Geometry Seminar

Organizers: Sema Salur and Robert Hladky

The seminar will generally be held on Tuesdays from 3:30 P. M. to 4:30 P. M. in Hylan 1106B. Any exceptions will be noted below.

Spring 2008 Seminars:

April 29: Albert Todd, U of R; "Deformations of Asymptotically Cylindrical Special Lagrangian Submanifolds with Fixed Boundary"

Abstract: We prove that the moduli space of nearby special Lagrangian deformations of an asymptotically cylindrical special Lagrangian submanifold $L$ in an asymptotically cylindrical Calabi-Yau 3-fold $X$ is a smooth manifold of dimension equal to the dimension of the image of $H^1_{cs}(L,R)$ in $H^1(L,R)$ under the canonical map.

April 24 (Colloquium, Thursday 4:50-5:50; Goergen 108): Jeffrey Brock, Brown University; "Beyond the geometrization conjecture"

April 24 (Thursday):Andreas Arvanitoyeorgos, University of Patras; "Riemannian flag manifolds with homogeneous geodesics"
Abstract: A geodesic in a Riemannian homogeneous manifold (M = G/K, g ) is called a homogeneous geodesic if it is an orbit of an one-parameter subgroup of the Lie group G. In this joint work with D.V. Alekseevsky we investigate G-invariant metrics with homogeneous geodesics (i.e. such that all geodesics are homogeneous) when M = G/K is a flag manifold, that is an adjoint orbit of a compact semisimple Lie group G. We use an important invariant of a flag manifold M = G/K, its T -root system, to give a simple necessary condition that M admits a non-standard G-invariant metric with homogeneous geodesics. Hence, the problem reduces substantially to the study of a short list of prospective flag manifolds. A common feature of these spaces is that their isotropy representation has two irreducible components. We prove that among all flag manifolds M = G/K of a simple Lie group G, only the manifold Com(R2) = SO(2)/U() of complex structures in R2 , and the complex projective space CP2 = Sp/U(1) . Sp(1) admit a non-naturally reductive invariant metric with homogeneous geodesics. In all other cases the only G-invariant metric with homogeneous geodesics is the metric which is homothetic to the standard metric (i.e. the metric associated to the negative of the Killing form of the Lie algebra of G). We also find explicitly homogeneous geodesics in some examples of flag manifolds.

April 22: David Blair, Michigan State University; "A Complex Geodesic Flow"
Abstract: When one poses a question like, what is the complex analogue of the geodesic flow, many objections may arise in one's mind, e.g. What is a complex geodesic? The home of the classical geodesic flow is the unit tangent bundle and an important special case is when the base manifold is a compact Riemannian manifold of negative curvature; in this case the geodesic flow is an Anosov flow. In contact metric geometry the tangent sphere bundle is an important example and the characteristic vector field (Reeb vector field) of the contact structure is twice the geodesic flow. So what is the complex analogue of the unit tangent bundle? Since there is no natural ordering of the complex numbers, how can one have a flow? We will address these objections and discuss a complex geodesic flow for complex space forms. We will conclude with a discussion of the relation of this problem to complex contact geometry.

April 16 (Colloquium, Wednesday, 5 pm - 6 pm, 101 Goergen Hall): John Morgan, Columbia University; "Geometry and Topology in Dimension 3"
Abstract: Topology of 3-dimensional manifolds has long been a focus of topology; indeed one could say that topology began as an independent subject with Poincaré's work in the early 20th century which culminated in his formulating his conjecture about characterizing the 3-sphere among all 3-dimensional manifolds. Over the intervening years this question about the simplest 3-dimensional manifold was generalized to a conjecture about the nature of all compact 3-dimensional manifolds, and also to manifolds in higher dimensions. Much progress was made, but through it all the 3-dimensional case, and Poincaré's original question remained unsolved. Recently, Grigory Perelman has successfully completed a program initiated by Richard Hamilton to resolve these long-standing questions. Hamilton's idea was to use a heat-type evolution equation, called Ricci flow, for a Riemannian metric on a 3-manifold. Just as ordinary heat flow smooths out temperature distributions, Hamilton's intuition was that Ricci flow should smooth out the curvature of the manifold, leading to a metric that was homogeneous, from which the results would follow directly. There are enormous difficulties in implementing this program, including the fact that the flow develops finite-time singularities. Hamilton showed how to deal with some of the difficulties, and Perelman, building on what Hamilton achieved, has recently succeeded in completely carrying out the program. This talk will discuss the nature of the conjectures, introduce curvature and the Ricci flow evolution equation, and give an overview of how this equation is used to resolve the conjectures.
April 11 (Friday): Rob Hladky, U of R; "Fredholm and regularity theorems for the bar{partial}_b-Neumann problem"
Abstract: The inhomogeneous equation $bar{partial}_b u =f$ for the tangential Cauchy-Riemann operator on domains with boundary is an important largely unsolved problem in the theory of CR geometry. One approach is to consider the Neumann problem for the associated Laplacian. Ideally one would like to establish a priori estimates, Fredholm theorems and sharp regularity results for this locally subelliptic operator. Unfortunately, many of the classical techniques fail in this instance. We shall outline the methods required to establish priori estimates, look at cases where Fredholm theorems can be established and provide examples for when they fail. Our sufficient Fredholm criteria is surprisingly geometric in nature and is based on some deep results in topology and geometry.

