## Robert Bryant : The affine Bonnet problem

- Geometry and Topology ( 296 Views )The classical Euclidean problem studied by Bonnet in the 19th century was to determine whether, and in how many ways, a Riemannian surface can be isometrically embedded into Euclidean 3-space so that its mean curvature is a prescribed function. He found that, generically, specifying a metric and mean curvature admitted no solution but that there are special cases in which, not only are there solutions, but there are even 1-parameter families of distinct (i.e., mutually noncongruent) solutions. Much later, these Â?Bonnet surfacesÂ? were found to be intimately connected with integrable systems and Lax pairs. In this talk, I will consider the analogous problem in affine geometry: To determine whether, and in how many ways, a surface endowed with a Riemannian metric g and a function H can be immersed into affine 3-space in such a way that the induced Blaschke metric is g and the induced affine mean curvature is H. This affine problem is, in many ways, richer and more interesting than the corresponding Euclidean problem. I will classify the pairs (g,H) that display the greatest flexibility in their solution space and explain what is known about the (suspected) links with integrable systems and Lax pairs.

## Curtis Porter : Spinning Black Holes and CR 3-Folds

- Geometry and Topology ( 276 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.

## Amit Einav : Entropic Inequality on the Sphere

- Geometry and Topology ( 222 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 Â?almostÂ? 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.

## Mark Stern : Nahm transforms and ALF Spaces

- Geometry and Topology ( 175 Views )In this talk we consider the moduli space of Yang-Mills instantons on the family of hyperkahler 4 manifolds known as multi-center TaubNUT spaces. We describe the Nahm transform for flat manifolds. Then we sketch its extension to the above hyperkahler family, where it defines an isometry between the moduli space of instantons on the multi-center TaubNUT and the moduli space of solutions of a rococo system of ordinary differential equations. This is joint work with Sergey Cherkis and Andres Larrain Hubach

## Adam Levine : Heegaard Floer Homology and Closed Exotic 4-Manifolds

- Geometry and Topology ( 171 Views )We discuss new methods for using the Heegaard Floer homology of hypersurfaces to distinguish between smooth closed 4-manifolds that are homeomorphic but non-diffeomorphic. Specifically, for a 4-manifold X with b_1(X)=1, the minimum rank of the reduced Heegaard Floer homology of any embedded 3-manifold X representing a generator of H_1(X) gives a diffeomorphism invariant of X. We use this invariant to distinguish certain infinite families of exotic 4-manifolds that cannot be distinguished by previously known techniques. Using related ideas, we also provide the first known examples of (non-simply-connected) exotic 4-manifolds with negative definite intersection form. This is joint work with Tye Lidman and Lisa Piccirillo.

## Vera VÃ©rtesi : Knots in contact 3--manifolds

- Geometry and Topology ( 168 Views )In this talk I will give a purely combinatorial description of Knot Floer Homology for knots in the three-sphere (Manolescu-OzsvÃ¡th-SzabÃ³-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-OzsvÃ¡th-Thurston). Moreover it leads to the construction of an infinite family of non-transversely simple knot-types (VÃ©rtesi).

## Ken Jackson : Numerical Methods for the Valuation of Synthetic Collateralized Debt Obligations (CDOs)

- Geometry and Topology ( 167 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.

## Ziva Myer : Product Structures for Legendrian Submanifolds with Generating Families

- Geometry and Topology ( 166 Views )In contact topology, invariants of Legendrian submanifolds in 1-jet spaces have been obtained through a variety of techniques. I will discuss how I am enriching one Morse-theoretic invariant, Generating Family Cohomology, to an A-infinity algebra by constructing product maps. The construction uses moduli spaces of Morse flow trees: spaces of intersecting gradient trajectories of functions whose critical points encode Reeb chords of the Legendrian submanifold. I will focus my talk on the construction of a 2-to-1 product and discuss how it lays the foundation for the A-infinity algebra.

