Geometry-Topology Archives for Academic Year 2014

Levi-Civita's Theorem in noncommutative geometry

When: Mon, September 8, 2014 - 3:00pm
Where: Math 1313
Speaker: Jonathan Rosenberg (UMd) -
Abstract: In Riemannian geometry, usually one starts with the notion of Riemannian metric, then one proves Levi-Civita's Theorem, that gives a canonical connection once the metric is fixed, and then proceeds to define curvature, etc. In noncommutative geometry, the obvious approach to doing all this fails. But we show how to define Riemannian metrics and connections on a noncommutative torus in such a way that an analogue of Levi-Civita's theorem on the existence and uniqueness of a Riemannian connection holds. The major novelty is that we need to use two different notions of noncommutative vector field.

Topology of the space of metrics of positive scalar curvature

When: Mon, September 15, 2014 - 3:00pm
Where: Math 1313
Speaker: Boris Botvinnik (University of Oregon) -
Abstract: We use recent results on the moduli spaces of manifolds, plus relevant
index and surgery theory, to study the index-difference map from the space
Riem$^+(W^d)$ of psc-metrics on a manifold $W$
to the space $\Omega^{d+1}KO$ representing real $K$-theory.
This is joint work with Johannes Ebert and Oscar Randal-Williams.

In particular, we show that the index map induces a nontrivial homomorphism in homotopy.

Uniqueness of Stenzel's metric

When: Mon, September 22, 2014 - 3:00pm
Where: Math 1313
Speaker: Hans-Joachim Hein (UMD)
Abstract: Stenzel's metric is an SO(n+1)-invariant Calabi-Yau metric on the cotangent bundle of S^n. It is complete and asymptotic to a cone at infinity. In joint work with Ronan Conlon, we show that, for n > 4, Stenzel's metric is the only complete Calabi-Yau metric asymptotic to this cone.

Geometry of the space of Kahler metrics

When: Mon, September 29, 2014 - 3:00pm
Speaker: Tamas Darvas (UMD)
Abstract: Given a Kahler manifold $(X,\omega)$, Mabuchi observed that one can endow the space of smooth Kahler metrics $H$ with a natural Riemannian structure. This space has received attention after Donaldson linked it to existence and uniqueness of constant scalar curvature metrics. The Riemannian structure induces a path length metric $d$ on $H$, however the resulting metric space $(H,d)$ is not geodesically convex, i.e. there may not be a geodesic connecting arbitrary points of $H$. In this talk we will identify the metric completion of $(H,d)$, and argue that this bigger space is not only geodesically convex, but also non-positively curved in the sense of Alexandrov.

Arithmetic quotients of the mapping class group (Joint with Algebra-Number Theory)

When: Mon, October 6, 2014 - 3:00pm
Where: Math 1313
Speaker: Justin Malestein (Universität Bonn) -
Abstract: In this talk, I will discuss a procedure for obtaining infinitely many “virtual” arithmetic quotients of mapping class groups, (surjective maps up to finite index). Specifically, for any irreducible rational representation of a finite group of rank less than g, we produce a corresponding virtual arithmetic quotient of the genus g mapping class group. Particular choices of irreducible representations of finite groups yield arithmetic quotients of type Sp(2m), SO(2m, 2m), and SU(m, m) for arbitrarily large m in every genus. Joint with F. Grunewald, M. Larsen, and A. Lubotzky.

Quaternionic structures on complexifications of Kahler manifolds

When: Mon, October 13, 2014 - 3:00pm
Speaker: Andy Sanders (UIC)
Abstract: It is known, by a result of Feix and Kaleiden, that a neighborhood of the zero section in the cotangent bundle of a real-analytic Kahler manifold M admits a hyper-Kahler metric which is compatible with the complex symplectic structure of the cotangent bundle. Utilizing some tools from symplectic geometry, we will give a construction of a different type of Riemannian quaternionic structure on a neighborhood of the diagonal in M x M. Due to integrability issues, these structures will usually not be hyper-Kahler, but they have interesting geometric properties in their own right. At the end, we will spend some time discussing the application of this structure to Teichmuller space sitting inside of quasi-Fuchsian space.

Conformal actions of PSL(2,R) on compact Lorentz manifolds

When: Mon, October 20, 2014 - 3:00pm
Where: Math 1313
Speaker: Vincent Pecastaing (Université Paris Sud, Orsay)
Abstract: If (M,g) is a pseudo-Riemannian manifold and G < Conf(M,g) is a
Lie group acting conformally on M, the action of G on M is said to be
essential if G does not act by isometries for any metric g' in the
conformal class of g.

