Vera Vértesi : Knots in contact 3--manifolds
- Geometry and Topology ( 160 Views )In this talk I will give a purely combinatorial description of Knot Floer Homology for knots in the three-sphere (Manolescu-Ozsváth-Szabó-Thurston). In this homology there is a naturally associated invariant for transverse knots. This invariant gives a combinatorial but still an effective way to distinguish transverse knots (Ng-Ozsváth-Thurston). Moreover it leads to the construction of an infinite family of non-transversely simple knot-types (Vértesi).
Zhou Zhang : Volume Form and Scalar Curvature for K\ahler-Ricci Flow over General Type Manifold
- Geometry and Topology ( 151 Views )It is an interesting project guided by Tian's conjecture to use K\"ahler-Ricci flow with changing cohomology class in the study of general type manifold. The locally smooth convergence leaves quite some freedom for the global geometry. Meanwhile, volume form and scalar curvature have shown different behavior in infinite and finite time cases.
Aaron Naber : Orbifold Regularity of Collapsed Spaces and applications to Einstein Manifolds.
- Geometry and Topology ( 142 Views )Let (M_i,g_i) be a sequence of Riemannian n-manifolds with uniformly bounded curvature such that (M_i,g_i)->(X,d), a metric space, in the Gromov Hausdorff sense. Then we show that there is a closed subset S of X with codimension at least 3 and dimension at most n-5 such that X-S is a Riemannian Orbifold. We use this and an \epsilon-regularity theorem to show that metric spaces in the closure of the moduli space of Einstein 4-manifolds are Riemannian Orbifolds away from a finite number of points. This is joint with G. Tian.
Jason Parsley : Helicity, Configuration Spaces, & Characteristic Classes
- Geometry and Topology ( 174 Views )The helicity of a vector field in R^3, an analog to linking number, measures the extent to which its flowlines coil and wrap around one another. Helicity turns out to be invariant under volume-preserving diffeomorphisms that are isotopic to the identity. Motivated by Bott-Taubes integration, we provide a new proof of this invariance using configuration spaces. We then present a new topological explanation for helicity, as a characteristic class. Among other results, this point of view allows us to completely characterize the diffeomorphisms under which helicity is invariant and give an explicit formula for the change in helicity under a diffeomorphism under which helicity is not invariant. (joint work with Jason Cantarella, U. of Georgia)
Dan Rutherford : Generating families and invariants of Legendrian knots
- Geometry and Topology ( 137 Views )Legendrian knots in standard contact R3 have in addition to their topological knot type two classical invariants known as the Thurston-Bennequin and rotation numbers. Over the past decade several invariants have been developed which are capable of distinguishing between knots with identical classical invariants. The purpose of this talk is to describe interesting relationships between some of these new invariants. Major players in this talk are the Chekanov-Eliashberg DGA (Legendrian contact homology) and related objects, as well as combinatorial structures on front diagrams and homological invariants arising from the theory of generating families (due to Chekanov-Pushkar, Fuchs, and Traynor). The main new result (joint with Fuchs) is that, when a Legendrian knot is defined by a generating family, homology groups obtained by linearizing the Chekanov-Eliashberg DGA are isomorphic to the homology of a pair of spaces associated with the generating family.
Jer-Chin Chuang : Subdivisions and Transgressive Chains
- Geometry and Topology ( 171 Views )Combinatorial transgressions are secondary invariants of a space admitting triangulations. They arise from subdivisions and are analogous to transgressive forms such as those in Chern-Weil theory. In this talk, I characterize transgressions that are path-independent of subdivision sequence. The result is obtained by using a cohomology on posets that is shown to be equivalent to higher derived functors of the inverse (or projective) limit over the opposite poset.
Mark Stern : Geometry of stable Yang-Mills connections
- Geometry and Topology ( 155 Views )On a compact 4-manifold, every self-dual connection and every anti-self-dual connection minimizes the Yang-Mills energy. In this talk, I will answer the converse question for compact homogeneous 4-manifolds. I will also survey related stability results in other dimensions.