## Justin Sawon : Lagrangian fibrations by Jacobians of low genus curves

- Geometry and Topology ( 157 Views )The Beauville-Mukai integrable system is a well-known Lagrangian fibration, i.e., a holomorphic symplectic manifold fibred by Lagrangian complex tori. It is constructed by beginning with a complete linear system of curves on a K3 surface, and then taking the compactified relative Jacobian of the family of curves. One may ask whether other families of curves yield Lagrangian fibrations in this way. Markushevich showed that this is not the case in genus two: a Lagrangian fibration by Jacobians of genus two curves must be a Beauville-Mukai system. We generalize his result to genus three curves, and also to non-hyperelliptic curves of genus four and five.

## Vladimir Matveev : Geodesically equivalent metrics in the large: Beltrami and Schouten problems.

- Geometry and Topology ( 155 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).

## Yu Wang : Quantitative stratification of stationary Yang-Mills and recent progress on global gauge problem

- Geometry and Topology ( 152 Views )Given a stationary Yang-Mills connection A, we are interested in studying its singular structure. In this talk we introduce a quantitative way to stratify the singular sets. Our main results include a Minkowski Volume estimate and the rectifiability of this quantitative stratification, which leads to the rectifiability of the classical stratifications S^k(A) for all integer k. To be precise, we first recall certain background preliminaries needed for this talk. After giving the statements of the main results, I will briefly describe the machinary used in the proof, and explain the new points and the major difficulty that we have faced. The main results in the talk are based on a work by myself last year. If time allows I will further discuss some open problems regarding global gauge in this field, and recent progress in those directions made jointly with Aaron Naber.

## Graeme Wilkin : Morse theory and stable pairs

- Geometry and Topology ( 151 Views )In the early 1980s Atiyah and Bott described a new approach to studying the cohomology of the moduli space of stable bundles: the equivariant Morse theory of the Yang-Mills functional. There are many other interesting moduli spaces that fit into a similar framework, however the catch is that the total space is singular, and it is not obvious how to construct the Morse theory of the appropriate functional. In this talk I will describe how to get around these difficulties for the moduli space of stable pairs, for which we prove a Kirwan surjectivity theorem and give a Morse-theoretic interpretation of the change in cohomology due to a flip. This builds upon earlier work with George Daskalopoulos, Jonathan Weitsman and Richard Wentworth for rank 2 Higgs bundles.

## Paul Norbury : Magnetic monopoles on manifolds with boundary

- Geometry and Topology ( 151 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.

## Mark Stern : Introduction to nonlinear harmonic forms.

- Geometry and Topology ( 151 Views )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.

## Jesse Madnick : The Mean Curvature of (Co)Associative Submanifolds

- Geometry and Topology ( 144 Views )In flat R^7, two classes of submanifolds stand out: the associative 3-folds and coassociative 4-folds, which enjoy the remarkable property of being area-minimizing in their homology class. In fact, these submanifolds make sense in any 7-manifold with a G2-structure, and it is natural to ask: Under what conditions to do they continue to be minimal? We answer this question by deriving pleasantly simple formulas for their mean curvature. Time permitting, we will explain how these formulas suggest new avenues for the construction of minimal submanifolds of high codimension. This is joint work with Gavin Ball.

## Marcos Jardim : On the spectrum of the Dirac operator and the Dolbeault Laplacian on Kahler manifolds

- Geometry and Topology ( 133 Views )There exists a large literature on lower bounds for the spectrum of the Dirac operator and the de Rham Laplacian on Riemannian manifolds. In this talk, we will consider the twisted Dirac operator and the twisted Dolbeault Laplacian on Kahler manifolds, and study how the spectrum changes with the coupling connection. We give lower bounds for their spectrum, showing that they are attained in the case of Riemann surfaces. This is joint work with Rafael F. Leao. The talk will be based on those two preprints, 0807.0813 and 0706.0878

## Kyle Hayden : Complex curves through a contact lens

- Geometry and Topology ( 124 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.)

