During the Fall of 2020, we will meet online on Thursdays, at 4pm. 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).
Those participating in the Geometric Analysis seminar may also be interested in the Nonlinear Analysis and PDEs seminar which meets Thursdays starting at 4:15pm.
Fall 2020:
Thursday, September 3, 2pm , Christian Lange (University of Koeln, Germany).
Zoll flows on surfaces
A Riemannian metric is called Zoll if all its geodesics are closed with the same period.
We discuss rigidity and flexibility phenomena of such Riemannian and more general Zoll systems.
In particular, we show that if a magnetic flow on a torus is Zoll at arbitrarily high energies, then the
torus is flat. The latter is joint work with Luca Asselle.
This will be a zoom virtual talk. The link for the talk will be sent to the seminar participants. To be added to the mailing list, please email Prof. Neil Katz at nkatz(NoSpamPlease)citytech.cuny.edu
Thursday, September 17, 4pm , Mariana Smit Vega Garcia (Western Washington University).
Almost minimizers for obstacle problems
In the applied sciences one is often confronted with free boundaries, which arise when the solution to a problem consists of a pair: a function u (often satisfying a partial differential equation), and a set where this function has a specific behavior. Two central issues in the study of free boundary problems and related problems in the calculus of variations and geometric measure theory are:
(1) What is the optimal regularity of the solution u?
(2) How smooth is the free boundary (or how smooth is a certain set related to u)?
The study of the classical obstacle problem, one of the most renowned free boundary problems, began in the ?60s with the pioneering works of G. Stampacchia, H. Lewy, and J. L. Lions. During the past five decades, it has led to beautiful and deep developments in the calculus of variations and geometric partial differential equations, and its study still presents very interesting and challenging questions. In contrast to the classical obstacle problem, which arises from a minimization problem, minimizing problems with noise lead to the notion of almost minimizes. Though deeply connected to "standard" free boundary problems, almost minimizers do not satisfy a PDE as minimizers do, requiring additional tools from geometric measure theory to address (1) and (2). In this talk, I will overview recent developments on obstacle type problems and almost minimizers for the thin obstacle problem, illustrating techniques that can be used to tackle questions (1) and (2) in various settings.
Thursday, September 24, 4pm , Ian Adelstein (Yale University) .
The length of the shortest closed geodesic on positively curved 2-spheres
We start with an intuitive introduction to the isosystolic inequalities. We then show that the shortest closed geodesic on a 2-sphere with non-negative curvature has length bounded above by three times the diameter. We prove a new isoperimetric inequality for 2-spheres with pinched curvature; this allows us to improve our bound on the length of the shortest closed geodesic in the pinched curvature setting. This is joint work with Franco Vargas Pallete.
Thursday, October 1, 4pm , Gonçalo Oliveira (UFF, Brazil) .
G2-monopoles (a summary)
This talk is aimed at reviewing what is known about $G_2$-monopoles and motivate their study. After this, I will mention some recent results obtained in collaboration with Ákos Nagy and Daniel Fadel which investigate the asymptotic behavior of $G_2$-monopoles. Time permitting, I will mention a few possible future directions regarding the use of monopoles in $G_2$-geometry.
Thursday, October 8, 4pm , Samuel Lin (Dartmouth College) .
Three-dimensional Geometric Structures and the Laplace Spectrum
The earliest examples of non-isometric Laplace-isospectral manifolds have the same local geometries. In fact, the first example of 16-tori given by Milnor and other isospectral pairs arising from the classical group theoretic method of Sunada have the same local geometries. However, examples from Gordon, Schueth, Sutton, and An-Yu-Yu demonstrate that in dimension four and higher, the local geometry is not a spectral invariant, even among locally homogeneous spaces. Thus, it is natural to ask whether the local geometry is a spectral invariant in dimension two and three. I will present our result in this direction, which provides strong evidence that the local geometry of a three-dimensional locally homogeneous space is a spectral invariant. Motivated by this problem in spectral geometry, I will also present a metric classification of all locally homogeneous three-manifolds covered by topological spheres. This talk is based on a joint work with Ben Schmidt and Craig Sutton.
Thursday, October 15, 4pm , Yueh-Ju Lin (Wichita State University).