Simon Brendle : Minimal Lagrangian diffeomorphisms between domains in the hyperbolic plane
- Geometry and Topology ( 148 Views )Let $\Omega$ and $\tilde{\Omega}$ be domains in the hyperbolic plane with smooth boundary. Assume that both domains are uniformly convex, and have the same area. We show that there exists an area-preserving, orientation-preserving diffeomorphism $f: \Omega \to \tilde{\Omega}$ such that the graph of $f$ is a minimal surface in $\mathbb{H}^2 \times \mathbb{H}^2$. Moreover, we show that the set of all such diffeomorphisms is parametrized by the circle.
Ken Jackson : Numerical Methods for the Valuation of Synthetic Collateralized Debt Obligations (CDOs)
- Geometry and Topology ( 156 Views )Our numerical computation group has studied several problems in computational finance over the past decade. One that we've looked at recently is the pricing of "collateralized debt obligations" (CDOs). The market for CDOs has grown rapidly to over US$1 trillion annually in 2006, since the appearance of JP Morgan's Bistro deal, the first synthetic CDO, in December 1997. Much of the turmoil in the financial markets recently has been due to such credit derivatives. As this suggests, there are still many open problems associated with the pricing and hedging of these complex financial instruments. I will talk briefly about some work that we have done recently in this area.
Matthew Hedden : On Floer homology and knots admitting lens space surgeries
- Geometry and Topology ( 170 Views )J. Berge discovered a simple condition on a knot, K, in the three-sphere which ensures that Dehn surgery on K yields a lens space. It is an open conjecture, known as the Berge conjecture, that any knot on which one can perform surgery and obtain a lens space satisfies his condition. I will discuss a strategy, developed jointly with Ken Baker and Eli Grigsby by which the knot Floer homology invariants of Ozsvath, Szabo, and Rasmussen could be used to prove this conjecture.
Anda Degeratu : Analysis on crepant resolutions of Calabi-Yau orbifolds
- Geometry and Topology ( 144 Views )A Calabi-Yau orbifold is locally modeled on C^n/G with G a finite subgroup of SU(n). If the singularity is isolated, then the crepant resolution (if it exists) is an ALE manifold, for which index-type results are well known. However, most of the time the singularity is not isolated, and for the corresponding crepant resolution there is no index theorem so far. In this talk, I present the first step towards obtaining such a result: I will introduce the class of iterated cone-edge singular manifolds and the corresponding quasi-asymptotically conical spaces (of which orbifolds and their resolutions of singularities are examples), and build-up the general set-up for studying Fredholm properties of geometrical elliptic operators on these spaces. This is joint work with Rafe Mazzeo.
Fédéric Rochon : On the uniqueness of certain families of holomorphic disks
- Geometry and Topology ( 133 Views )A Zoll metric is a metric whose geodesics are all circles of equal length. In this talk, we will first review the definition of the twistor correspondence of LeBrun and Mason for Zoll metrics on the sphere $S^{2}$. It associates to a Zoll metric on $S^{2}$ a family of holomorphic disks in $CP_{2}$ with boundary in a totally real submanifold $P\subset CP_{2}$. For a fixed $P\subset CP_{2}$, we will indicate how one can show that such a family is unique whenever it exists, implying that the twistor correspondence of LeBrun and Mason is in some sense injective. One of the key ingredients in the proof will be the blow-up and blow-down constructions in the sense of Melrose.
Vestislav Apostolov : Old and new trends in Bihermitian geometry
- Geometry and Topology ( 158 Views )A bihermitian structure is a Riemannian metric compatible with two distinct orthogonal complex structures. In the mathematical literature this notion appeared in 90's in the study of the curvature of conformal 4-manifolds. However, bihermitian metrics were already studied in the physics literature in the 80's, as a building bloc of what Gates, Hull and Rocek call `the target space for a (2,2) super-symmetric sigma model'. There has been a great deal of interest in bihermitian geometry more recently, motivated by its link with the notion of generalized Kaehler geometry, introduced by Gualtieri and Hitchin. In this talk I will explain some main features of 4-dimensional bihermitian manifolds, as developed in the 90's, and report on recent classification results that I obtained with M. Gualtieri and G. Dloussky.