In Riemannian signature, a result due to J. Ferrand and M. Obata (1971)
says that up to conformal diffeomorphism, the round sphere is the only
compact Riemannian manifold admitting a conformal essential action of some
Lie group. In Lorentz signature, the analogous result is no longer true
and there are numerous counterexamples (C. Frances, 2002). However, all
the examples given by Frances are conformally flat (ie locally conformally
equivalent to the Minkowski space), and it is now conjectured that when a
compact Lorentz manifold admits an essential action of some Lie group, it
is conformally flat.

In this talk, I will present some results on conformal actions of simple
Lie groups that partially confirm this conjecture, and I will mainly focus
on the case of Lie groups locally isomorphic to PSL(2,R), which represents
the core of the problem.

Coordinates for representations of 3-manifold groups

When: Mon, October 27, 2014 - 3:00pm
Where: Math 1313
Speaker: Christian Zickert (UMD)
Abstract: We discuss the shape and Ptolemy coordinates, which are coordinates on representation varieties coming from triangulations. The coordinates are 3-dimensional analogues of coordinates on higher Teichmuller spaces due to Fock and Goncharov.

Boundary-value problems for the Ricci flow

When: Mon, November 3, 2014 - 3:00pm
Speaker: Artem Pulemotov (Queensland)
Abstract: The Ricci flow is a second-order partial differential equation describing the evolution of a Riemannian metric on a manifold. This equation is particularly famous for its key role in the proof of the Poincare Conjecture. Understanding the Ricci flow on manifolds with boundary is a difficult problem with applications to a variety of fields, such as topology and mathematical physics. The talk will survey the current progress towards the resolution of this problem. In particular, we will discuss new results concerning spaces with symmetries.

Topology of Free Group Character Varieties

When: Mon, November 10, 2014 - 3:00pm
Where: Math 1313
Speaker: Sean Lawton (George Mason University) -
Abstract: We will discuss new results concerning the homotopy groups of moduli spaces of G-valued representations of free groups, for a reductive algebraic group G.

Spatial Asymptotic Behavior of the Kahler-Ricci Flow and Kahler-Einstein Metric

When: Mon, November 17, 2014 - 3:00pm
Where: Math 1313
Speaker: Zhou Zhang (University of Sydney) -
Abstract: In this talk, we mainly consider the Kahler-Ricci flow over a complete non-compact manifold. The spatial asymptotic behavior is a natural and important new ingredient in the study comparing with the closed manifold setting. For the finite volume case, we have approached this topic from both geometric and analytic points of view. The analysis of the corresponding Kahler-Einstein metric is included, illustrating the difference. We have also studied the infinite volume case in a geometric fashion more recently. Overall, the main focus is on the quasi-projective setting. This talk is based on joint works with John Lott and Frederic Rochon.

Flows on irreducible parabolic geometries

When: Mon, November 24, 2014 - 3:00pm
Speaker: Karin Melnick (University of Maryland)
Abstract: Irreducible parabolic geometries are a family of geometric structures including conformal semi-Riemannian structures, projective structures, and many more. Automorphisms of these structures need not be linearizable around a fixed point, as for semi-Riemannian isometries or affine transformations of a connection. I will present rigidity theorems for structures of this type admitting special nonlinearizable flows by automorphisms. A consequence will be that in many cases, if the geometry is not flat---that is, not locally equivalent to the homogeneous model---then automorphisms are determined by their 1-jet at a point. This is joint work with K. Neusser.

Bifurcation of periodic solutions to the singular Yamabe problem on spheres

When: Mon, December 1, 2014 - 3:00pm
Where: Math 1313
Speaker: Bianca Santoro (City College of New York) -
Abstract: In this talk, we describe how to obtain uncountably many periodic solutions to the singular Yamabe problem on a round sphere, that blow up along a great circle. These
are (complete) constant scalar curvature metrics on the complement of S^1
inside S^m, m ≥ 5, that are conformal to the (incomplete) round metric and
periodic in the sense of being invariant under a discrete group of conformal
transformations. These solutions come from bifurcating branches of constant
scalar curvature metrics on compact quotients of S^m \ S^1. This is a joint work with
R. Bettiol (University of Notre Dame) and P. Piccione (USP-Brazil).