Volume comparison of Q-curvature
Classical volume comparison for Ricci curvature is a fundamental result in Riemannian geometry. In general, scalar curvature as the trace of Ricci curvature, is too weak to control the volume. However, with the additional stability assumption on the closed Einstein manifold, one can obtain a volume comparison for scalar curvature. In this talk, we investigate a similar phenomenon for $Q$-curvature, a fourth-order analogue of scalar curvature. In particular, we prove a volume comparison result of $Q$-curvature for metrics near stable Einstein metrics by variational techniques and a Morse lemma for infinite dimensional manifolds. This is a joint work with Wei Yuan.
Thursday, October 22, 4pm , Curtis Pro (Cal State Stanislaus).
Extending a diffeomorphism finiteness theorem to dimension 4.
Cheeger's Finiteness Theorem says: Given numbers k < K in R and v, D > 0, there are at most finitely many differentiable structures on the class of n-manifolds M that support metrics with k<= sec M <= K, vol(M)>=v,$ and diam(M) <= D.$ In the early 90s, Grove, Petersen, Wu, and (independently) Perelman showed in all dimensions, except possibly n=4, this conclusion still holds for the larger class that has no upper bound on sectional curvature. In this talk, I'll present recent work with Fred Wilhelm that shows this conclusion is also true in dimension 4.
Thursday, October 29, 4pm , Ernani Ribeiro Jr. (UFC, Brazil) .
Four-dimensional gradient shrinking Ricci solitons
In this talk, we will discuss 4-dimensional complete (not necessarily compact) gradient shrinking Ricci solitons. We will show that a 4-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual or anti-self-dual part of the Weyl tensor is either Einstein, or a finite quotient of either the Gaussian shrinking soliton $\Bbb{R}^4,$ or $\Bbb{S}^{3}\times\Bbb{R}$, or $\Bbb{S}^{2}\times\Bbb{R}^{2}.$ In addition, we will present some curvature estimates for 4-dimensional complete gradient Ricci solitons. Some open problems will be also discussed. This is a joint work with Huai-Dong Cao and Detang Zhou.
Thursday, November 5, 4pm , Jason Ledwidge (U Tuebingen, Germany) .
The sharp Li-Yau equality on shrinking Ricci solitons
In this talk we will prove a sharp Li-Yau equality on shrinking Ricci solitons and use this equality to prove the existence of a minimiser for Perelman's W functional on shrinking Ricci solitons. By a result of Haslhofer-Mueller, the uniqueness of the minimisier of the W functional leads to the classification of Type I singularity models to the Ricci flow in four dimensions. If time permits, we will also show how the Li-Yau equality leads to a global Isoperimetric inequality on shrinkig Ricci solitons. We will be more interested in the importance of the conjugate heat semigroup and its estimates on shrinking Ricci solitons and hence our aim is for the talk not to be too technical.
This will be a zoom virtual talk. The link for the talk will be sent to the seminar participants. To be added to the mailing list, please email Prof. Neil Katz at nkatz(NoSpamPlease)citytech.cuny.edu
Thursday, November 12, 4pm , Jonathan Zhu (Princeton University).
Explicit Lojasiewicz inequalities for mean curvature flow shrinkers
Lojasiewicz inequalities are a popular tool for studying the stability of geometric structures. For mean curvature flow, Schulze used Simon?s reduction to the classical Lojasiewicz inequality to study compact tangent flows. Colding and Minicozzi instead used a direct method to prove Lojasiewicz inequalities for round cylinders. We?ll discuss similarly explicit Lojasiewicz inequalities and applications for other shrinking cylinders and Clifford shrinkers.
Thursday, November 19, 4pm , Robin Neumayer (Northwestern University).
d_p- Convergence and epsilon-regularity theorems for entropy and scalar curvature lower bounds
In this talk, we consider Riemannian manifolds with almost non-negative scalar curvature and Perelman entropy. We establish an $\epsilon$-regularity theorem showing that such a space must be close to Euclidean space in a suitable sense. Interestingly, such a result is false with respect to the Gromov-Hausdorff and Intrinsic Flat distances, and more generally the metric space structure is not controlled under entropy and scalar lower bounds. Instead, we introduce the notion of the $d_p$ distance between (in particular) Riemannian manifolds, which measures the distance between $W^{1,p}$ Sobolev spaces, and it is with respect to this distance that the $\epsilon$ regularity theorem holds. We will discuss various applications to manifolds with scalar curvature and entropy lower bounds, including a compactness and limit structure theorem for sequences, a uniform $L^\infty$ Sobolev embedding, and a priori $L^p$ scalar curvature bounds for $p<1$ This is joint work with Man-Chun Lee and Aaron Naber.
This will be a zoom virtual talk. The link for the talk will be sent to the seminar participants. To be added to the mailing list, please email Prof. Neil Katz at nkatz(NoSpamPlease)citytech.cuny.edu
Thursday, November 26, No Seminar.
Happy Thanksgiving!
FRIDAY, December 4, 10am , Lashi Bandara (U. Potsdam)
The world of rough metrics
Rough metrics are measurable coefficient Riemannian structures.
They capture a very large class of natural geometries, with the
quintessential example being Lipschitz pullbacks of smooth metrics.
Although they have implicitly appeared for a very long time,
particularly in the context of bounded-measurable coefficient divergence
form equations, they have only been studied explicitly recently.
The aim of this talk would be to introduce these metrics, motivated by
an important example - their connection to the geometric Kato square
root problem.
Their salient features would be described, along with recent results,
such as the existence of heat kernels and Weyl asymptotics for
associated Laplacians in compact settings.
This will be a zoom virtual talk. The link for the talk will be sent to the seminar participants. To be added to the mailing list, please email Prof. Neil Katz at nkatz(NoSpamPlease)citytech.cuny.edu
Summer 2020:
Thursday, May 28, Eduardo Longa (USP, Brazil).
Sharp systolic inequalities for 3-manifolds with boundary
Systolic Geometry dates back to the late 1940s, with the work of Loewner and his doctoral student Pu. This branch of differential geometry received more attention after the seminal work of Gromov, where he proved his famous systolic inequality and defined many important concepts. In this talk I will recall the notion of systole and present some sharp systolic inequalities for free boundary surfaces in 3-manifolds.
Here is the video of the talk:
Thursday, June 4, 2pm , Klaus Kroencke (Univ. Hamburg, Germany).
L^p-stability and positive scalar curvature rigidity of Ricci-flat ALE manifolds
We will establish long-time and derivative estimates for the heat semigroup of various natural Schrödinger operators on asymptotically locally Euclidean (ALE) manifolds. These include the Lichnerowicz Laplacian of a Ricci-flat ALE manifold, provided that it is spin and admits a parallel spinor. The estimates will be used to prove its L^p-stability under the Ricci flow for p less than n. A positive scalar curvature rigidity theorem will also be deduced. This is joint work with Oliver Lindblad Petersen.
Here is the video of the talk:
Thursday, June 11, Ricardo Mendes (University of Oklahoma).
The isometry group of spherical quotients
A special class of Alexandrov metric spaces are the quotients X=S^n/G of the round spheres by isometric actions of compact subgroups G of O(n+1). We will consider the question of how to compute the isometry group of such X, the main result being that every element in the identity component of Isom(X) lifts to a G-equivariant isometry of the sphere. The proof relies on a pair of important results about the "smooth structure" of X.
Here is the video of the talk:
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Thursday, June 18, Shih-Kai Chiu (Notre Dame) .
A Liouville type theorem for harmonic 1-forms
The famous Cheng-Yau gradient estimate implies that on a
complete Riemannian manifold with nonnegative Ricci curvature, any
harmonic function that grows sublinearly must be a constant. This is
the same as saying the function is closed as a 0-form. We prove an
analogous result for harmonic 1-forms. Namely, on a complete
Ricci-flat manifold with Euclidean volume growth, any harmonic 1-form
with polynomial sublinear growth must be the differential of a
harmonic function. We prove this by proving an L^2 version of the
"gradient estimate" for harmonic 1-forms. As a corollary, we show that
when the manifold is Ricci-flat Kähler with Euclidean volume growth,
then any subquadratic harmonic function must be pluriharmonic. This
generalizes the result of Conlon-Hein.
Here is the video of the talk:
Thursday, June 25, Clara Aldana (Universidad del Norte, Colombia).
Strong A_\infty weights and compactness of conformal metrics.
In the talk I will introduce A_{\infty}-weights and strong A_{\infty}-weights and present some of their properties. I will show how, using these weights, we can prove compactness of conformal metrics with critical integrability conditions on the scalar curvature. This relates to two problems in differential geometry: Pinching of the curvature and finding geometrical conditions under which a sequence of conformal metrics admits a convergent subsequence. The results presented here are joint work with Gilles Carron and Samuel Tapie, both of them from University of Nantes.
