive Mass Theorem. Since the notion of nonnegative scalar curvature is not strong enough to persist alone, we propose that one pair a lower scalar curvature bound with a lower bound on the area of a closed minimal surface when taking sequences as this will prevent the sewing of manifolds and possibly also the existence of counter examples. This is joint work with Christina Sormani.
- Thursday, March 16: Dan Ketover (Princeton University)
Variational constructions of minimal surfaces
Abstract: I will describe how variational methods can be applied to produce many new examples of minimal surfaces with various symmetries. IÕll focus on the construction of free boundary minimal surfaces resembling a desingularization of the critical catenoid and flat disk, which were first conjectured by Fraser-Schoen.
- Thursday, March 23: Jacob Bernstein (Johns Hopkins University)
Surfaces of Low Entropy
Abstract: Following Colding and Minicozzi, we consider the entropy of (hyper)-surfaces in Euclidean space. This is a numerical measure of the geometric complexity of the surface. In addition, this quantity is intimately tied to to the singularity formation of the mean curvature flow which is a natural geometric heat flow of submanifolds. In the talk, I will discuss several results that show that closed surfaces for which the entropy is small are simple in various senses. This is all joint work with L. Wang.
- Thursday, March 30: Florentin Munch (Potsdam University)
Rigidity properties of the hypercube via discrete Ricci curvature
Abstract: We give rigidity results for discrete Bonnet-Myers diameter bound
and Lichnerowicz eigenvalue estimate. Both inequalities are sharp if
and only if the underlying graph is a hypercube. The proofs use
well-known semigoup methods as well as new direct methods which
translate curvature to combinatorial properties. The results can be
seen as first known discrete analogues of Cheng's
and Obata's rigidity theorems.
- Thursday, April 6: Ronan Conlon (Florida International University)
New examples of gradient expanding Kahler-Ricci solitons
Abstract: A complete Kahler metric g on a Kahler manifold M is a gradient expanding Kahler-Ricci soliton if there exists a smooth real-valued function f:M -> R with $\nabla^{g}f$ holomorphic such that Ric(g)-Hess(f)+g=0. I will present new examples of such metrics on the total space of certain holomorphic vector bundles. This is joint work with Alix Deruelle (Universite Paris-Sud).
- Thursday, April 27: Liming Sun (Rutgers University)
Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant
Abstract: We study one of Yamabe problems on compact manifolds with boundary. For negative Yamabe type manifolds, we are able to prescribe scalar curvature and boundary mean curvature. Precisely, given any negative smooth functions f in M and h on the boundary $\partial M$, there exists a unique conformal metric of g_0 such that its scalar curvature equals f and mean curvature curvature equals h. One family of flow with dynamic boundary mean curvature is constructed to prove this result. At the end, I will mention some ongoing results on positive type manifold using flow approach.
- Thursday, May 4: TBA
TBA
Abstract: TBA
- Thursday, May 11: Jiayin Pan (Rutgers University)
A proof of Milnor conjecture in dimension 3
Abstract: In this talk, we present a proof of Milnor conjecture in dimension 3 based on the Cheeger-Colding theory on limit spaces of manifolds with Ricci curvature
bounded below.
Fall 2016: