## Yang Li : On the Donaldson-Scaduto conjecture

- Geometry and Topology ( 708 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 hyperkÃ¤hler manifold. We prove this conjecture by solving a real Monge-AmpÃ¨re 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.

## Calvin McPhail-Snyder : Making the Jones polynomial more geometric

- Geometry and Topology ( 370 Views )The colored Jones polynomials are conjectured to detect geometric information about knot complements, such as hyperbolic volume. These relationships ("volume conjectures") are known in a number of special cases but are in general quite mysterious. In this talk I will discuss a program to better understand them by constructing holonomy invariants, which depend on both a knot K and a representation of its knot group into SL_2(C). By defining a version of the Jones polynomial that knows about geometric data, we hope to better understand why the ordinary Jones polynomial does too. Along the way we can obtain more powerful quantum invariants of knots and other topological objects.

## Subhankar Dey : Cable knots are not thin

- Geometry and Topology ( 246 Views )Thurston's geometrization conjecture and its subsequent proof for Haken manifolds distinguish knots in S^3 by the geometries in the complement of the knots. While the definition of alternating knots make use of nice knot diagrams, Knot Floer homology, a knot invariant toolbox, defined by Ozsvath-Szabo and Rasumussen, generalizes the definition of alternating knots in the context of knot Floer homology and defines family of quasi-alternating knots which contains all alternating knots. Using Lipshitz-Ozsvath-Thurston's bordered Floer homology, we prove a partial affirmation of a folklore conjecture in knot Floer theory, which bridges these two viewpoints of looking at knots.

## Isaac Sundberg : The Khovanov homology of slice disks

- Geometry and Topology ( 236 Views )To a cobordism between links, Khovanov homology assigns a linear map that is invariant under boundary-preserving isotopy of the cobordism. In this talk, we study those maps arising from surfaces in the 4-ball and apply our findings to existence and uniqueness questions regarding slice disks bounding a given knot. This reflects joint works with Jonah Swann and Kyle Hayden.

## Jimmy Petean : On the Yamabe invariant of Riemannian products

- Geometry and Topology ( 219 Views )The Yamabe invariant of a closed manifold appears naturally when studying the total scalar curvature functional on the space of Riemannian metrics on the manifold. Computations are difficult, in particular in the positive case (when the manifold admits metrics of positive scalar curvarture, and there is no unicity of metrics of constant scalar curvature on a conformal class). In this talk I will review a little of what is known about the computation of the invariant and discuss some recent joint work with K. Akutagawa and L. Florit on the Yamabe constants of Riemannian products.

## Christina Tonnesen-Friedman : Canonical classes on admissible bundles

- Geometry and Topology ( 203 Views )For each KÂ¨ahler class on a compact KÂ¨ahler manifold there is a lower bound of the Calabi functional, which we call the ``potential energy''. Fixing the volume and letting the KÂ¨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 KÂ¨ahler class whose extremal KÂ¨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.

## Lev Rozansky : A categorification of the stable Witten-Reshetikhin-Turaev invariant of links in S2 x S1

- Geometry and Topology ( 182 Views )This work was done in close collaboration with M. Khovanov. The Witten-Reshetikhin-Turaev invariant Z(M,L;r) of a link L in a 3-manifold M is a seemingly random function of an integer r. However, for a small class of 3-manifolds constructed by identical gluing of two handlebodies (e.g., for S3 and for S2 x S1) and for sufficiently large values of r the ratio Z(M,L;r)/Z(M;r) is equal to a rational function J(M,L;q) of q evaluated at the first 2r-th root of unity. If M = S3, then J is the Jones polynomial. Khovanov categorified J(S3,L), that is, to a link L in S3 he assigned a homology H(L) with an extra Z-grading such that its graded Euler characteristic equals J(S3,L). We extend Khovanov's construction to links in S2 x S1 thus categorifying J(S2xS1,L). In his work on categorification of the Jones polynomial, Khovanov introduced special algebras H_n and assigned a H_m x H_n module to every (2m,2n)-tangle. We show that if a link L in S2 x S1 is presented as a closure of a (2n,2n)-tangle, then the Hochschild homology of its H_n bimodule is determined by the link itself and serves as a categorificaiton of J(S2xS1,L). Moreover, we show that this Hochschild homology can be approximated by Khovanov homology of the circular closure of the tangle within S3 by a high twist torus braid, thus providing a practical method of its computation.

## 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

## 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.

## Jason Lotay : Hyperkaehler metrics on a 4-manifold with boundary

- Geometry and Topology ( 158 Views )An oriented hypersurface in a hyperkaehler 4-manifold naturally inherits a coclosed coframing. Bryant showed that, in the real analytic case, any oriented 3-manifold with a coclosed coframing can always be locally Â?thickenedÂ? to a hyperkaehler 4-manifold, in an essentially unique way. This raises the natural question: when can these 3-manifolds with this structure arise as the boundary of a hyperkaehler 4-manifold? In particular, starting from a compact hyperkaehler 4-manifold with boundary, which deformations of the boundary structure can be extended to a hyperkaehler deformation of the interior? I will discuss recent progress on this problem, which is joint work with Joel Fine and Michael Singer.

## Ioana Suvaina : ALE Ricci flat Kahler surfaces

- Geometry and Topology ( 142 Views )The talk presents an explicit classification of the ALE Ricci flat K\"ahler surfaces, generalizing previous classification results of Kronheimer. The manifolds are related to a special class of deformations of quotient singularities of type $\mathbb C^2/G$, with $G$ a finite subgroup of $U(2)$. I will also explain the relation with the Tian-Yau construction of complete Ricci flat Kahler manifolds.

## Eylem Zeliha Yildiz : Braids in planar open books and fillable surgeries.

- Geometry and Topology ( 129 Views )We'll give a useful description of braids in $\underset{n}{\#}(S^1\times S^2)$ using surgery diagrams, which will allow us to address families of knots in lens spaces that admit fillable positive contact surgery. We also demonstrate that smooth $16$ surgery to the knot $P(-2,3,7)$ bounds a rational ball, which admits a Stein handlebody. This answers a question left open by Thomas Mark and BÃ¼lent Tosun.

## Henri Roesch : Proof of a Null Penrose Conjecture using a new Quasi-local Mass

- Geometry and Topology ( 122 Views )We define an explicit quasi-local mass functional which is nondecreasing along all foliations of a null cone (satisfying a convexity assumption). We use this new functional to prove the Null Penrose Conjecture under fairly generic conditions.

## Bulent Tosun : Legendrian and transverse knots in cabled knot types

- Geometry and Topology ( 121 Views )In this talk we will exhibit many new phenomena in the structure of Legendrian and transverse knots by giving a complete classification of all cables of the positive torus knots. We will also provide two structural theorems to ensure when cable of a Legendrian simple knot type is also Legendrian simple. Part of the results are joint work with John Etnyre and Douglas LaFountain

## Wenzhao Chen : Negative amphicheiral knots and the half-Alexander polynomial

- Geometry and Topology ( 115 Views )In this talk, we will study strongly negative amphicheiral knots - a class of knots with symmetry. These knots provide torsion elements in the knot concordance group, which are less understood than infinite-order elements. We will introduce the half-Alexander polynomial, an equivariant version of the Alexander polynomial for strongly negative amphicheiral knots, focusing on its applications to knot concordance. In particular, I will show how it facilitated the construction of the first examples of non-slice amphicheiral knots of determinant one. This talk is based on joint work with Keegan Boyle.