Here is the video of the talk:
Thursday, July 2, Raquel Perales (UNAM, Mexico).
Convergence of manifolds under volume convergence and a tensor bound
[Based on join work with Allen-Sormani and Cabrera Pacheco-Ketterer]
Given a Riemannian manifold $M$ and a pair of Riemannian tensors $g_0 \leq g_j$ on $M$
we have $\vol_0(M) \leq \vol_j(M)$ and the volumes are equal if and only if $g_0=g_j$.
In this talk I will show that if we have a sequence of Riemmanian tensors $g_j$ such that
$g_0\leq g_j$ and $\vol_j(M)\to \vol_0(M)$ then $(M,g_j)$ converge to $(M,g_0)$ in the
volume preserving intrinsic flat sense. I will present examples demonstrating that under these
conditions we do not necessarily obtain smooth, $C^0$ or Gromov-Hausdorff convergence.
Furthermore, this result can be applied to show stability of graphical tori.
Here is the video of the talk:
Thursday, July 9, 2pm, Ilaria Mondello (Université de Paris Est Créteil, France).
Non-existence of Yamabe metrics in a singular setting
The existence of Yamabe metrics, that is, metrics which minimize the Einstein-Hilbert functional in a conformal class, has been proven for compact smooth manifolds thanks to the celebrated work of Yamabe, Trudinger, Aubin and Schoen. When considering manifolds with singularities, the situation is quite different: while an existence result due to Akutagawa, Mazzeo and Carron is available, Viaclovsky had constructed in 2010 an example of 4-manifold, with one orbifold isolated singularity, for which a Yamabe metric does not exists. After briefly presenting the singularities we deal with, we will discuss a new non-existence result for a class of examples with non isolated singularities, not necessarily orbifold. This is based on a joint work with Kazuo Akutagawa.
Here is the video of the talk:
Thursday, July 16, Hyun Chul Jang (UConn).
Mass rigidity of asymptotically hyperbolic manifolds
In this talk, we present the rigidity of positive mass theorem for asymptotically hyperbolic (AH) manifolds. That is, if the total mass of a given AH manifold is zero, then the manifold is isometric to hyperbolic space. The proof of the rigidity used a variational approach with the scalar curvature constraint. It also involves an investigation of a type of Hessian equation, which leads to recent splitting results with G. J. Galloway. We will briefly discuss them as well. This talk is based on the joint works with L.-H. Huang and D. Martin, and with G. J. Galloway.
Here is the video of the talk:
Due to some technical problems, the recording starts only after the fourth slide. Here are the original slides of the talk.
Thursday, July 23, Dan Lee (Queens College and Graduate Center, CUNY).
Bartnik minimizing initial data sets
We will review what is known about Bartnik minimizing initial data sets in the time-symmetric case, and then discuss new results on the general case obtained in joint work with Lan-Hsuan Huang of the University of Connecticut. Bartnik conjectured that these minimizers must be vacuum and admit a global Killing vector. We make partial progress toward the conjecture by proving that Bartnik minimizers must arise from so-called ?null dust spacetimes? that admit a global Killing vector field. In high dimensions, we find examples that contradict Bartnik?s conjecture, as well as the ?strict? positive mass theorem, though these examples have "sub-optimal" asymptotic decay rates.
Here is the video of the talk:
Thursday, July 30, 2pm, Martin Kerin ( NUI Galway).
A pot-pourri of non-negatively curved 7-manifolds
Manifolds with non-negative sectional curvature are rare and difficult to find, with interesting topological phenomena traditionally being restricted by a dearth of methods of construction. In this talk, I will describe a large family of seven-dimensional manifolds with non-negative curvature, leading to examples of exotic diffeomorphism types, non-standard homotopy types, and fake versions of familiar non-simply connected friends. This is based on joint work with Sebastian Goette and Krishnan Shankar.
Here is the video of the talk:
Thursday, August 6, Shubham Dwivedi (University of Waterloo, Canada).
