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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.)

Series: Geometry Topology Seminar

I will describe new
techniques for computing the homology of braid groups with coefficients
in certain exponential coefficient systems. An unexpected side of this
story (at least to me) is a connection with the cohomology of certain
braided Hopf
algebras — quantum shuffle algebras and Nichols algebras — which are
central to the classification of pointed Hopf algebras and quantum
groups. We can apply these tools to get a bound on the growth of the
cohomology of Hurwitz moduli spaces of branched covers
of the plane in certain cases. This yields a weak form of Malle’s
conjecture on the distribution of fields with prescribed Galois group in
the function field setting. This is joint work with Jordan Ellenberg
and TriThang Tran.

Series: Geometry Topology Seminar

We consider two knot diagrams to be equivalent if they are isotopic without Reidemeister moves, and prove a method for determining if the equivalence class of a knot diagram contains a representative that is the Lagrangian projection of a Legendrian knot. This work gives us a new tool for determining if a Legendrian knot can be de-stabilized.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

In this talk we first introduce a new "singularity-free" approach to the proof of Seidel's long exact sequence, including the fixed-point version. This conveniently generalizes to Dehn twists along Lagrangian submanifolds which are rank one symmetric spaces and their covers, including RPn and CPn, matching a mirror prediction due to Huybrechts and Thomas. The idea of the proof can be interpreted as a "mirror" of the construction in algebraic geometry, realized by a new surgery and cobordism construction. This is a joint work with Cheuk-Yu Mak.

Series: Geometry Topology Seminar

Birman and Hilden ask: given finite branched cover X over the 2-sphere, does every homeomorphism of the sphere lift to a homeomorphism of X? For covers of degree 2, the answer is yes, but the answer is sometimes yes and sometimes no for higher degree covers. In joint work with Ghaswala, we completely answer the question for cyclic branched covers. When the answer is yes, there is an embedding of the mapping class group of the sphere into a finite quotient of the mapping class group of X. In a family where the answer is no, we find a presentation for the group of isotopy classes of homeomorphisms of the sphere that do lift, which is a finite index subgroup of the mapping class group of the sphere. Our family introduces new examples of orbifold Picard groups of subloci of Teichmuller space that are finitely generated but not cyclic.