In Spring 2011, we will meet on Tuesdays at 3:00pm in Room 3207, unless otherwise noted. The organizers of this seminar are Dan Lee, Zheng Huang, and Marcello Lucia. Please email Dan at dan.lee(NoSpamPlease)qc.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).

Spring 2011:

Tuesday, 2/8: Sajjad Lakzian (CUNY Graduate Center)
Distorted Radially-Conformal Change of Riemannian Manifolds and Gnomonic Compactification
Abstract: The Distorted Radially-Conformal change of a metric on a Riemannian manifold will be defined and the changes in curvatures will be investigated. We will also study a specific type of distorted radially-conformal change which is called the Gnomonic Projection that gives a compactification of asymptotically flat manifolds modeled on $\R^n$ and at the end we will mention what happens to mass under Gnomonic compactification.
Tuesday, 2/15: Michael Eichmair (MIT)
Isoperimetric structure of initial data sets
Abstract: I will present recent joint work with Jan Metzger. A basic question in mathematical relativity is how geometric properties of an asymptotically flat manifold (or initial data set) encode information about the physical properties of the space time that it is embedded in. For example, the square root of the area of the outermost minimal surface of an initial data with non-negative scalar curvature provides a lower bound for the "mass" of its associated space time, as was conjectured by Penrose and proven by Bray and Huisken-Ilmanen. Other special surfaces that have been studied in this context include stable constant mean curvature surfaces or isoperimetric surfaces. I will explain why positive mass works to the effect that large stable constant mean curvature surfaces are always isoperimetric. This answers a question of Bray's and complements the results by Huisken-Yau and Qing-Tian on the "global uniqueness problem for stable CMC surfaces" in initial data sets with positive scalar curvature.
Tuesday, 2/22: no seminar
2/21 is Presidents Day
Tuesday, 3/1: no seminar
Tuesday, 3/8: Leili Shahriyari (Johns Hopkins University)
Existence of a capillary surface with prescribed contact angle in $M\times\R$
Abstract: We study the prescribed mean curvature equation with a prescribed boundary contact angle condition in $M\times\R$ where $M^n$ is a Riemannian submanifold in $\R^{n+1}$. The main purpose is to establish a priori gradient estimates for solutions, from which the long time existence of the solution are derived.
Tuesday, 3/15: no seminar
FRG workshop at Princeton
Tuesday, 3/22: Luc Nguyen (Princeton University)
Regularity near the axis for axially symmetric stationary electro-vaccum spacetimes
Abstract: According to the Ernst-Geroch reduction, in an axially symmetric stationary electrovac spacetime, the Einstein-Maxwell equations reduce to a harmonic map problem with singular boundary data. I will discuss the ``regularity'' of the reduced harmonic maps near the boundary and its implication on the regularity of the corresponding spacetimes.
Tuesday, 3/29 at 4pm in Math Thesis Room: Valentino Tosatti (Columbia University)
Symplectic forms and almost-complex structures in 4 dimensions.
Abstract: Consider a closed 4-manifold with a given almost-complex structure. A natural question to ask is: under what conditions can we find a symplectic form compatible with the almost-complex structure? A necessary condition is that there exists a symplectic form "taming" it (in the sense of Gromov), and a conjecture of Donaldson says that this should be also sufficient. I will discuss a possible approach to this conjecture that involves a generalization of the classical Calabi-Yau theory on Kahler manifolds, and report on some recent work on this (joint with B. Weinkove and partly with S.-T. Yau).
Tuesday, 4/5: Jun Masamune (Penn State Altoona)
Conservation property of jump-processes and its application to graphs
Abstract: In 1959, M.P. Gaffney discovered a volume-growth condition for the Brownian motion on a Riemannian manifold to be conservative; namely, non-explosive. Since then, this condition has been improved in various settings for a strongly-local Dirichlet form. Recently, R.K. Wojciechowski constructed a stochastically incomplete graph that satisfies this condition with respect to the graph distance. In this talk, I will introduce a new volume-growth condition for a jump-process to be conservative on a metric measure space, then I will apply that result to a graph. Wojciechowski's example confirms that our condition for a graph is sharp. This result was obtained in a collaboration with Alexander Grigor`yan and Xeuping Huang (Bielefeld, Germany). If time permits, I will discuss the same problem for a jump-diffusion on a disconnected media.
Tuesday, 4/12: Sagun Chanillo (Rutgers University)
Embedding CR manifolds
Abstract: This is joint work with Hung-Lin Chiu and Paul Yang. We recall the global embedding problem for three dimensional CR manifolds and give a CR invariant condition that ensures embedding in C^n.