Deformation theory of nearly G_2 manifolds
We will discuss the deformation theory of nearly G_2 manifolds. After defining nearly G_2 manifolds, we will describe some identities for 2 and 3-forms on such manifolds. We will introduce a Dirac type operator which will be used to completely describe the cohomology of nearly G_2 manifolds. Along the way, we will give a different proof of a result of Alexandrov-Semmelman on the space of infinitesimal deformation of nearly G_2 structures. Finally, we will prove that the infinitesimal deformations of the homogeneous nearly G_2 structure on the Aloff--Wallach space are obstructed to second order. The talk is based on a joint work with Ragini Singhal (University of Waterloo)
Here is the video of the talk:
Spring 2020:
Thursday, February 6, 3pm, Double Header: Nan Li (CUNY City Tech) Gluing of multiple Alexandrov spaces
It was proved by Petrunin that the space glued by an intrinsic isometry between the boundaries of two Alexandrov spaces is an Alexandrov space with the same lower curvature bound. We will discuss some generalizations of this theorem, in which the partial gluing of multiple Alexandrov spaces are included.
Thursday, February 6, 4:15pm, room 6496, Double Header: Yehuda Pinchover (The Technion - Israel Institute of Technology) . This is a joint event with the Nonlinear Analysis and PDEs seminar.
How large can Hardy-weight be?
In the first part of the talk we will discuss the existence of optimal Hardy-type inequalities with 'as large as possible' Hardy-weight for a general second-order elliptic operator defined on noncompact Riemannian manifolds and discrete graphs, while the second part of the talk will be devoted to a sharp answer to the question: "How large can Hardy-weight be?"
Thursday, February 20, 3pm, Double Header: Marco Guaraco (University of Chicago) Minimal surfaces and mean curvature flow in hyperbolic
quasi-Fuchsian 3-manifolds
We show that every quasi-Fuchsian manifold has a canonical
minimal surface bounding all incompressible minimal surfaces and one
end. We use this canonical surface to study the possibility of
parametrizing the space of quasi-Fuchsian manifolds using formal
minimal surfaces, revisiting a problem studied by K. Uhlenbeck in the
80s. In addition, as an application of the mean curvature flow with
surgery, we establish the existence of incompressible mean-convex
foliations in general three-manifolds. Combining this result with
ideas from min-max theory, we show that generic quasi-Fuchsian
manifolds admit entire foliations by smooth mean-convex surfaces.
(This is joint work with V. Lima and F. Vargas Pallete)
Thursday, February 20, 4:15pm, room 6496, Double Header: Liming Sun (Johns Hopkins University) Some convexity theorems of translating solitons in the mean curvature flow.
I will be talking about the translating solitons (translators) in the mean curvature flow. Convexity theorems of translators play fundamental roles in the classification of them. Spruck and Xiao proved any two dimensional mean convex translator is actually convex. Spruck and I proved a similar convex theorem for higher dimensional translators, namely the 2-convex translating solitons are actually convex. Our theorem implies 2-convex translating solitons have to be the bowl soliton. Our second theorem regards the solutions of the Dirichlet problem for translators in a bounded convex domain . We proved the solutions will be convex under appropriate conditions. This theorem implies the existence of n-2 family of locally strictly convex translators in higher dimension. In the end, we will show that our method could be used to establish a convexity theorem for constant mean curvature graph equation.
Thursday, February 27: Jackson Goodman (U Penn) Moduli spaces of Ricci positive metrics in dimension five
Invariants related to the spectra of Dirac operators can be used to determine when two Riemannian metrics cannot be connected with a path of metrics maintaining a certain curvature condition. We use the $\eta$ invariant of spin$^c$ Dirac operators to distinguish connected components of moduli spaces of Riemannian metrics with positive Ricci curvature. We then find infinitely many non-diffeomorphic five dimensional manifolds for which these moduli spaces each have infinitely many components. The manifolds are total spaces of principal $S^1$ bundles over $\#^a\mathbb{C}P^2\#^b\overline{\mathbb{C}P^2}$ and the metrics are lifted from Ricci positive metrics on the base.
Thursday, March 12, 3pm, Double Header: Xiaowei Wang (Rutgers University). Compact Moduli space of Fano Kahler-Einstein varieties
In this talk, I will survey our construction of a compact moduli space of Fano KE varieties, this is based on the joint work with Chi Li and Chenyang Xu.
This talk is postponed to Fall 2020.
Thursday, March 12, 4:15pm, room 6496, Double Header: Zuoqin Wang (USTC/MIT). This is a joint event with the Nonlinear Analysis and PDEs seminar.
