Pankaj K. Agarwal : Union of geometric objects, epsilon-nets, and hitting sets
- Geometry and Topology ( 112 Views )The combinatorial complexity of the union of a set of geometric objects is the total number of faces of all dimensions that lie on the boundary of the union. We review some recent results on the complexity of the union of geometric objects in 2d and 3D satisfying various natural conditions and on computing the union. We then discuss the critical roles they play in computing an epsilon net and a hitting set of a set system.
Adam Levine : Heegaard Floer invariants for homology S^1 x S^3s
- Geometry and Topology ( 99 Views )Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented 4-manifold X with the homology of S^1 x S^3. Specifically, we show that for any smoothly embedded 3-manifold Y representing a generator of H_3(X), a suitable version of the Heegaard Floer d invariant of Y, defined using twisted coefficients, is a diffeomorphism invariant of X. We show how this invariant can be used to obstruct embeddings of certain types of 3-manifolds, including those obtained as a connected sum of a rational homology 3-sphere and any number of copies of S^1 x S^2. We also give similar obstructions to embeddings in certain open 4-manifolds, including exotic R^4s. This is joint work with Danny Ruberman.
Yang Li : On the Donaldson-Scaduto conjecture
- Geometry and Topology ( 666 Views )Motivated by G2-manifolds with coassociative fibrations in the adiabatic limit, Donaldson and Scaduto conjectured the existence of associative submanifolds homeomorphic to a three-holed 3-sphere with three asymptotically cylindrical ends in X \times R^3, where X is an A2-type ALE hyperkähler manifold. We prove this conjecture by solving a real Monge-Ampère equation with singular right hand side. The method produces many other asymptotically cylindrical U(1)-invariant special Lagrangians in X \times R^2, where X arises from the Gibbons-Hawking construction. This is joint work in progress with Saman Habibi Esfahani.
Shubham Dwivedi : Geometric flows of $G_2$ structures
- Geometry and Topology ( 202 Views )We will start by discussing a flow of isometric $G_2$ structures. We consider the negative gradient flow of the energy functional restricted to the class of $G_2$ structures inducing a given Riemannian metric. We will discuss various analytic aspects of the flow including global and local derivative estimates, a compactness theorem and a monotonicity formula for the solutions. After defining an entropy functional we will prove that low entropy initial data lead to solutions that exist for all time and converge smoothly to a $G_2$ structure with divergence free torsion. We will also discuss finite time singularities and the singular set of the solutions. Finally, we will discuss the isometric flow "coupled” with the Ricci flow of the underlying metric, which again is a flow of $G_2$ structures, and discuss some of its properties. This is a based on two separate joint works with Panagiotis Gianniotis (University of Athens) and Spiro Karigiannis (University of Waterloo).
Lisa Piccirillo : The Conway knot is not slice
- Geometry and Topology ( 140 Views )Surgery-theoretic classifications fail for 4-manifolds because many 4-manifolds have second homology classes not representable by smoothly embedded spheres. Knot traces are the prototypical example of 4-manifolds with such classes. Ill give a flexible technique for constructing pairs of distinct knots with diffeomorphic traces. Using this construction, I will show that there are knot traces where the minimal genus smooth surface generating second homology is not the obvious one, resolving question 1.41 on the Kirby problem list. I will also use this construction to show that Conway knot does not bound a smooth disk in the four ball, which completes the classification of slice knots under 13 crossings and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot.
Curtis Porter : Spinning Black Holes and CR 3-Folds
- Geometry and Topology ( 254 Views )Some physically significant solutions to Einstein's field equations are spacetimes which are foliated by a family of curves called a shear-free null geodesic congruence (SFNGC). Examples include models of gravitational waves that were recently detected, and rotating black holes. The properties of a SFNGC induce a CR structure on the 3-dimensional leaf space of the foliation. The Kerr Theorem says that the family of metrics associated to a SFNGC contains a conformally flat representative iff the corresponding CR structure is embeddable in a real hyperquadric. Using Cartan's method of moving frames, we can classify which Levi-nondegenerate CR 3-folds are embeddable in the hyperquadric.
Jacques Hurtubise : Isomonodromy deformations of connections
- Geometry and Topology ( 145 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.
Daniel Ruberman : Periodic-end Dirac operators and Seiberg-Witten theory
- Geometry and Topology ( 105 Views )We study the Seiberg-Witten equations on a 4-manifold X with the homology of S1 x S3. The count of such solutions, called the Seiberg-Witten invariant, depends on choices of Riemannian metric and perturbation of the equations to make a smooth moduli space. A similar issue is resolved in dimension 3 by relating the jumps in the Seiberg-Witten invariant to the spectral flow of the Dirac operator. In dimension 4, we use Taubes' theory of periodic-end operators to relate the jumps in the Seiberg-Witten invariant to the index theory of the Dirac operator on the infinite cyclic cover of X. This circle of ideas has applications to classification of smooth manifolds homeomorphic to S1 x S3, to questions about positive scalar curvature, and to homology cobordisms. This is joint work with Tomasz Mrowka and Nikolai Saveliev.
