Geometry and Topology

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I will give a brief survey of concordance in high-dimensional knot theory and how slice results have classically been obtained in this setting with the aid of surgery theory. Time permitting, I will then discuss an example of how some non-abelian slice obstructions come into the picture for 1-knots, as intuition for the seminar talk about L^2 invariants.

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The question of which high-dimensional knots are slice was entirely solved by Kervaire and Levine. Compared to this, the question of which knots are doubly slice in high-dimensions is still a largely open problem. Ruberman proved that in every dimension, some version of the Casson-Gordon invariants can be applied to obtain algebraically doubly slice knots that are not doubly slice.

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The classical degree-genus formula computes the genus of a nonsingular algebraic curve in the complex projective plane. The well-known Thom conjecture posits that this is a lower bound on the genus of smoothly embedded, oriented and connected surface in CP^2. The conjecture was first proved twenty-five years ago by Kronheimer and Mrowka, using Seiberg-Witten invariants. In this talk, we will describe a new proof of the conjecture that combines contact geometry with the novel theory of bridge trisections of
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The semi-cubical cusp which is formed in the bottom of a mug when you shine a light on it is an everyday example of a caustic. In this talk we will become familiar with the singularities of Lagrangian and Legendrian fronts, also known as caustics in the mathematics literature, which have played an important role in symplectic and contact topology since the work of Arnold and his collaborators.
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We will present an h-principle for the simplification of singularities of Lagrangian and Legendrian fronts. The h-principle says that if there is no homotopy theoretic obstruction to simplifying the singularities of tangency of a Lagrangian or Legendrian submanifold with respect to an ambient foliation by Lagrangian or Legendrian leaves, then the simplification can be achieved by means of a Hamiltonian isotopy. We will also discuss applications of the h-principle to symplectic and contact topology.
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The soul of a complete, noncompact, connected Riemannian manifold (M; g) of non-negative sectional curvature is a compact, totally convex, totally geodesic submanifold such that M is diffeomorphic to the normal bundle of the soul. Hence, understanding of the souls of M can reduce the study of M to the study of a compact set. Also, souls are metric invariants, so understanding how they behave under deformations of the metric is useful to analyzing the space of metrics on M. In particular, little is understood about the case when M = R2 .
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In this talk a curve complex HC(S) closely related to the "Cyclic Cycle Complex" (Bestvina-Bux-Margalit) and the "Complex of Cycles" (Hatcher) is defined for an orientable surface of genus g at least 2. The main result is a simple algorithm for calculating distances and constructing quasi-geodesics in HC(S). Distances between two vertices in HC(S) are related to the "Seifert genus" of the corresponding link in S x R, and behave quite differently from distances in other curve complexes with regards to subsurface projections.
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Weinstein cobordisms give a natural relationship on the set of Weinstein domains. Flexible Weinstein domains are minimal with respect to this relationship. In this talk, I will use these minimal domains to construct maximal Weinstein domains: any two high-dimensional Weinstein domains with the same topology are Weinstein subdomains of a maximal Weinstein domain also with the same topology. Using this construction, a wide range of new Weinstein domains can be produced, for example exotic cotangent bundles of spheres containing many different closed exact Lagrangians.
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I will describe the h-principle philosophy and explain some recent developments on the flexible side of symplectic topology, including Murphy's h-principle for loose Legendrians and Cieliebak and Eliashberg's construction of flexible symplectic manifolds in high-dimensions.

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