Seminars and Colloquia by Series

Joint GT-UGA Seminar at UGA

Series
Geometry Topology Seminar
Time
Monday, February 6, 2017 - 14:30 for 2.5 hours
Location
UGA Room 303
Speaker
Dan Cristofaro-Gardiner and John Etnyre – Harvard and Georgia Tech
John Etnyre: "Embeddings of contact manifolds" Abstract: I will discuss recent results concerning embeddings and isotopies of one contact manifold into another. Such embeddings should be thought of as generalizations of transverse knots in 3-dimensional contact manifolds (where they have been instrumental in the development of our understanding of contact geometry). I will mainly focus on embeddings of contact 3-manifolds into contact 5-manifolds. In this talk I will discuss joint work with Ryo Furukawa aimed at using braiding techniques to study contact embeddings. Braided embeddings give an explicit way to represent some (maybe all) smooth embeddings and should be useful in computing various invariants. If time permits I will also discuss other methods for embedding and constructions one may perform on contact submanifolds. Dan Cristofaro-Gardiner: "Beyond the Weinstein conjecture" Abstract: The Weinstein conjecture states that any Reeb vector field on a closed manifold has at least one closed orbit. The three-dimensional case of this conjecture was proved by Taubes in 2007, and Hutchings and I later showed that in this case there are always at least 2 orbits. While examples exist with exactly two orbits, one expects that this lower bound can be significantly improved with additional assumptions. For example, a theorem of Hofer, Wysocki, and Zehnder states that a generic nondegenerate Reeb vector field associated to the standard contact structure on $S^3$ has either 2, or infinitely many, closed orbits. We prove that any nondegenerate Reeb vector field has 2 or infinitely many closed orbits as long as the associated contact structure has torsion first Chern class. This is joint work with Mike Hutchings and Dan Pomerleano.

Turaev-Viro invariants of links and the colored Jones polynomial

Series
Geometry Topology Seminar
Time
Wednesday, January 25, 2017 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Renaud Detcherry – Michigan State University –
In a recent conjecture by Tian Yang and Qingtao Chen, it has been observedthat the log of Turaev-Viro invariants of 3-manifolds at a special root ofunity grow proportionnally to the level times hyperbolic volume of themanifold, as in the usual volume conjecture for the colored Jonespolynomial.In the case of link complements, we derive a formula to expressTuraev-Viro invariants as a sum of values of colored Jones polynomial, andget a proof of Yang and Chen's conjecture for a few link complements. Theformula also raises new questions about the asymptotics of colored Jonespolynomials. Joint with Effie Kalfagianni and Tian Yang.

Point-pushing in the mapping class group

Series
Geometry Topology Seminar
Time
Monday, January 23, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Victoria Akin – University of Chicago
The point-pushing subgroup of the mapping class group of a surface with a marked point can be considered topologically as the subgroup that pushes the marked point about loops in the surface. Birman demonstrated that this subgroup is abstractly isomorphic to the fundamental group of the surface, \pi_1(S). We can characterize this point-pushing subgroup algebraically as the only normal subgroup inside of the mapping class group isomorphic to \pi_1(S). This uniqueness allows us to recover a description of the outer automorphism group of the mapping class group.

Cosmetic surgeries on homology spheres

Series
Geometry Topology Seminar
Time
Monday, January 9, 2017 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Huygens Ravelomanana – University of Georgia
Dehn surgery is a fundamental tool for constructing oriented 3-Manifolds. If we fix a knot K in an oriented 3-manifold Y and do surgeries with distinct slopes r and s, we can ask under which conditions the resulting oriented manifold Y(r) and Y(s) might be orientation preserving homeomorphic. The cosmetic surgery conjecture state that if the knot exterior is boundary irreducible then this can't happen. My talk will be about the case where Y is an homology sphere and K is an hyperbolic knot.

Polynomial functors and algebraic K-theory

Series
Geometry Topology Seminar
Time
Monday, December 5, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Akhil Mathew – Harvard University
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.

The universal quantum invariant and colored ideal triangulations

Series
Geometry Topology Seminar
Time
Friday, December 2, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
S. Suzuki – RIMS, Kyoto University
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

The Homfly skein and elliptic Hall algebras

Series
Geometry Topology Seminar
Time
Wednesday, November 30, 2016 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Peter Samuelson – Edinburgh –
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.)

Reconstructing polytopes from projections

Series
Geometry Topology Seminar
Time
Monday, November 28, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sergii Myroshnychenko – Kent State University –
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.

Homology cobordism and Dehn surgery

Series
Geometry Topology Seminar
Time
Monday, November 21, 2016 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Francesco Lin – Princeton
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.

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