Thomas Ivey : Cable knot solutions of the vortex filament flow
- Geometry and Topology ( 134 Views )The simplest model of vortex filament motion in an ideal fluid leads to an integrable nonlinear evolution equation, known as the localized induction approximation or the vortex filament flow, closely related to the cubic focusing nonlinear Schroedinger equation. For closed finite-gap solutions of this flow, certain geometric and topological features of the evolving curves appear to be correlated with the algebro-geometric data used to construct them. In this talk, I will briefly discuss this construction, and some low-genus examples (in particular, Kirchhoff elastic rod centerlines) where this correlation is well understood. I will mainly discuss recent joint work with Annalisa Calini, describing how to generate a family of closed finite-gap solutions of increasingly higher genus via a sequence of deformations of the multiply covered circle. We prove that every step in this sequence corresponds to constructing a cable on previous filament; moreover, the cable's knot type (which is invariant under the evolution) can be read off from the deformation sequence.
Jim Isenberg : Construcing solutions of the Einstein constraint equations
- Geometry and Topology ( 173 Views )The first step in finding a spacetime solution to the Einstein gravitational field equations via the inital value formulation is to construct initial data which satisfy the Einstein constraint equations. There are three ways of carrying out this construction which have been found to be useful: the conformal and conformal thin sandwich methods, the gluing techniques, and the quasi-spherical approaches. We describe each of these, we discuss their advantages and disadvantages, we outline some of their recent successful applications, and we present some of the outstanding questions remaining to be solved from each of these perspectives.
Christina Tonnesen-Friedman : Canonical classes on admissible bundles
- Geometry and Topology ( 187 Views )For each K¨ahler class on a compact K¨ahler manifold there is a lower bound of the Calabi functional, which we call the ``potential energy''. Fixing the volume and letting the K¨ahler classes vary, the energy defines a functional which may be studied in it?s own right. Any critical point of the energy functional is then a K¨ahler class whose extremal K¨ahler metrics (if any) are so-called strongly extremal metrics. We take the well-studied case of Hirzebruch surfaces and generalize it in two different directions; along the dimension of the base and along the genus of the base. In the latter situation we are able to give a very concrete description of the corresponding dynamical system (as defined first by S. Simanca and L. Stelling). The talk is based on work in progress with Santiago Simanca.
Paul Norbury : Magnetic monopoles on manifolds with boundary
- Geometry and Topology ( 139 Views )Kapustin and Witten introduced interesting boundary value problems for magnetic monopoles on a Riemann surface times an interval. They described the moduli space of such solutions in terms of Hecke modifications of holomorphic bundles over the Riemann surface. I will explain this and prove existence and uniqueness for such monopoles.
Michael Eichmair : Non-variational Plateau problems and the spacetime positive mass theorem in general relativity
- Geometry and Topology ( 176 Views )In this talk I will introduce some new ideas to the existence theory for a class of non-variational existence problems arising naturally in geometry and analysis. I will discuss some applications (and potential applications) to positive mass-type and Penrose-type theorems in general relativity.
Jacques Hurtubise : Isomonodromy deformations of connections
- Geometry and Topology ( 144 Views )The link between meromorphic connections on a Riemann surface and their monodromy is a very classical one, indeed so classical that it was the subject of one of Hilbert?s problems. The deformation theory of these connections, and when these deformations preserve the monodromy, is almost equally ancient. I will give an overview of some results in the area, some ancient, and some quite recent.
Ruxandra Moraru : Moduli spaces of stable bundles on certain non-Kaehler surfaces.
- Geometry and Topology ( 122 Views )In this talk, I will examine the geometry of moduli spaces of stable bundles on Hopf and Kodaira surfaces, which are compact complex surfaces that do not admit Kaehler metrics. In particular, I will show that these moduli admit interesting geometric structures such as hypercomplex structures and strong HKT-metrics, in the case of Hopf surfaces, as well as algebraic integrable systems.
Larry Guth : Area-contracting maps between rectangles
- Geometry and Topology ( 136 Views )The k-dilation of a map measures how much the map stretches k-dimensional volumes. The 1-dilation is the usual Lipschitz constant. We consider the problem of finding the smallest k-dilation among all degree 1 maps from one rectangle to another rectangle. (These are n-dimensional rectangles.) In general the linear map is far from optimal.
