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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

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

The compact transverse cross-sections of a cylinder over a central
ovaloid in Rn, n ≥ 3, with hyperplanes are central ovaloids. A similar result holds
for quadrics (level sets of quadratic polynomials in Rn, n ≥ 3). Their compact
transverse cross-sections with hyperplanes are ellipsoids, which are central ovaloids.
In R3, Blaschke, Brunn, and Olovjanischnikoff found results for compact convex
surfaces that motivated B. Solomon to prove that these two kinds of examples
provide the only complete, connected, smooth surfaces in R3, whose ovaloid cross sections are central. We generalize that result to all higher dimensions, proving: If M^(n-1), n >= 4, is a complete, connected smooth hypersurface of R^n, which intersects at least one hyperplane transversally along an ovaloid, and every such ovaloid on M is central, then M is either a cylinder over a central ovaloid or a quadric.

Series: Geometry Topology Seminar

We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show that the minimal weak symplectic filling is unique up to homeomorphism. (This is a joint work with Youlin Li.)