Amit Einav : Entropic Inequality on the Sphere
- Geometry and Topology ( 208 Views )It is an interesting well known fact that the relative entropy with respect to the Gaussian measure on $\mathbb{R}^N$ satisfies a simple subadditivity property. Namely, if $\Pi_1^{(i)}(F_N)$ is the first marginal of the density function F_N in the i-th variable then \begin{equation} \sum_{i=1}^N H(\Pi_1^{(i)}(F_N) | \gamma_1) \leq H(F_N | \gamma_N), \end{equation} where $\gamma_k$ is the standard Gaussian on $\mathbb{R}^k$. Surprisingly enough, when one tries to achieve a similar result on $\mathbb{S}^{N-1}(\sqrt{N})$ a factor of 2 appears in the right hand side of the inequality (a result due to Carlen, Lieb and Loss), and the constant is sharp. Besides a deviation from the simple equivalence of ensembles principle in equilibrium Statistical Mechanics, this entropic inequality on the sphere has interesting ramifications in other fields, such as Kinetic Theory. In this talk we will present conditions on the density function F_N, on the sphere, under which we can get an almost subaditivity property; i.e. the factor 2 can be replaced with a factor of $1+\epsilon_N$, with $\epsilon_N$ given explicitly and going to zero. The main tools to be used in order to proved this result are an entropy conservation extension of F_N to $\mathbb{R}^N$ together with comparison of appropriate transportation distances such as the entropy, Fisher information and Wasserstein distance between the marginal of the original density and that of the extension. Time permitting, we will give an example, one that arises naturally in the investigation of the so-called Kac Model, to many families of functions that satisfy these conditions.
Jeremy Van Horn-Morris : Fiber genus and the topology of symplectic fillings
- Geometry and Topology ( 119 Views )Work of Donaldson, Giroux, and many others shows how to associate a singular surface fibration to a symplectic 4-manifold, either closed or with boundary, as well as to a contact 3-manifold. These are Lefschetz pencils, fibrations and open books, resp. It was asked by Stipsicz, Ozbagci, Korkmaz and others, whether the genus (or genus and self intersection) of the fiber of these fibrations gave an a priori bound on the topological complexity of the symplectic manifold. This question is equivalent to asking for a bound on the length of a factorization of a mapping class element of the fiber into right handed Dehn twists. We will discuss some of the known conditions which can produce such a bound, as well as present examples where such a bound does not exist. This is joint work with I. Baykur.
Christina Tonnesen-Friedman : Canonical classes on admissible bundles
- Geometry and Topology ( 190 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.
Richard Hain : Fundamental groups of branched coverings of certain Kaehler manifolds
- Geometry and Topology ( 119 Views )In this talk I will discuss a result that describes (under certain conditions) the fundamental group of certain branched coverings of quasi-projective varieties that admits a complete Kaehler metric with non-positive curvature. When applied to the period map for genus 3 curves, it implies that the Torelli group in genus 3 is finitely generated. Combined with recent work of Putman and Hatcher-Margalit, it gives a new proof of Dennis Johnson's result that the Torelli group is genus g is finitely generated for all g > 2.
Laura Fredrickson : The asymptotic geometry of the Hitchin moduli space
- Geometry and Topology ( 111 Views )Hitchin's equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichmuller theory, and the geometric Langlands correspondence. The Hitchin moduli space carries a natural hyperkahler metric. A conjectural description of its asymptotic structure appears in the work of Gaiotto-Moore-Neitzke and there has been a lot of progress on this recently. I will discuss some recent results.
Nelia Charalambous : On the $L^p$ Spectrum of the Hodge Laplacian on Non-Compact Manifolds
- Geometry and Topology ( 246 Views )One of the central questions in Geometric Analysis is the interplay between the curvature of the manifold and the spectrum of an operator. In this talk, we will be considering the Hodge Laplacian on differential forms of any order $k$ in the Banach Space $L^p$. In particular, under sufficient curvature conditions, it will be demonstrated that the $L^p\,$ spectrum is independent of $p$ for $1\!\leq\!p\!\leq\! \infty.$ The underlying space is a $C^{\infty}$-smooth non-compact manifold $M^n$ with a lower bound on its Ricci Curvature and the Weitzenb\"ock Tensor. The further assumption on subexponential growth of the manifold is also necessary. We will see that in the case of Hyperbolic space the $L^p$ spectrum does in fact depend on $p.$ As an application, we will show that the spectrum of the Laplacian on one-forms has no gaps on certain manifolds with a pole and on manifolds that are in a warped product form. This will be done under weaker curvature restrictions than what have been used previously; it will be achieved by finding the $L^1$ spectrum of the Laplacian.