Tameness of Margulis Spacetimes (dynamics seminar)

When: Thu, December 11, 2014 - 2:10pm
Where: 1311 (not 1313; note also that the seminar starts at 2pm, not 3pm)
Speaker: Bill Goldman
Abstract: A complete affine manifold is a quotient of Euclidean space by
a discrete group of affine transformations acting properly.
A Margulis spacetime is a 3-dimensional complete affine
3-manifold with free fundamental group of finite rank at least 2.
Such a flat manifold corresponds to a deformation of a noncompact hyperbolic surface S. It is conjectured that such 3-manifolds are homeomorphic to solid handlebodies. We apply dynamical properties of the geodesic flow on S to prove this conjecture when S has no cusps. Not all quotient manifolds of Euclidean 3-space by free groups
are solid handlebodies, however.

An algebraic model for the infinitesimal bialgebra of string topology

When: Thu, January 15, 2015 - 3:00pm
Where: Math 1311
Speaker: Manuel Rivera (CUNY) -
Abstract: The homology of the free loop space of a manifold has a rich non trivial algebraic structure organized by an action of the homology of a compactification of the moduli space of Riemann surfaces with input and output boundary. The Hochschild homology of a Frobenius algebra has an analogous algebraic structure organized by a moduli space of graphs called Sullivan diagrams. In both the free loop space and Frobenius algebra stories we encounter an involutive infinitesimal bialgebra which induces an involutive Lie bialgebra (originally discovered by Chas-Sullivan and Goresky-Hingston) on the S^1-equivariant and cyclic Hochschild homologies respectively. The two algebra structures (the Chas-Sullivan loop product and the Hochschild cup product) have been identified over the rationals and for simply connected manifolds by Felix, Thomas, Vigué-Poirrier and by Merkulov. It is strongly believe that these identifications can be extended to all the algebraic structure present in both stories. However, there are several subtleties concerning Poincare duality at the chain level that one has to take into account when identifying further operations such as the Goresky-Hingston coproduct on the free loop space with its analogous structure on the Hochschild side. I will describe how to associate functorially to a manifold an algebraic object that expresses Poincaré duality at the chain level as two operations which are formally Frobenius compatible. Then, I will outline how this construction can be used to show how the Goresky-Hingston coproduct on the homology of free loop space can be recovered algebraically. As a corollary, it will follow that such coalgebra structure on the homology of the free loop space of a manifold is an invariant of the homotopy type of the underlying manifold, settling a longstanding question posed by D. Sullivan. Time permitting, I will finish by describing current projects regarding algebraic models for string topology operations for the non simply connected case and by posing certain questions concerning geometric variants of string topology that are not homotopy invariants.


Duality for Elliptic Curve Orientifolds and Twisted KR-Theory

When: Mon, February 2, 2015 - 3:00pm
Where: Math 1313
Speaker: Jonathan Rosenberg (UMd) -
Abstract: An amazing discovery of physicists is that there are many seemingly quite different physical quantum field theories that lead to the same observable predictions. Such theories are said to be related by dualities. A duality leads to interesting mathematical consequences; for example, certain K-theory groups on the two spacetime manifolds have to be isomorphic. Mirror symmetry of Calabi-Yau manifolds was also discovered this way. We will discuss the special case of elliptic curve orientifolds, (complex) elliptic curves equipped with a holomorphic or anti-holomorphic involution. In this case, the relevant kind of K-theory is Atiyah's KR-theory, which we will define and explain, but certain additional twisting has to be taken into account.
It turns out that predictions from physics do match very nicely with calculations into topology. Some of this work is joint with Stefan Mendez-Diez and Chuck Doran.

The largest "eigenvalue" of a convex function, duality, and a theorem of Slodkowski

When: Mon, February 23, 2015 - 3:00pm
Where: Math 1313
Speaker: Matt Dellatorre (UMD) -
Abstract: Given a convex function u on Rn, differentiable at x_0, many generalized second-order directional derivatives can be defined. Slodkowski (1985) introduces one such derivative, a maximal one, and proves that at a nontrivial fraction of nearby points this derivative must take on a similar value. A lowerbound (in terms of density) on this set is quantified precisely in terms of the value at x_0 and a larger comparison value. Being maximal, this second-order derivative is of particular interest because it corresponds to the largest eigenvalue of the Hessian when defined, and gives a useful quantity to work with otherwise, especially in the context of C^{1,1} estimates. Given the almost-everywhere 2nd differentiability of convex functions this result can be quite useful. In particular, it allows one to extend an a.e. lower bound on the largest eigenvalue to a bound holding everywhere. Additionally, this derivative has an especially nice geometric interpretation in terms of spheres tangent to the graph of the function. Although this theorem was proved 30 years ago, it has been key to recent progress on the fully nonlinear, elliptic Dirchlet problem and the accompanying Dirichlet duality theory of Harvey and Lawson.
I will give a sketch of the proof, an introduction to the Legendre-Fenchel transform, and show how a bound on this largest "eigenvalue" transforms to give an alternative proof to a key step.