April 1: McKenzie Wang, McMaster University; "A cohomogeneity one framework for Ricci solitons"
Abstract: Many explict Einstein metrics are constructed by reducing the Einstein equation to a system of ordinary differential equations under symmetry assumptions. I will discuss how a similar framework works for Ricci solitons and use it to give a uniform construction of many known Ricci solitons and their natural generalizations.

March 20 (Faculty Colloquium, Thursday 1:30pm Hylan 1106A): Rob Hladky, U of R; "SubRiemannian geometry"
Abstract: SubRiemannian geometry arises in situations where movement is only permitted in certain directions. We'll discuss the basic definitions and properties of such spaces, look at some motivating examples and consider the connection with the analysis of subelliptic PDE. We'll conclude by looking at progress that has been made on the minimal surface and isoperimetric problems in the subRiemannian setting and briefly discuss an application to neurobiology. 21 (Friday)

February 19: S.G. Rajeev, U of R; "Physics, Topology, Geometry and Analysis of the Skyrme Soliton"
Abstract: A Skyrme soliton is a continuous map U:R3-> S3 which tends to a constant at infinity which minimizes energy. Such maps are topologically equivalent to S3->S3, so have a degree (winding number) equal to an integer. If the energy of the map is defined in the usual way ( the integral of the square of the gradient) the minimum would be a harmonic map. But there are no harmonic maps of degree one. Skyrme proposed a physically justified modification of energy that is should have minima with non-zero winding number. This is of great importance in physics, because the degree one minimizer ( Skyrme soliton ) describes the proton, one of the fundamental particles of nature. The fact that the fourth homotopy group of S3 is Z_2 explains why the proton is a fermion (I will explain what that means). The mathematically rigorous proof that there is a regular solution to the Skyrme PDE was only found recently. I will propose an interesting problem on the hyperbolic analogue of this PDE, which is also of physical importance. Thus the Skyrme model is a rare point where physics, geometry, topology and analysis intersect in an interesting way. I emphasize that the physics involved is not speculative: it is based on firm experimental evidence.
January 31: (Faculty Colloquium, Thursday, 12:30 - 1:30pm, Hylan 1106A)): Sema Salur, U of R;"An Application of Calibrated Geometries : Mirror Symmetry in String Theory"
Abstract: Calibrated submanifolds are distinguished classes of minimal submanifolds and their moduli spaces are expected to play an important role in geometry, low dimensional topology and theoretical physics. Examples of these submanifolds are special Lagrangian 3-folds for Calabi-Yau, associative 3-folds and coassociative 4-folds for G_2, and Cayley 4-folds for Spin(7) manifolds. String theorists believe that every Calabi-Yau 3-fold X has a quantization, which is a Super Conformal Field Theory (SCFT) - a Hilbert space H with a collection of operators satisfying some relations - to be interpreted as the quantum theory of strings moving in X. Two different Calabi-Yau manifolds X and X' may have the same SCFT and in this case there are powerful relationships between the (topological) invariants of X and X'. This is the idea behind the Mirror Symmetry. In this talk, I will first give brief introductions to Calabi-Yau and G_2 manifolds, and then a survey of my recent research on relations between calibrated geometries and the Mirror Symmetry.
January 30 (Wednesday): Yi Lin, University of Toronto; "Morse-Bott Theory and Moment Maps in Generalized Complex Geometry"
Abstract: Generalized complex geometry was introduced by Hitchin and further developed by his student Gualtieri. It is a simultaneous generalization of both symplectic geometry and complex geometry and so is well-suited to study things related to both, e.g., Mirror symmetry. In this talk, first we will review the notions of generalized moment maps and Hamiltonian actions in generalized compley geometry which were introduced by Tolman and the speaker a couple of years ago. Then we will explain the equivariant Morse theory behind the geometry of generalized moment maps. We are going to see that many central results in Hamiltonian symplectic geometry, for instance equivariant formality, Kirwan injectivity and surjectivity, extend to Hamiltonian actions on generalized complex manifolds.