Bong H. Lian : Riemann-Hilbert problem for period integrals
- Geometry and Topology ( 106 Views )Period integrals of an algebraic variety are transcendental objects that describe, among other things, deformations of the variety. They were originally studied by Euler, Gauss and Riemann, who were interested in analytic continuation of these objects. In this lecture, we will discuss a number of problems on period integrals in connection with mirror symmetry and Calabi-Yau geometry. We will see how the theory of D-modules have led us to solutions and insights into some of these problems.
Steven Sivek : A contact invariant in sutured monopole homology
- Geometry and Topology ( 112 Views )Kronheimer and Mrowka recently used monopole Floer homology to define an invariant of sutured manifolds, following work of Juhász in Heegaard Floer homology. Contact 3-manifolds with boundary are natural examples of such manifolds. In this talk, I will construct an invariant of a contact structure as an element of the associated sutured monopole homology group. I will discuss several interesting properties of this invariant, including gluing maps which are analogous to the Heegaard Floer sutured gluing maps of Honda, Kazez, and Matić and applications to Legendrian knots. This is joint work with John Baldwin.
Michael Abel : HOMFLY-PT homology of general link diagrams and its decategorification
- Geometry and Topology ( 124 Views )In the construction of HOMFLY-PT homology, one must start with a link presented as a braid closure. This restriction was expected by Khovanov and Rozansky to be required for the homology to be an isotopy invariant. In this talk, after reviewing the construction of the HOMFLY-PT polynomial and homology, we explore the consequences of dropping this requirement and allowing general link diagrams. We explicitly show that the Reidemeister IIb move (where the strands have opposite orientations) fails. Finally we will show that the Euler characteristic of this homology theory is a deformed version of the HOMFLY-PT polynomial which detects "braidlike" isotopy of tangles and links. This new polynomial agrees with the HOMFLY-PT polynomial on link diagrams which are presented as closed braid diagrams.
Colleen Robles : Degeneration of Hodge structure
- Geometry and Topology ( 126 Views )I will describe how representation theory and the geometry of homogeneous spaces may be used to determine the degenerations of a given Hodge structure. This work is part of a larger program to understand the degenerations of a smooth variety that is being pursued, in various subset of collaboration, by Mark Green, Phillip Griffiths, Matt Kerr, Greg Pearlstein and me.
Ken Jackson : Numerical Methods for the Valuation of Synthetic Collateralized Debt Obligations (CDOs)
- Geometry and Topology ( 157 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.
Mark Stern : Introduction to p-harmonic forms, L^p Hodge theory, and L^p cohomology
- Geometry and Topology ( 11 Views )In this talk I will lay the foundations of the geometry of p-harmonic forms and L^p-Hodge theory. As an application, I will give strong evidence for (half of) a conjecture of Gromov on the L^p cohomology of negatively curved symmetric spaces.
Gábor Székelyhidi : On the positive mass theorem for manifolds with corners.
- Geometry and Topology ( 99 Views )A problem originally studied by P. Miao is whether the positive mass theorem holds on manifolds with certain singularities along a hypersurface. I will discuss an approach to this problem which uses the Ricci flow to smooth out the metric, so that one can apply the usual positive mass theorem. This allows for extending the rigidity statement in the zero mass case to higher dimensions, which was only known in the 3 dimensional case previously. This is joint work with D. McFeron.
Carla Cederbaum : The Newtonian Limit of Geometrostatics
- Geometry and Topology ( 99 Views )Geometrostatics is an important subdomain of Einstein's General Relativity. It describes the mathematical and physical properties of static isolated relativistic systems such as stars, galaxies or black holes. For example, geometrostatic systems have a well-defined ADM-mass (Chrusciel, Bartnik) and (if this is nonzero) also a center of mass (Huisken-Yau, Metzger) induced by a CMC-foliation at infinity. We will present surface integral formulae for these physical properties in general geometrostatic systems. Together with an asymptotic analysis, these can be used to prove that ADM-mass and center of mass 'converge' to the Newtonian mass and center of mass in the Newtonian limit $c\to\infty$ (using Ehler's frame theory). We will discuss geometric similarities of geometrostatic and classical static Newtonian systems along the way.
Kyle Hayden : Complex curves through a contact lens
- Geometry and Topology ( 118 Views )Every four-dimensional Stein domain has a height function whose regular level sets are contact three-manifolds. This allows us to study complex curves in the Stein domain via their intersection with these contact level sets, where we can comfortably apply three-dimensional tools. We use this perspective to characterize the links in Stein-fillable contact manifolds that bound complex curves in their Stein fillings. (Some of this is joint work with Baykur, Etnyre, Hedden, Kawamuro, and Van Horn-Morris.)
Anda Degeratu : Analysis on crepant resolutions of Calabi-Yau orbifolds
- Geometry and Topology ( 147 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.