## Bianca Santoro : Complete Kahler metrics on crepant resolutions of singular Calabi-Yau spaces

- Geometry and Topology ( 123 Views )In this talk, we plan to explain some existence results for complete Ricci-flat \kahler metrics on crepant resolutions of singularities. The method allows us to provide a wider class of examples of complete Ricci-flat Kahler metrics with richer topoplogy at infinity.

## Michael Lipnowski : The Seiberg-Witten equations and the length spectrum of hyperbolic three-manifolds

- Geometry and Topology ( 120 Views )We exhibit examples of hyperbolic three-manifolds for which the Seiberg-Witten equations do not admit any irreducible solution. Our approach relies hyperbolic geometry in an essential way; it combines an explicit upper bound for the first eigenvalue on coexact 1-forms \lambda_1^* on rational homology spheres which admit irreducible solutions together with a version of the Selberg trace formula relating the spectrum of the Laplacian on coexact 1-forms with the volume and complex length spectrum of a hyperbolic three-manifold. Using these relationships, we also provide precise certified numerical bounds on \lambda_1^* for several hyperbolic rational homology spheres.

## David Rose : Quantum knot invariants and Howe dualities

- Geometry and Topology ( 119 Views )We'll review the quantum sl_n knot invariants and their description via MOY calculus, as well as work of Cautis-Kamnitzer-Licata-Morrison showing how these invariants arise naturally from a duality between sl_n and sl_m called skew Howe duality. We'll then discuss work (joint with Aaron Lauda and Hoel Queffelec) categorifying this result to give elementary constructions of Khovanov and Khovanov-Rozansky knot homology. Time permitting, we'll also discuss work (joint with Daniel Tubbenhauer) relating symmetric Howe duality to the colored Jones polynomial, and giving a new diagrammatic method for computing this invariant.

## Akram Alishahi : Trivial tangles, compressible surfaces and Floer homology

- Geometry and Topology ( 118 Views )Heegaard Floer homology has different extensions for 3-manifolds with boundary. In this talk, we will recall some basics of these extensions and explain how they can be used to give a computationally effective way for detecting boundary parallel components of tangles, and existence of homologically essential compressing disks. The fact that these are checkable by computer, is based on the factoring algorithm of Lipshitz-Ozsvath-Thurston for computing bordered Floer homology, and our extension of it to compute bordered-sutured Floer homology. This is joint work with Robert Lipshitz.

## Matt Hogancamp : Categorical diagonalization of the full twist.

- Geometry and Topology ( 118 Views )I will discuss recent joint work with Ben Elias in which we introduce a theory of diagonalization of functors. Our main application is the diagonalization of the the Rouquier complex associated to full-twist braid, acting on the category of Soergel bimodules. The ``eigenprojections'' yield categorified Young symmetrizers, which are related to the flag Hilbert scheme by a beautiful recent conjecture of Gorsky-Rasmussen. Finally, I will mention a relationship with stable homology of torus links, which was recently investigated by myself and Michael Abel.

## John Baldwin : A combinatorial spanning tree model for delta-graded knot Floer homology

- Geometry and Topology ( 117 Views )I'll describe a new combinatorial method for computing the delta-graded knot Floer homology of a link in S^3. Our construction comes from iterating an unoriented skein exact triangle discovered by Manolescu, and yields a chain complex for knot Floer homology which is reminiscent of that of Khovanov homology, but is generated (roughly) by spanning trees of the black graph of the link. This is joint work with Adam Levine.