Semi-classical isotropic functions and applications
Rapidly oscillating functions associated with Lagrangian submanifolds play a fundamental role in semi-classical analysis. In this talk I will describe how to associate spaces of semi-classical oscillatory functions to isotropic submanifolds of phase space, and sketch their symbol calculus. As a special case we obtain the semi-classical version of the Hermite distributions of Boutet the Monvel and Guillemin. I will also discuss a couple applications of the theory. This is based on joint works with Victor Guillemin and Alejandro Uribe.
This talk is postponed to Fall 2020.
Thursday, March 19, 4:15pm, room 6496: Jie Qing (University of California at Santa Cruz). This is a joint event with the Nonlinear Analysis and PDEs seminar.
On asymptotically hyperbolic Einstein manifolds
In this talk I will introduce the research on asymptotically hyperbolic Einstein manifolds and its relation to conformal geometry in the spirit of AdS/CFT correspondence proposed in the study of quantum gravity in theoretical physics. We will report some recent works regarding rigidity and compactness on asymptotically hyperbolic manifolds.
This talk is postponed to Fall 2020.
Thursday, March 26: Peter Smilie(Caltech) This talk is postponed to Fall 2020.
Thursday, April 2: Ravi Shankar (University of Oklahoma) Recent developments in non-negative sectional curvature
In this talk we will present some recent progress in the
study of non-negatively curved manifolds. We begin with a survey of
some of the main theorems and obstructions in the least several
decades as well as statements of some open problems and conjectures.
This includes the seminal theorems of Gromov on the total Betti
number, the Soul theorem of Cheeger and Gromoll for open manifolds as
well as the Bott conjecture on the rational ellipticity of
non-negatively curved manifolds. We then present results on recent
progress in constructing non-negatively curved manifolds beginning
with the work of Grove and Ziller in 2000. We conclude with a recent
generalization of the Grove-Ziller construction as well as related
results.
This talk is postponed to Fall 2020.
Thursday, April 23: CUNY symposium This event is postponed to Fall 2020.
Thursday, April 30: Anusha Krishnan (Syracuse University) Diagonalizing the Ricci tensor
We will discuss some recent work on diagonalizing the Ricci tensor of invariant metrics on compact Lie groups, homogeneous spaces and cohomogeneity one manifolds, and connections to the Ricci flow.
Here is the video of the talk:
Thursday, May 7, 4pm: Tim Buttsworth (Cornell University) The prescribed Ricci curvature problem on manifolds with large symmetry groups.
The prescribed Ricci curvature problem continues to be of fundamental interest in Riemannian geometry. In this talk, I will describe some classical results on this topic, as well as some more recent results that have been achieved with homogeneous and cohomogeneity-one assumptions.
Here is the video of the talk:
Thursday, May 14, 4pm: Ronan Conlon (Florida International University). This is a joint event with the Nonlinear Analysis and PDEs seminar.
Classification Results for Expanding and Shrinking gradient Kahler-Ricci solitons
A complete Kahler metric g on a Kahler manifold M is a "gradient Kahler-Ricci soliton" if there exists a smooth real-valued function f:M-->R with \nabla f holomorphic such that Ric(g)-Hess(f)+\lambda g=0 for \lambda a real number. I will present some classification results for such manifolds. This is joint work with Alix Deruelle (Université Paris-Sud) and Song Sun (UC Berkeley).
Here is the video of the talk:
Thursday, May 14, Jonathan Zhu(Princeton) This event is postponed to Fall 2020.
Fall 2019:
Thursday, September 12, room 3212 (please note the room change): Markus Upmeier (University of Oxford)
Spin structures and orientation data for CY3
We introduce the idea of spin structures on differential geometric moduli spaces and apply it to `orientation dataÕ for Calabi-Yau 3-folds X in algebraic geometry, in the sense of Kontsevich-Soibelman. Then we explain the connection to 7d; in particular, that the derived moduli space of coherent sheaves on X connects to the G2-instanton equation. Our main result is a transgression theorem relating spin structures in 6d to canonical orientations in 7d, and we sketch the proof based on a spectral analysis of the appearing PDEs. The required input from 7d has been established in previous work on G2-instantons.
Thursday, September 19, 3pm, room 6496, Double Header: Charles Ouyang (Rice University) High Energy Harmonic Maps and Degeneration of Minimal Surfaces
A classical theorem of Schoen says that given any two closed hyperbolic surfaces of the same genus, there is a unique minimal lagrangian between them, which is isotopic to the identity. The graph with the induced metric is hence a minimal surface in the product of the two hyperbolic surfaces with the product metric, and lagrangian with respect to the difference of the two symplectic forms. We study the limits of the induced metrics as one or both of the hyperbolic
surfaces degenerate and show the limits are hybrid structures, where part of the surface has a flat metric coming from a holomorphic quadratic differential and part of the surface is a measured lamination.
