Seminars & Events for Differential Geometry & Geometric Analysis Seminar

September 12, 2014
3:00pm - 4:00pm
Coarse Ricci curvature and the manifold learning problem

We use the framework exploited by Bakry and Emery for the study of logarithmic Sobolev inequalities to consider the statistical problem of estimating the Ricci curvature (and more generally the Bakry-Emery tensor of a manifold with density) of an embedded submanifold of Euclidean space from a point cloud drawn from the submanifold. Our method leads to a definition of Coarse Ricci curvature alternative to the one proposed by Y. Ollivier. This is joint work with Micah Warren. 

Speaker: Antonio Ache , Princeton University
Location:
Fine Hall 314
September 19, 2014
3:00pm - 4:00pm
On the topology and index of minimal surfaces

We show that for an immersed two-sided minimal surface in R^3, there is a lower bound on the index depending on the genus and number of  ends. Using this, we show the nonexistence of an embedded minimal surface
in R^3 of index 2, as conjectured by Choe. Moreover, we show that the index of an immersed two-sided minimal surface with embedded ends is bounded from above and below by a linear function of the total curvature of the surface.  (This is joint work with Otis Chodosh) 

Speaker: Davi Maximo, Stanford University
Location:
Fine Hall 314
September 26, 2014
3:00pm - 4:00pm
Umbilicity and characterization of Pansu spheres in the Heisenberg group

For n≥2 we define a notion of umbilicity for hypersurfaces in the Heisenberg group H_{n}. We classify umbilic hypersurfaces in some cases, and prove that Pansu spheres are the only umbilic spheres with positive constant p(or horizontal)-mean curvature in H_{n} up to Heisenberg translations. This is joint work with Hung-Lin Chiu, Jenn-Fang Hwang, and Paul Yang. 

Speaker: Jih-Hsin Cheng , Academia Sinica, Taiwan
Location:
Fine Hall 314
October 3, 2014
2:00pm - 3:00pm
Regularity of manifolds with bounded Ricci curvature and the codimension $4$ conjecture

This talk will concern joint work with Aaron Naber on the regularity of noncollapsed Riemannian manifolds $M^n$ with bounded Ricci curvature $|{\rm Ric}_{M^n}|\leq n-1$, as well as their Gromov-Hausdorff limit spaces, $(M^n_j,d_j)\stackrel{d_{GH}}{\longrightarrow}(X,d)$, where $d_j$ denotes the Riemannian distance. We will explain a proof of the conjecture that $X$ is smooth away from a closed subset of  codimension $4$. By combining this with the ideas of quantitative stratification, we obtain a priori $L^q$ estimates on the full curvature tensor, for all $q<2$.  We also prove a conjecture of Anderson stating that for all $v>0$, $ <\infty$, the collection of $4$-manifolds $(M^4,g)$ with $|{\rm Ric}_{M^n}|\leq 3$, ${\rm Vol}(M^3)\geq v$, ${\rm diam}(M^4)\leq d$, contains a most a finite number of diffeomorphism types.

Speaker: Jeff Cheeger, NYU
Location:
Fine Hall 314
October 10, 2014
3:00pm - 4:00pm
Nonexistence results for solitons in the mean curvature flow

In this talk I will give a more refined quantitative understanding of some of the important known solitons in the n-dimensional mean curvature flow in R^{n+1}. The global estimates in question follow by iteration of monotonicity formulae - an idea and technique which, while quite elementary in nature, appears to be particularly useful in several of such situations. The applications of the explicit estimates are many. I will focus mostly on the new nonexistence theorems that follow. Time permitting, I will also mention other, related consequences for the analysis of the soliton PDEs for the flow. 

Speaker: Niels Moeller, Princeton University
Location:
Fine Hall 314
October 17, 2014
3:00pm - 4:00pm
Regularity of semi-calibrateed integral $2$-cycles

Semi-calibrated currents naturally appear when dealing with several geometric questions, some aspects of which require a deep understanding of regularity properties of semi-calibrated currents. We will focus on the case of dimension $2$, where it turns out that semi-calibrated cycles are actually pseudo holomorphic. By using an analysis implementation of the algebro-geometric blowing up of a point we study the regularity of semi-calibrated $2$-cycles from the point of view of uniqueness of tangent cones and of local smoothness.