## Melissa Zhang : Annular Khovanov homology and 2-periodic links

- Geometry and Topology ( 116 Views )I will exhibit a spectral sequence from the annular Khovanov homology of a 2-periodic link to that of its quotient, which in turn proves rank inequalities and decategorifies to polynomial congruences. While previous work used heavier algebraic machinery to prove this rank inequality in a particular sl_2 weight space grading, we instead mimic Borel's construction of equivariant cohomology and employ grading considerations to give a combinatorial proof of the rank inequality for all quantum and sl_2 weight space gradings. Curiously, the same methods suggest a similar spectral sequence relating the Khovanov homology of a 2-periodic link and the annular Khovanov homology of its quotient link. We'll discuss partial results on this front.

## Shmuel Weinberger : Aspherical manifolds whose fundamental groups have center

- Geometry and Topology ( 115 Views )I will discuss a conjecture of Conner and Raymond that any aspherical manifold whose fundamental group has center possesses a circle action, and put it into the context of earlier work and conjectures of Borel and others.

## Christopher Cornwell : Polynomials, grid diagrams, and Legendrian knots in lens spaces

- Geometry and Topology ( 112 Views )We discuss a HOMFLY polynomial invariant for links in lens spaces. We then show how this polynomial is related to the contact invariants of Legendrian and transverse links in lens spaces having a certain tight contact structure. In fact, we generalize a result of Ng, casting Bennequin-type inequalities in such contact lens spaces into a general framework.

## Douglas LaFountain : Deligne-Mumford and the space of filtered screens

- Geometry and Topology ( 112 Views )For a genus g surface with s > 0 punctures and 2g+s > 2, decorated Teichmuller space (DTeich) is a trivial R_+^s-bundle over the usual Teichmuller space, where the fiber corresponds to families of horocycles peripheral to each puncture. As proved by R. Penner, DTeich admits a mapping class group-invariant cell decomposition, which then descends to a cell decomposition of Riemann's moduli space. In this talk we introduce a new cellular bordification of DTeich which is also MCG-invariant, namely the space of filtered screens. After an appropriate quotient, we obtain a cell decomposition for a new compactification of moduli space, which is shown to be homotopy equivalent to the Deligne-Mumford compactification. This work is joint with R. Penner.

## Adam Levine : Heegaard Floer invariants for homology S^1 x S^3s

- Geometry and Topology ( 109 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.

## Chris Kottke : Partial compactification and metric asymptotics of monopoles

- Geometry and Topology ( 108 Views )I will describe a partial compactification of the moduli space, M_k, of SU(2) magnetic monopoles on R^3, wherein monopoles of charge k decompose into widely separated `monopole clusters' of lower charge going off to infinity at comparable rates. The hyperkahler metric on M_k has a complete asymptotic expansion, the leading terms of which generalize the asymptotic metric discovered by Bielawski, Gibbons and Manton in the case that the monopoles are all widely separated. This is joint work with M. Singer, and is part of a larger work in progress with R. Melrose and K. Fritzsch to fully compactify the M_k as manifolds with corners and determine their L^2 cohomology.

## Dan Rutherford : Cellular compuation of Legendrian contact homology in dimension 2.

- Geometry and Topology ( 107 Views )This is joint work with Mike Sullivan. We consider a Legendrian surface, $L$, in $R^5$ (or more generally in the 1-jet space of a surface). Such a Legendrian can be conveniently presented via its front projection which is a surface in $R^3$ that is immersed except for certain standard singularities. We associate a differential graded algebra (DGA) to $L$ by starting with a cellular decomposition of the base projection (to $R^2$) of $L$ that contains the projection of the singular set of $L$ in its 1-skeleton. A collection of generators is associated to each cell, and the differential is determined in a formulaic manner by the nature of the singular set above the boundary of a cell. Our motivation is to give a cellular computation of the Legendrian contact homology DGA of $L$. In this setting, the construction of Legendrian contact homology was carried out by Etnyre-Ekholm-Sullvan with the differential defined by counting holomorphic disks in $C^2$ with boundary on the Lagrangian projection of $L$. Equivalence of our DGA with LCH may be established using work of Ekholm on gradient flow trees. Time permitting, we will discuss constructions of augmentations of the cellular DGA from two parameter families of functions.