Thursday, September 19, 4:15pm, room 6496, Double Header: Mohameden Ahmedou (University of Giessen) . This is a joint event with the Nonlinear Analysis and PDEs seminar.
Morse theory and the resonant $Q$-curvature problem
The Q-curvature is a scalar quantity which plays a central role in conformal geometry, in particular in the search of high order conformal invariants. In this talk we address the problem of finding conformal metrics of prescribed Q-curvature on four riemannian manifolds in the so called resonant case, that is when the total integral of the Q-curvature is a multiple of the one of the four-dimensional round sphere. This geometric problem has a variational structure with a lack of compactness. Using some topological tools of the theory of critical points at infinity of Abbas Bahri, combined with a refined blow-up analysis, we extend the full Morse theory, including Morse inequalities, to this non-compact geometric variational problem and derive existence and multiplicity results. This is a joint work with C.B. Ndiaye (Howard University)
Thursday, October 3: Florian Johne (Columbia University) Surgery for an extended Ricci flow system
List flow is a geometric flow for a pair (g,u), where g is a Riemannian
metric and u a smooth function. This extended Ricci flow system has
applications to static vacuum solutions of the Einstein equations and to
Ricci flow on warped products.
The coupling in this flow induces additional difficulties compared to
Ricci flow, which we overcome by proving an improved bound on the
Hessian. This allows us to prove a convergence result, a singularity
classification result and a surgery result in three dimensions.
Thursday, October 10: Peter McGrath (University of Pennsylvania) Generalizing Linearized Doubling and Applications
I will discuss recent work (with N. Kapouleas) on generalizing the linearized doubling (LD) approach for constructions of doubled minimal surfaces. By generalizing the methodology of earlier work, we construct high-genus families of: complete minimal surfaces in the Euclidean space by doubling the catenoid, free boundary minimal surfaces in the unit ball by doubling the critical catenoid, and self-shrinkers of the mean curvature flow by doubling the spherical self-shrinker.
Thursday, October 17: Frederick Tsz-Ho Fong (Hong Kong University of Science and Technology) Curvature Estimates of Long-Time Solutions to the Kahler-Ricci Flow
The speaker will discuss the local curvature estimates of the Kahler-Ricci flow on compact Kahler manifolds with semi-ample canonical line bundles. On such a manifold, the Kahler-Ricci flow has long-time solutions and its convergence and singular behaviors have been widely studied by various authors. In this talk, the speaker will discuss his works on this topic, in particular showing that the set of fibers (either singular or regular) on which the Riemann curvature blow up along the flow is an invariant set independent of the choice of initial Kahler metric. The talk is based on two joint works, one with Zhou Zhang, another with Yashan Zhang.
Thursday, October 24: Christine Breiner (Fordham University) Harmonic maps to metric spaces
We consider harmonic maps from a compact Riemann surface to a metric space with upper curvature bounds in the sense of Alexandrov. We will discuss some existence results. We will also prove an analogue of the Measurable Riemann Mapping Theorem for the singular setting.
Thursday, November 7: Chen-Yun Lin (Lehman College) Data and Curse of Dimensionality
High dimensional data is increasingly available in many fields. However, the analysis of such data suffers the so-called curse of dimensionality. One powerful approach to nonlinear dimensionality reduction is the diffusion-type maps. Its continuous counterpart is the embedding of a manifold using the eigenfunctions of the Laplace-Beltrami operator. Accordingly, one may ask, how many eigenfunctions are required in order to embed a given manifold. In this talk, I will give some background regarding the dimensionality reduction problem, spectral geometry, and show theoretical results for a generalized diffusion map. Specifically, I will show a closed Riemannian manifold can be embedded into a finite dimensional Euclidean space by maps constructed based on the connection Laplacian at a certain time. This time and the embedding dimension can be bounded in terms of the dimension and geometric bounds of the manifold. Furthermore, the map based on heat kernels can be made arbitrarily close to an isometry. In addition, I will give a ÔÕreal worldÓ example pertaining to paleontology, that demonstrates how heat kernels and diffusion maps can be used to quantify the similarity of shapes. The empirical results suggest that this framework is better than the metric commonly used in biological morphometrics.