Speaker: Costante Bellettini, Cambridge University
Location:
Fine Hall 314
November 7, 2014
3:00pm - 4:00pm
Shear-free asymptotically hyperboloidal initial data in general relativity

One of the most useful tools for studying isolated gravitational systems is conformal compactification, which transforms ``infinity'' into a finite boundary by multiplying the metric by a suitable smooth function. For studying outgoing radiation effects, an effective approach is to set up an initial-value problem on an asymptotically hyperboloidal spacelike hypersurface that intersects future conformal null infinity transversely.  The initial data for the Einstein equations consists of a Riemannian metric and a second fundamental form on the initial hypersurface, together with some matter and energy fields. These data are not freely specifiable, but instead must satisfy the Einstein constraint equations, which boil down to a nonlinear elliptic system on the chosen initial hypersurface.

Speaker: John Lee, University of Washington, Seattle
Location:
Fine Hall 314
November 7, 2014
4:15pm - 5:15pm
Asymptotic geometry and large isoperimetric regions

This is an additional talk on this date.  In spite of the long history of the isoperimetric problem, the list of manifolds where isoperimetric regions are well understood is remarkably short. In this talk, I will discuss the following question: "If a Riemannian manifold (M,g) is asymptotic at infinity to a space in which the isoperimetric regions are understood, can we determine the large isoperimetric regions in (M,g)?" I'll discuss some positive and negative answers to this question for various asymptotic geometries. Some of these results are joint work with Michael Eichmair and Alex Volkmann. 

Speaker: Otis Chodosh, Stanford University
Location:
Fine Hall 314
November 21, 2014
3:00pm - 4:00pm
Quantitative uniqueness, doubling lemma and nodal sets

Based on a variant of frequency function, we improve the vanishing order of solutions for Schr\"{o}dinger equations which describes quantitative behavior of strong uniqueness continuation property. For the first time, we investigate the quantitative uniqueness of higher order elliptic equations and show the vanishing order of solutions.  Using the Carleman estimates, we obtain the doubling estimates and optimal vanishing order of Steklov eigenfunctions, which  is the eigenfunctions of the Dirichlet-to-Neumann map. A lower bound of nodal sets of Steklov eigenfunctions is also derived.

Speaker: Jiuyi Zhu, Johns Hopkins
Location:
Fine Hall 314
December 5, 2014
3:00pm - 4:00pm
Pluriclosed flow and generalized Kahler geometry

In joint work with G. Tian I introduced a natural evolution equation generalizing the Kahler Ricci flow to complex, non-Kahler manifolds.  Moreover we showed that this equation preserves "generalized Kahler geometry."  In this talk I will discuss further results on this flow in the generalized Kahler setting, including a sharp long time existence result for complex surfaces.  These results lead to strong rigidity and classification results for generalized Kahler structures. 

Speaker: Jeff Streets, University of California Irvine
Location:
Fine Hall 314
December 12, 2014
3:00pm - 4:00pm
Sign of Green's function of Paneitz operator and the Q curvature equations

I will review some recent works with Paul on the fourth order Paneitz operators and Q curvature equations. Among other things, we will discuss the positivity of Green's function of Paneitz operator and its applications to finding constant Q curvature metrics by variational methods.

Speaker: Fengbo Hang, NYU
Location:
Fine Hall 314
January 16, 2015
3:00pm - 4:00pm
Global existence for fully nonlinear stochastic hyperbolic equations : the Nash-Moser approach

In 1979, Klainerman proved the first global existence result for fully nonlinear hyperbolic equations using Nash-Moser's method. The result was a breakthrough in the field.
In the present talk, I shall report on the corresponding study for fully nonlinear stochastic hyperbolic equations driven by a Wiener process. The main result of the research is stochastic counterpart of Klainerman's global existence theorem. The main tool for our investigation is Nash-Moser's iteration scheme combined with stochastic calculus and the theory of enlargement of filtrations. 

Speaker: Mamadou Sango, University of Pretoria
Location:
Fine Hall 314
January 20, 2015
3:00pm - 4:00pm
Minimal surfaces in H2xR

PLEASE NOTE SPECIAL DAY AND LOCATION:  Talk #1.  In the classical theory of minimal surfaces of the Euclidean space, the better known ones are those with finite total curvature. Laurent Hauswirth and Harold Rosenberg started in 2006 the corresponding theory of complete minimal surfaces with finite total curvature in H2xR, the product space of the hyperbolic plane and the real line. We will give an overview of the subject, including the construction of some examples and
classification results. 