## 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.

## Robert Bryant : A Weierstrass representation for affine Bonnet surfaces

- Geometry and Topology ( 107 Views )Ossian Bonnet (1819â??1892) classified the surfaces in Euclidean 3-space that can be isometrically deformed without changing the mean curvature function H, showing that there are two types: the surfaces of constant mean curvature and a 4-dimensional â??exceptional familyâ?? (with variable mean curvature) that are now known as Bonnet surfaces. The corresponding problem in affine 3-space is much more difficult, and the full classification is still unknown. More than 10 years ago, I classified the affine surfaces that can isometrically deformed (with respect to the induced Blaschke metric) while preserving their affine mean curvature in a 3-dimensional family (the maximum dimension possible), showing that they depend on 2 functions of 1 variable in Cartanâ??s sense. When I gave a talk* in this seminar about these results on September 10, 2013, I only knew that these surfaces corresponded to pseudoholomorphic curves in a certain almost-complex surface. However, I have recently shown that the structure equations for these mysterious surfaces can be interpreted as describing holomorphic Legendrian curves in CP^3 subject to a natural positivity condition, and the integration corresponds to a flat sp(2,R) connection, i.e., they can be interpreted as a Lax pair, but of a very special kind, for which the integration can be effected explicitly. Iâ??ll explain these results and use them to show how the classical problem of determining the affine surfaces with constant affine mean curvature and constant Gauss curvature of the Blaschke metric can be explicitly integrated, which, heretofore, was unknown. * https://www4.math.duke.edu/media/watch_video.php?v=6948e657e69cadbaa1a6915335e9ea87

## David Shea Vela-Vick : Triple linking numbers, ambiguous Hopf invariants and integral formulas for three-component links

- Geometry and Topology ( 106 Views )To each three-component link in the 3-dimensional sphere we associate a characteristic map from the 3-torus to the 2-sphere, and establish a correspondence between the pairwise and Milnor triple linking numbers of the link and the Pontryagin invariants that classify its characteristic map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two- component link is the degree of its associated Gauss map from the 2- torus to the 2-sphere. When the pairwise linking numbers are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of the 3-sphere.

## Saman Habibi Esfahani : Gauge theory, from low dimensions to higher dimensions and back

- Geometry and Topology ( 105 Views )We start by recalling gauge theory and some of its applications in low-dimensional topology. We briefly discuss Donaldson-Thomas program to extend the methods of gauge theory to study higher-dimensional manifolds, specially Calabi-Yau 3-folds and G2-manifolds. Finally, we will see that the study of gauge theory in higher dimensions motivates new ideas and questions in low-dimensional topology.

## Yao Xiao : Equivariant Lagrangian Floer theory on compact toric manifolds

- Geometry and Topology ( 103 Views )We define an equivariant Lagrangian Floer theory on compact symplectic toric manifolds for the subtorus actions. We prove that the set of Lagrangian torus fibers (with weak bounding cochain data) with non-vanishing equivariant Lagrangian Floer cohomology forms a rigid analytic space. We can apply tropical geometry to locate such Lagrangian torus fibers in the moment map. We show that these Lagrangian submanifolds are nondisplaceable by equivariant Hamiltonian diffeomorphisms.

## Anna Skorobogatova : Area-minimizing currents: structure of singularities and uniqueness of tangent cones

- Geometry and Topology ( 79 Views )The problem of determining the size and structure of the interior singular set of area-minimizing surfaces has been studied thoroughly in a number of different frameworks, with many ground-breaking contributions. In the framework of integral currents, when the surface has higher codimension than 1, the presence of singular points with flat tangent cones creates an obstruction to easily understanding the interior singularities. Until recently, little was known in this direction, particularly for surfaces of dimension higher than two, beyond Almgrenâ??s celebrated dimension estimate on the interior singular set. In this talk I will discuss joint works with Camillo De Lellis and Paul Minter, where we establish (m-2)-rectifiability of the interior singular set of an m-dimensional area-minimizing integral current and classify tangent cones at \mathcal{H}^{m-2}-a.e. interior point.

## Lenny Ng : New algebraic invariants of Legendrian links

- Geometry and Topology ( 77 Views )For the past 25 years, a key player in contact topology has been the Floer-theoretic invariant called Legendrian contact homology. I'll discuss a package of new invariants for Legendrian knots and links that builds on Legendrian contact homology and is derived from rational symplectic field theory. This includes a Poisson bracket on Legendrian contact homology and a symplectic structure on augmentation varieties. Time permitting, I'll also describe an unexpected connection to cluster theory for a family of Legendrian links associated to positive braids. Parts of this are joint work in progress with Roger Casals, Honghao Gao, Linhui Shen, and Daping Weng.