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Series: Geometry Topology Seminar

The Grothendieck group K_0 of a commutative ring is well-known to be a \lambda-ring: although the exterior powers are non-additive, they induce maps on K_0 satisfying various universal identities. The \lambda-operations are known to give homomorphisms on higher K-groups. In joint work in progress with Barwick, Glasman, and Nikolaus, we give a general framework for such operations. Namely, we show that the K-theory space is naturally functorial with respect to polynomial functors, and describe a universal property of the extended K-theory functor. This extends an earlier algebraic result of Dold for K_0.

Series: Geometry Topology Seminar

The Drinfeld double of a finite dimensional Hopf algebra is a
quasi-triangular Hopf algebra with the canonical element as the universal R
matrix, and we obtain a ribbon Hopf algebra by adding the ribbon element.
The universal quantum invariant is an invariant of framed links, and is
constructed diagrammatically using a ribbon Hopf algebra. In that
construction, a copy of the universal R matrix is attached to each positive
crossing, and invariance under the Reidemeister III move is shown by the
quantum Yang-Baxter equation of the universal R matrix.
On the other hand, R. Kashaev showed that the Heisenberg double has the
canonical element (the universal S matrix) satisfying the pentagon
relation. In this talk we reconstruct the universal quantum invariant using
Heisenberg double, and extend it to an invariant of colored ideal
triangulations of the complement. In this construction, a copy of the
universal S matrix is attached to each tetrahedron and the invariance under
the colored Pachner (2,3) move is shown by the pentagon equation of the
universal S matrix

Series: Geometry Topology Seminar

The Homfly skein algebra of a surface is defined using links in
thickened surfaces modulo local "skein" relations. It was shown by
Turaev that this quantizes the Goldman symplectic structure on the
character varieties of the surface. In this talk we give a complete
description of this algebra for the torus. We also show it is
isomorphic to the elliptic Hall algebra of Burban and Schiffmann,
which is an algebra whose elements are (formal sums of) sheaves on an
elliptic curve, with multiplication defined by counting extensions of
such sheaves. (Joint work with H. Morton.)

Series: Geometry Topology Seminar

We are going to discuss one of the open problems of geometric tomography about projections. Along with partial previous results, the proof of the problem below will be investigated.Let $2\le k\le d-1$ and let $P$ and $Q$ be two convex polytopes in ${\mathbb E^d}$. Assume that their projections, $P|H$, $Q|H$, onto every $k$-dimensional subspace $H$, are congruent. We will show that $P$ and $Q$ or $P$ and $-Q$ are translates of each other. If the time permits, we also will discuss an analogous result for sections by showing that $P=Q$ or $P=-Q$, provided the polytopes contain the origin in their interior and their sections, $P \cap H$, $Q \cap H$, by every $k$-dimensional subspace $H$, are congruent.

Series: Geometry Topology Seminar

We discuss a few applications of Pin(2)-monopole Floer homology to problems in homology cobordism. Our main protagonists are (connected sums of) homology spheres obtained by surgery on alternating and L-space knots with Arf invariant zero.

Series: Geometry Topology Seminar

Khovanov homology is a powerful and computable homology theory for links which extends to tangles and tangle cobordisms. It is closely, but perhaps mysteriously, related to many flavors of Floer homology. Szabó has constructed a combinatorial spectral sequence from Khovanov homology which (conjecturally) converges to a Heegaard Floer-theoretic object. We will discuss work in progress to extend Szabó’s construction to an invariant of tangles and surfaces in the four-sphere.

Series: Geometry Topology Seminar

We show that an embedding of a (small) ball into a contact manifold is contact if and only if it preserves the (modified) shape invariant. The latter is, in brief, the set of all cohomology classes that can be represented by the pull-back (to a closed one-form) of a contact form by a coisotropic embedding of a fixed manifold (of maximal dimension) and of a given homotopy type. The proof is based on displacement information about (non)-Lagrangian submanifolds that comes from J-holomorphic curve methods (and gives topological invariants), and the construction of a coisotropic torus whose image (under a given embedding that is not contact) admits a transverse contact vector field (i.e. a convex surface in dimension 3). The definition of shape preserving does not involve derivatives and is preserved by uniform convergence (on compact subsets). As a consequence, we prove C^0-rigidity of contact embeddings (and diffeomorphisms). The underlying ideas are adaptations of symplectic techniques to contact manifolds that, in contrast to symplectic capacities, work well in the contact setting; the heart of the proof however uses purely contact topological methods.

Series: Geometry Topology Seminar

Trisections of 4-manifolds relative to their boundary were introduced by Gay and Kirby in 2012. They are decompositions of 4-manifolds that induce open book decomposition in the bounding 3-manifolds. This talk will focus on diagrams of relative trisections and will be divided in two. In the first half I will focus on trisections as fillings of open book decompositions and I will present different fillings of different open book decompositions of the Poincare homology sphere. In the second half I will show examples of trisections of pieces of some of the surgery techniques that result in exotic 4-manifolds.

Series: Geometry Topology Seminar

A Stein manifold is a complex manifold with particularly nice convexity properties. In real dimensions above 4, existence of a Stein structure is essentially a homotopical question, but for 4-manifolds the situation is more subtle. An important question that has been circulating among contact and symplectic topologist for some time asks: whether every contractible smooth 4-manifold admits a Stein structure? In this talk we will provide examples that answer this question negatively. Moreover, along the way we will provide new evidence to a closely related conjecture of Gompf, which asserts that a nontrivial Brieskorn homology sphere, with either orientation, cannot be embedded in complex 2-space as the boundary of a Stein submanifold.

Series: Geometry Topology Seminar