Speaker: Magdalena Rodriguez, Universidad de Granada
Location:
Fine Hall 224
January 20, 2015
4:15pm - 5:15pm
A positive mass theorem for asymptotically flat manifolds with a noncompact boundary and an application to a Yamabe-type flow

PLEASE NOTE SPECIAL DAY AND LOCATION:  Talk #2.  First I will discuss a positive mass theorem for noncompact manifolds with boundary and ends asymptotic to the Euclidean half-space. This theorem is a joint work with Ezequiel Barbosa and Levi de Lima. Then I will show how this result can be applied to prove the convergence of a certain Yamabe-type flow on compact manifolds with boundary.

Speaker: Sergio Almaraz, Universidade Federal Fluminense
Location:
Fine Hall 224
January 22, 2015
4:00pm - 5:00pm
Mean curvature flow without singularities

Please note special day, time and location.  We study graphical mean curvature flow of complete solutions defined on subsets of Euclidean space. We obtain smooth long time existence. The projections of the evolving graphs also solve mean curvature flow. Hence this approach allows to smoothly flow through singularities by studying graphical mean curvature flow in one additional dimension. 

Speaker: Mariel Saez, Universidad Catolica de Chile
Location:
Fine Hall 214
February 6, 2015
3:00pm - 4:00pm
Kähler-Einstein metrics: from cones to cusps

In this talk, I will explain the following result and outline its proof: Let $X$ be a compact Kähler manifold and $D$
a smooth divisor such that $K_X+D$ is ample. Then the negatively curved Kähler-Einstein metric with cone angle $\beta$ along $D$ converges to the cuspidal Kähler-Einstein metric of Tian-Yau when $\beta$ tends to zero.

Speaker: Henri Guenacia, SUNY Stony Brook
Location:
Fine Hall 314
February 6, 2015
4:15pm - 5:30pm
Duality in Boltzmann Equation and its Applications.

In this paper we will survey a quantitative and qualitative
development on the Boltzmann equation. This development reveals the dual
natures of the Boltzmann
equation: The particlelike nature and the fluidlike nature. This dual
nature property gives rise to the precise construction of the Green's
function for Boltzmann equation around a  global Maxwellian state. With the
precise structure of the Green's function, one can implement the Green's
function to study various problems such as invariant manifolds for the
steady Boltzmann flows, time asymptotic nonlinear stability of Boltzmann
shock  layers and Boltzmann boundary layers, Riemann Problem, and
bifurcation problem of boundary layer problem, etc.

Speaker: Shih-Hsien Yu, National University of Singapore
Location:
Fine Hall 314
February 13, 2015
3:00pm - 4:00pm
Gromov-Haudorff convergence of Kahler manifolds and the finite generation conjecture

We study the uniformization conjecture of Yau by using the Gromov-Haudorff convergence. As a consequence, we confirm Yau's finite generation conjecture. More precisely, on a complete noncompact Kahler manifold ith nonnegative bisectional curvature, the ring of polynomial growth holomorphic functions is finitely generated.  During the course, we prove if M is a complete noncompact Kahler manifold with nonnegative bisectional curvature and maximal volume growth, then it is biholomorphic to an affine algebraic variety. We also confirm a conjecture of Ni on the equivalence of several conditions on complete Kahler manifolds with nonnegative bisectional curvature. 

Speaker: Gang Liu, U.C. Berkeley
Location:
Fine Hall 314
February 20, 2015
3:00pm - 4:00pm
Minimal Surfaces with Arbitrary Topology in H^2xR

In this talk, we show that any open orientable surface can be embedded in H^2xR as a complete area minimizing surface. Furthermore, we will discuss the asymptotic Plateau problem in H^2xR, and give a fairly complete solution.

Speaker: Baris Coskunuzer, MIT
Location:
Fine Hall 314
March 13, 2015
3:00pm - 4:00pm
Levi-flat hypersurfaces in complex manifolds

Levi-flat hypersurfaces arise naturally from complex foliation theory and complex dynamics. In this talk we will discuss the Cauchy-Riemann equations on domains in complex manifolds with Levi-flat boundary. We give an example of a pseudoconvex domain in a complex manifold whose $L^2$-Dolbeault cohomology is non-Hausdorff, yet the domain is Stein with real-analytic Levi-flat boundary (Joint work with Debraj Chakrabarti).

Speaker: Mei-Chi Shaw, University of Nortre Dame
Location:
Fine Hall 314

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