Tuesday, 4/19: no seminar
Spring Break
Tuesday, 4/26: no seminar
Spring Break
Tuesday, 5/3: Chi Li (Princeton University)
On the continuity method in Kahler-Einstein problem for toric Fano manifolds
Abstract: In the problem of existence of Kahler-Einstein metric on any Fano manifold, the continuity method was used. The maximal possible value of the parameter in the continuity method gives the greatest lower bound of Ricci curvature among Kahler metrics in the first Chern class. This can be seen as an invariant of the Fano manifold. For any toric Fano manifold, we can determine this invariant. We can also get information for the limit metric with maximal parameter. It satisfies a singular complex Monge-Ampere equation and should have conic type singularities. These information can be read out from the geometry of the momentum polytope.
Tuesday, 5/10: Adam Jacob (Columbia University)
Approximate Hermitian-Einstein structures and the Yang-Mills flow
Abstract: After a brief introduction to Kahler manifolds, I will talk about the Yang-Mills flow in the case where it does not converge. I will prove that a semi-stable vector bundle over a compact Kahler manifold admits an approximate Hermitian-Einstein structure, and then explain how this structure can be used to identify the limit of the Yang-Mills flow in the semi-stable case.
Tuesday, 5/31 (Room 4214:03 at 4pm): Pedro Solorzano (Stony Brook University)
Limits of Tangent Bundles under GH convergence
Tuesday, 6/7 (Room 4214:03 at 4pm): Jozef Dodziuk (CUNY Graduate Center)
Combinatorial Laplacian I
Abstract: In the first talk I will attempt to motivate the definition of the combinatorial Laplace operator on a graph and show that it is a natural concept. I will also discuss analogs of familiar results in analysis (Harnack's inequality, the maximum principle, Cheeger's inequality) that have powerful application to the study of graphs.
Tuesday, 6/14 (Room 4214:03 at 4pm): Jozef Dodziuk (CUNY Graduate Center)
Combinatorial Laplacian II
Abstract: In the second talk, I will present a new result obtained jointly with Tullio Ceccherini-Silberstein and Michel Coornaert to the effect that the combinatorial Laplacian acting on the space of all real-valued functions on vertices is surjective for every infinite connected graph.
Tuesday, 6/21 (Room 4214:03 at 4pm): Pedro Solorzano (Stony Brook University)
Riemannian maps and the Sasaki metric
Abstract: A Riemannian map is a map between Riemannian manifolds that is locally the composition of a Riemannian submersion and an isometric immersion. The Sasaki metric is a natural Riemannian metric on the total space tangent bundle of a Riemannian manifold that makes the projection a Riemannian submersion. In the talk I will address the following question: Is the total derivative of a Riemannian map again a Riemannian map? The answer is negative in general. In particular, this yields the problem of determining when is the total derivative of an isometric immersion or a Riemannian submersion again an isometric immersion or a Riemannian submersion. I will give a very natural condition to obtain a positive answer for certain types of Riemannian maps.
Tuesday, 7/12 (Room 4214:03 at 4pm): Joseph Malkevitch (York College CUNY)
Representing Convex Three-Dimensional Polyhedra in the Plane
Abstract: This talk will survey the issue of representing convex 3-dimensional polyhedra in the plane. When is a graph (in the dots and line segments sense) drawn in the plane isomorphic to the vertex-edge graph of some 3-dimensional convex polyhedon? Furthermore, there is the issue of when it is possible to cut the surface of a convex 3-dimensional polyhedron and flatten it out into a plane polygon. The question, due to Geoffrey Shephard, as to whether this can be done by cutting only edges of the polyhedron is still unsolved. What is known are conditions under which a polygon can be folded up to form a convex polyhedron. This area of geometry abounds in unsolved problems.
Tuesday, 7/19 (Room 4214:03 at 4pm): Penny Smith (Lehigh University)
Regularity of Mass Minimizing Rectifiable Sections of Fiber Bundles
Abstract: Let B be a Fiber Bundle with compact Fiber F over a compact Riemannian n-manifold M. There is a natural Riemannian metric on the total space B consistent with the metric on M. With respect to that metric, the volume of a rectifiable section \sigma \colon M \rightarrow B is the mass of the image \sigma (M) as a rectifiable n-current in B. I will my discuss some of my published joint work with David Johnson, on the partial regularity problem of such sections. In particular: Theorem: For any homology class of sections of B, there is a mass minimizing rectifiable current T representing that homology class which is the graph of a C^{1} section on an open dense set of M.