Thursday, November 14, 3pm, Double Header: Ruobing Zhang (Stony Brook University) Degenerations of collapsing Calabi-Yau metrics
In this talk, we will focus on some recent progress in understanding the geometric structures of collapsing Calabi-Yau manifolds. Specifically, we will describe the metric collapsing and singularity behaviors as the complex structures of the underlying spaces are degenerating. This is recent joint work with Song Sun.
Thursday, November 14, 4:15pm, same room, Double Header: Jinggang Xiong (Beijing Normal University/Rutgers University) . This is a joint event with the Nonlinear Analysis and PDEs seminar.
Optimal boundary regularity for fast diffusion equations in bounded domains
We prove optimal boundary regularity for bounded positive weak solutions of fast diffusion equations in smooth bounded domains. This solves a problem raised by Berryman and Holland in 1980 for these equations in the subcritical and critical regimes. Our proof of the a priori estimates uses a geometric type structure of the fast diffusion equations, where an important ingredient is an evolution equation for a curvature-like quantity. This is a joint work with Tianling Jin.
Thursday, November 21: Blair Davey(CCNY) How to obtain parabolic theorems from their elliptic counterparts
Experts have long realized the parallels between elliptic and parabolic theory of partial differential equations. It is well-known that elliptic theory may be considered a static, or steady-state, version of parabolic theory. And in particular, if a parabolic estimate holds, then by eliminating the time parameter, one immediately arrives at the underlying elliptic statement. Producing a parabolic statement from an elliptic statement is not as straightforward. In this talk, we demonstrate a method for producing parabolic theorems from their elliptic analogues. Specifically, we show that an $L^2$ Carleman estimate for the heat operator may be obtained by taking a high-dimensional limit of $L^2$ Carleman estimates for the Laplacian. Other applications of this technique will be discussed.
Thursday, November 28: No seminar Happy Thanksgiving
Thursday, December 5: Freid Tong (Columbia University) Degenerations of asymptotically conical Calabi-Yau metrics
Asymptotically conical Calabi-Yau metrics are complete Ricci-flat Kahler metrics which is asymptotic to a metric cone at infinity, many results have been obtained regarding the existence of these metrics, starting with the fundamental work of Tian and Yau. We will discuss some results concerning how the sequence of asymptotically conical Ricci-flat metrics can degenerates as we change the Kahler class. As a consequence, this allows us to construct singular Ricci-flat Kahler metrics with conical behaviour at infinity on certain quasi-projective Calabi-Yau varieties. This is join work with Bin Guo and Tristan Collins.
Thursday, December 12, 3pm, Double Header: Si Li (IAS) Singularities: from L^2 Hodge theory to Seiberg-Witten geometry
Let X be a non-compact Calabi-Yau manifold and f be a holomorphic function on X with compact critical locus, satisfying a general asymptotic condition. We establish a version of twisted L^2 Hodge theory for the pair (X,f) and prove the corresponding Hodge-to-de Rham degeneration property. It can be viewed as a generalization of Kyoji Saito's higher residue theory and primitive forms for isolated singularities. In the second part of the talk, I will explain a connection between primitive period maps and 4d N=2 Seiberg-Witten geometry.
Thursday, December 12, 4:15pm, same room, Double Header: Yizhong Zheng (Graduate Center, CUNY) . This is a joint event with the Nonlinear Analysis and PDEs seminar.
Uniform Lipschitz continuity of isoperimetric profile on compact surfaces under normalized Ricci flow.
We study the isoperimetric profile function h(\xi,g(t)) on a compact Riemannian manifold M under varying of metrics g(t), where \xi is the volume ratio. We show that h(\xi,g(t)) is jointly continuous when metrics g(t) vary continuously. We also show that h^2(\xi,g(t)) is uniform Lipschitz when M is a compact surface and g(t) is evolving under the normalized Ricci flow.
Thursday, December 19, 6:30pm, room C415 AChristina Sormani (Lehman College and Graduate Center, CUNY) .
Rigidity and Almost Rigidity Theorems in Geometry
In geometry a rigidity theorem is a theorem that states that a metric space satisfying certain hypotheses must be isometric to a specific metric space. An almost rigidity theorem states that if the metric space almost satisfies this hypothesis then the metric space is close to being the specific metric space where that closeness might be measured in a variety of ways. I will survey a variety of rigidity and almost rigidity theorems and conjectures.