January 22: John Harper, University of Rochester; "Statement Concerning the Differential Geometry of Finite H-Spaces"

Fall 2007 Seminars:

December 4: Ibrahim Unal, University of Rochester; "Strictly Phi-Convex Domains in Calibrated Manifolds"
Abstract: I will talk about strictly phi-convex domains which are introduced by Harvey and Lawson. These are generalizations of Stein manifolds in complex geometry to calibrated manifolds. By using Morse Theory, we prove results about the topology of strictly phi-convex domains in H^n with quaternion calibration, in R^7 with associative or coassociative calibration, and in R^8 with Cayley calibration, similar to the result proved by Andreotti and Frankel for Stein manifolds. We use phi-free submanifolds which are the analogues of totally real submanifolds to find examples of strictly phi-convex domains with every topological type allowed by Morse Theory.

November 29 (Thursday): Anar Ahmadov, Georgia Institute of Technology; "Exotic 4-Manifolds with Small Euler Characteristics (Part I)"
Abstract: It is known that many simply connected, smooth topological 4-manifolds admit infinitely many smooth structures. The smaller the Euler characteristic, the harder it is to construct exotic smooth structure. In this talk we present examples of symplectic 4-manifolds with same integral cohomology as S^2 x S^2. We also discuss the generalization of these examples to #_{2n-1}(S^2 x S^2) for n > 1. As an application of these symplectic building blocks, we construct exotic smooth structure on small 4-manifolds such as CP^2#k(-CP^2) for k = 2, 3, 4, 5 and 3CP^2#l(-CP^2) for l = 4, 5, 6, 7. We will also discuss an interesting applications to the geography of minimal symplectic 4-manifolds.

November 27: Marianty Iionel, University of Toledo; "Constructions of Special Lagrangian Submanifolds"
Abstract: Special Lagrangian submanifolds are a particular class of minimal submanifolds, introduced by Harvey & Lawson in the wider context of calibrated geometries, using the notion of a calibration. After presenting some main properties and examples in special Lagrangian geometry, I will describe some explicit constructions of symmetric special Lagrangian submanifolds that we obtained.

November 20: Selman Akbulut, Michigan State University; "Mirror Duality in a Joyce Manifold"
Abstract: Previously we defined a notion of dual Calabi - Yau manifolds in a G2 manifold, and described a process to obtain them. Here we apply this process to a compact G2 manifold, constructed by Joyce, and as a result we obtain a pair of Borcea-Voisin Calabi-Yau manifolds, which are known to be mirror duals of each other.

November 13: Semail Ulgen Yildirim, Northwestern University; "K-Exact Groups and Coarsely Embeddable Groups"
Abstract: I will talk about the K-exact groups and mention some properties. Gromov introduced coarse embeddability into a Hilbert space, a geometric property of metric spaces with important consequences such as the coarse Baum Connes Conjecture and the Novikov Conjecture. Indeed, every coarsely embeddable group satisfies the Coarse Baum-Connes conjecture, and therefore the Novikov Conjecture. This result suggests that K-exact groups are the next setting in which this fundamental conjecture should be studied. I will talk about the result that shows the relationship between K-exactness and coarsely embeddable groups.

October 30: Fred Cohen, University of Rochester; "On Moduli Spaces Associated to Surfaces with Marked Points"
Abstract: The purpose of this lecture is twofold: (1) Give natural, elementary geometric constructions for certain moduli spaces associated to surfaces with marked points. (2) Give the real cohomology of these moduli spaces. --- (1) The elementary geometry arises from groups acting on spaces of lines together with classical incidence relations. The exposition will be a leisurely stroll with natural, classical constructions. (2) If there is time, the real cohomology will be described in several cases: 1. genus 0 for which classes arise from free Lie algebras, 2. genus 1 for which classes are given in terms of classical modular forms while a stabilization result gives the ranks of cohomology groups in terms of those of certain natural Jacoby forms, 3. genus 2 for which classes are given in terms of Chern classes of two-plane bundles, and 4. extensions to surfaces of genus greater than 2.

October 23: Mohan Ramachandran, S. U. N. Y. at Buffalo; "Filtered Ends of Kahler Manifolds"
Abstract: I will discuss the following theorem of T.Napier and myself. Theorem:If a complete Kahler manifold with bounded geometry has atleast three filtered ends then it maps properly holomorphically onto a Riemann surface. Filtered ends are a refinement of the usual notion of ends. Applications will be given to restrictions on fundamental group of compact Kahler manifolds.

October 16: Ozgur Ceyhan, University of Montreal; "Quantum Cohomology of Real Varieties"
Abstract: In this talk we will discuss Gromov-Witten-Welschinger (GWW) classes and their applications. In particular, we will revisit Horava's definition of quantum cohomology of real algebraic varieties by using GWW-classes and discuss how it is introduced as a DG-operad. In light of this definition, we speculate about mirror symmetry for real varieties.

October 9: Sema Salur, University of Rochester; "Manifolds with Special Holonomy"