## Luya Wang : Deformation inequivalent symplectic structures and Donaldsons four-six question

- Geometry and Topology ( 0 Views )Studying symplectic structures up to deformation equivalences is a fundamental question in symplectic geometry. Donaldson asked: given two homeomorphic closed symplectic four-manifolds, are they diffeomorphic if and only if their stabilized symplectic six-manifolds, obtained by taking products with CP^1 with the standard symplectic form, are deformation equivalent? I will discuss joint work with Amanda Hirschi on showing how deformation inequivalent symplectic forms remain deformation inequivalent when stabilized, under certain algebraic conditions. This gives the first counterexamples to one direction of Donaldson??s ??four-six? question and the related Stabilizing Conjecture by Ruan. In the other direction, I will also discuss more supporting evidence via Gromov-Witten invariants.

## Zhenyi Chen : A-infinity Sabloff Duality via the LSFT Algebra

- Geometry and Topology ( 0 Views )The Chekanov-Eliashberg dga is a powerful invariant for Legendrian links. Using augmentations of this dga, one can truncate its differential to produce linearized contact homology. About two decades ago, Sabloff established a duality in this setting, closely linked to the Poincaré duality of Lagrangian fillings. This truncation has since been generalized into a unital A-infinity category, Aug_+. In this talk, I will present new results that extend Sabloff duality from the level of cochain complexes to A-infinity bimodules over Aug_+. The key tool in this extension is Ng's LSFT algebra, which enlarges the Chekanov-Eliashberg dga. If time permits, I will also discuss how the LSFT algebra encodes additional homotopy coherent data, providing further insights into Sabloff duality.

## Calvin McPhail-Snyder : Towards quantum complex Chern-Simons theory

- Geometry and Topology ( 0 Views )I will discuss recent joint work (with N. Reshetikhin) defining invariants ? of knot (and link and tangle) exteriors with flat ???? connections. The construction is via a geometric version of the Reshetikhin-Turaev construction: it is algebraic and relies on the representation theory of quantum groups. In this talk I will instead focus on the properties of these invariants and explain why I think they are a good candidate for quantum Chern-Simons theory with noncompact gauge group SL??(??). I will also discuss a connection with (and a generalization of) the Volume Conjecture.

## Sergey Cherkis : Gravitational Instantons: the Tesseron Landscape

- Geometry and Topology ( 0 Views )Since their introduction in Euclidean quantum gravity in mid-70??s, hyperkaehler Gravitational Instantons (aka tesserons) found their use in string theory and in supersymmetric quantum field theory. Their classification was recently completed and now their parameter space is being explored. We propose a systematic program of realizing each of these spaces as a moduli space of monopoles: the monopolization program. Monopolization reveals the combinatorial and geometric structure of the parameter space of all these spaces, equips each space with various natural structures (tautological bundles, Dirac-type operators, etc), and connects different types of integrable systems associated to these gravitational instantons.

## Laura Wakelin : Finding characterising slopes for all knots

- Geometry and Topology ( 0 Views )A slope p/q is characterising for a knot K if the oriented homeomorphism type of the 3-manifold obtained by performing Dehn surgery of slope p/q on K uniquely determines the knot K. For any knot K, there exists a bound C(K) such that any slope p/q with |q|?C(K) is characterising for K. This bound has previously been constructed for certain classes of knots, including torus knots, hyperbolic knots and composite knots. In this talk, I will give an overview of joint work with Patricia Sorya in which we complete this realisation problem for all remaining knots.

## Tye Lidman : Cosmetic surgeries and Chern-Simons invariants

- Geometry and Topology ( 0 Views )Dehn surgery is a fundamental construction in topology where one removes a neighborhood of a knot from the three-sphere and reglues to obtain a new three-manifold. The Cosmetic Surgery Conjecture predicts two different surgeries on the same knot always gives different three-manifolds. We discuss how gauge theory, in particular, the Chern-Simons functional, can help approach this problem. This technique allows us to solve the conjecture in essentially all but one case. This is joint work with Ali Daemi and Mike Miller Eismeier.

## 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 Bülent Tosun.

## Mark Stern : Introduction to p-harmonic forms, L^p Hodge theory, and L^p cohomology

- Geometry and Topology ( 114 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.

## 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.

## 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 Almgren??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.

## 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 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.

## 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.

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

- Geometry and Topology ( 107 Views )Ossian Bonnet (1819??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 ??exceptional family?? (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 Cartan??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. I??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

## Kai Xu : pi_2-systolic inequalities for 3-manifolds with positive scalar curvature

- Geometry and Topology ( 100 Views )We discuss the following recent result of the speaker. Suppose a closed 3-manifold M has scalar curvature at least 1, and has nontrivial second homotopy group, and is not covered by the cylinder (S^2)*R. Then the pi_2-systole of M (i.e. the minimal area in the second homotopy group) is bounded by a constant that is approximately 5.44pi. If we include quotients of cylinder into consideration, then the best upper bound is weakened to 8_pi. This shows a topological gap in the pi_2-systolic inequality. We will discuss the ideas behind this theorem, as well as the proof using Huisken and Ilmanen??s weak inverse mean curvature flow.

## 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.

## 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.

## Igor Zelenko (Texas A&M U) : Gromovs h-principle for corank two distribution of odd rank with maximal first Kronecker index

- Geometry and Topology ( 150 Views )While establishing various versions of the h-principle for contact distributions (Eliashberg (1989) in dimension 3, Borman-Eliashberg-Murphy (2015) in arbitrary dimension, and even-contact contact (D. McDuff, 1987) distributions are among the most remarkable advances in differential topology in the last four decades, very little is known about analogous results for other classes of distributions, e.g. generic distributions of corank 2 or higher. The smallest dimensional nontrivial case of corank 2 distributions is Engel distributions, i.e. the maximally nonholonomic rank 2 distributions on $4$-manifolds. This case is highly nontrivial and was treated recently by Casals-Pérez-del Pino-Presas (2017) and Casals-Pérez-Presas (2017). In my talk, I will show how to use the method of convex integration in order to establish all versions of the h-principle for corank 2 distributions of arbitrary odd rank satisfying a natural generic assumption on the associated pencil of skew-symmetric forms. During the talk, I will try to give all the necessary background related to the method of convex integration in principle. This is the joint work with Milan Jovanovic, Javier Martinez-Aguinaga, and Alvaro del Pino.