Niall O'Murchadha : The Liu-Yau mass as a good quasi-local energy in general relativity
- Geometry and Topology ( 127 Views )A quasi-local mass has been a long sought after quantity in general relativity. A recent candidate has been the Liu-Yau mass. One can show that the Liu-Yau mass of any two-surface is the maximum of the Brown-York energy for that two-surface. This means that it has significant disadvantages as a mass. It is much better interpreted as an energy and I will show one way of doing so. The Liu-Yau mass is especially interesting in spherical geometries, where mass and energy are indistinguishable. For a spherical two-surface, it equals the minimum of the amount of energy at rest that one needs to put inside the two-surface to generate the given surface geometry. Thus it gives interesting information about the interior, something no other mass or energy function does.
Fernando Schwartz : On the topology of black holes
- Geometry and Topology ( 144 Views )An important special case of the general construction of black holes translates into a problem in Riemannian geometry, since a totally geodesic slice of spacetime is an asymptotically flat Riemannian manifold with nonnegative scalar curvature, and the restriction of the event horizon to the slice is the apparent horizon in the slice. In this talk we show how to construct new examples of Riemannian manifolds with nonspherical apparent horizon, in dimensions four and above. More precisely, for any $n,m\ge 1$, we construct asymptotically flat, scalar flat Riemannian manifolds with apparent horizon that is a smooth outermost minimal hypersurface with topology $S^n\times S^{m+1}$.
Vladimir Matveev : Geodesically equivalent metrics in the large: Beltrami and Schouten problems.
- Geometry and Topology ( 139 Views )Two metrics are geodesically equivalent if they have the same (unparameterized) geodesics. During my talk I describe geodesically equivalent metrics on closed manifolds (which is an answer to Beltrami's question) and explain the proof of Lichnerowicz-Obata conjecture (which is an answer on the infinitesimal version of the Beltrami question known as Schouten problem).
Emma Carberry : Conformal Surface Geometry: an algebro-geometric approach.
- Geometry and Topology ( 143 Views )A number of classical integrable systems, for example harmonic maps of the plane to a compact Lie group or symmetric space, can be transformed into a \{\\em linear\} flow on a complex torus. This torus is the Jacobian of an algebraic curve, called the spectral curve. Recently several authors have produced an analogous one-dimensional analytic variety for conformal 2-tori in $S4$ (which are not in general integrable!) using the geometry of the quaternions. It is hoped that this new development will lead to progress on the Willmore conjecture for reasons that I will explain. However this variety is at present quite mysterious; very little is known about it. I will discuss the simplest case, namely constant mean curvature tori in $\mathbb{R}3$. I will demonstrate that in this case the variety is not at all mysterious and interpret its points geometrically in terms of transformations generalising the classical transform of Darboux. This is joint work with Katrin Leschke and Franz Pedit.
Richard Hain : The Lie Algebra of the Mapping Class Group, Part 2
- Geometry and Topology ( 197 Views )In this talk I will review the construction of the Lie algebra associated to the mapping class group of a (possibly decorated) surface and explain how this generalizes the Lie algebra associated to the pure braid group. I will also explain the analogue of the KZ-equation in the mapping class group case. In the second talk I will discuss filtrations of this Lie algebra associated to curve systems on the surface and their relation to handlebody groups.
Richard Hain : The Lie Algebra of the Mapping Class Group, Part 1
- Geometry and Topology ( 190 Views )In this talk I will review the construction of the Lie algebra associated to the mapping class group of a (possibly decorated) surface and explain how this generalizes the Lie algebra associated to the pure braid group. I will also explain the analogue of the KZ-equation in the mapping class group case. In the second talk I will discuss filtrations of this Lie algebra associated to curve systems on the surface and their relation to handlebody groups.
Marcus Khuri : On the Penrose Inequality
- Geometry and Topology ( 160 Views )The cosmic censorship conjecture roughly states that singularities in the evolution of spacetime are always hidden from the outside world by event horizons. As a test for this conjecture Penrose proposed the inequality M >= (A/16pi)^1/2, relating the total ADM mass M of a spacetime to the area A of an event horizon. For time symmetric initial data sets of Einstein's equations this inequality has been confirmed, independently by Huisken and Ilmanen (for one black hole) and by Bray (for multiple black holes). The purpose of this talk is to show how the time symmetric proofs can be generalized to apply to general initial data, assuming existence for a canonical degenerate elliptic system of equations. This is joint work with Hubert Bray.