Two-point functions in geometry

When: Mon, March 9, 2015 - 3:00pm
Where: Math 1313
Speaker: Hans-Joachim Hein (UMD) -
Abstract: Several major recent results in geometric analysis rely on the rather unusual idea to study a function u on a manifold M by deriving and analyzing a PDE for the "two-point function" u(x) - u(y) on M x M. I will review these results and describe one elementary application in detail (a very slick proof by Ben Andrews of a well-known eigenvalue estimate due to Zhong and Yang). This is an expository talk.

Introduction to nonlinear harmonic forms

When: Mon, March 30, 2015 - 3:00pm
Where: Math 1313
Speaker: Mark Stern (Duke University) -
Abstract: We motivate and introduce nonlinear harmonic forms. These are de Rham representatives $z$ of cohomology classes which minimize the energy $\|z\|_{L_2}^2$ subject to a nonlinear constraint. We give basic existence results for quadratic constraints, discuss the rich Euler Lagrange equations, and ask many regularity questions.

Einstein metrics, harmonic forms, and symplectic four-manifolds

When: Mon, April 6, 2015 - 3:00pm
Where: Math 1313
Speaker: Claude LeBrun (Stony Brook University) -
Abstract: If M is the underlying smooth oriented four-manifold of a Del Pezzo surface, we consider the set of Riemannian metrics h on M such that W_+(ω,ω) > 0, where W_+ is the self-dual Weyl curvature of h, and ω is a non-trivial self-dual harmonic two-form on (M, h). While this open region in the space of Riemannian metrics contains all the known Einstein metrics on M, we show that it contains no others. Consequently, it contributes exactly one connected component to the moduli space of Einstein metrics on M.

Kahler-Einstein potentials on convex simple polytopes

When: Mon, April 13, 2015 - 3:00pm
Where: Math 1313
Speaker: Eveline Legendre (Universite de Toulouse) -
Abstract: In this talk, I would like to bring at your attention that some simple observations in toric geometry imply that any compact convex simple lattice polytope is the moment polytope of a Kahler-Einstein orbifold, unique up to orbifold covering and homothety. Moreover, after extending the Wang-Zhu Theorem in a generalized setting we interpret our result in terms of existence of singular Kahler-Einstein metrics on toric manifolds.

Curvatures of direct image sheaves of vector bundles and applications

When: Mon, April 27, 2015 - 3:00pm
Where: Math 1313
Speaker: Xiaokui Yang (Northwestern University) -
Abstract: In this talk, we present several curvature formulas for direct image sheaves of vector bundles. We also introduce some applications on the positivity and vanishing theorems for vector bundles over Kahler and non-Kahler manifolds, new examples on nef but not semi-positive bundles, and curvature properties of moduli spaces of curves.

Topological insulators and Noncommutative Geometry

When: Wed, May 13, 2015 - 11:00am
Where: Math 1313
Speaker: Dan Li (Purdue University) -
Abstract: I will try to explain what a topological insulator is and the Z/2Z topological invariant. The physical picture is a parity anomaly, and the mathematical picture is a generalized index theorem. Noncommutative geometry (NCG) comes into the game when there exist disorders or impurities in material. If time permits, I will talk about possible generalizations of the Z/2Z topological invariant in NCG.

Projective and c-projective metric geometries: why they are so similar

When: Tue, June 9, 2015 - 3:00pm
Where: Math 1313
Speaker: Vladimir Matveev (Friedrich-Schiller-Universität Jena) -
Abstract: I will show an unexpected application of the standard techniques of integrable systems in projective and c-projective geometry (I will explain what they are and why they were studied). I will show that c-projectively equivalent metrics on a closed manifold generate a commutative isometric R^k-action on the manifold. The quotients of the metrics w.r.t. this action are projectively equivalent, and the initial metrics can be uniquely reconstructed by the quotients. This gives an almost algorithmic way to obtain results in c-projective geometry starting with results in much better developed projective geometry. I will give many application of this algorithmic way including local description, proof of Yano-Obata conjecture for metrics of arbitrary signature, and describe the topology of closed manifolds admitting strictly nonproportional c-projectively equivalent metrics.

Most results are parts of two projects: one is joint with D. Calderbank, M. Eastwood and K. Neusser, and another is joint with A. Bolsinov and S. Rosemann.