Seminars and Colloquia by Series

Strict hyperbolization and special cubulation

Series
Geometry Topology Seminar
Time
Monday, April 25, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
skies 006
Speaker
Ruffoni, Lorenzo Tufts University

Abstract: Gromov introduced some “hyperbolization” procedures, i.e. some procedures that turn a given polyhedron into a space of non-positive curvature. Charney and Davis developed a refined “strict hyperbolization” procedure that outputs a space of strictly negative curvature. Their procedure has been used to construct new examples of manifolds and groups with negative curvature, and other prescribed features. We construct actions of the resulting groups on CAT(0) cube complexes. As an application, we obtain that they are virtually special, hence linear over the integers and residually finite. This is joint work with J. Lafont.

Relating the untwisting and surgery description numbers

Series
Geometry Topology Seminar
Time
Monday, April 18, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Samantha AllenUGA

The untwisting number of a knot K is the minimum number of null-homologous full twists required to unknot K. The surgery description number of K can be defined similarly, allowing for multiple full twists in a single twisting region. We can find no examples of knots in the literature where these two invariants are not equal. In this talk, I will provide the first known example where untwisting number and surgery description number are not equal and discuss challenges to distinguishing these invariants in general.  This will involve an exploration of the existing obstructions (often Heegaard-Floer theoretic) as well as the algebraic versions of these invariants.  In addition, we show the surprising result that the untwisting number of a knot is at most three times its surgery description number.  This work is joint with Kenan Ince, Seungwon Kim, Benjamin Ruppik, and Hannah Turner.

Upsilon invariant for graphs and homology cobordism group of homology cylinders

Series
Geometry Topology Seminar
Time
Monday, April 11, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
skies 006
Speaker
Akram AlishahiUGA

Upsilon is an invariant of knots defined using knot Floer homology by Ozsváth, Szabó and Stipsicz. In this talk, we discuss a generalization of their invariant for embedded graphs in rational homology spheres satisfying specific properties. Our construction will use a generalization of Heegaard Floer homology for “generalized tangles” called tangle Floer homology. As a result, we get a family of homomorphisms from the homology cobordism group of homology cylinders (over a surface of genus 0), which is an enlargement of the mapping class group defined by Graoufaldis and Levine. 

Connected sum formula of embedded contact homology

Series
Geometry Topology Seminar
Time
Monday, April 4, 2022 - 14:00 for
Location
Skiles 006
Speaker
Luya WangUniversity of California, Berkeley

The contact connected sum is a well-understood operation for contact manifolds. I will discuss work in progress on how pseudo-holomorphic curves behave in the symplectization of the 3-dimensional contact connected sum, and as a result the connected sum formula of embedded contact homology. 
 

Complex Ball Quotients and New Symplectic 4-Manifolds with Nonnegative Signatures

Series
Geometry Topology Seminar
Time
Tuesday, March 29, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Sumeyra SakalliUniversity of Arkansas

Please Note: Note this talk is at a different time and day

We first construct a complex surface with positive signature, which is a ball quotient. We obtain it as an abelian Galois cover of CP^2 branched over the Hesse arrangement. Then we analyze its fibration structure, and by using it we build new symplectic and also non-symplectic exotic 4-manifolds with positive signatures.

 

In the second part of the talk, we discuss Cartwright-Steger surfaces, which are also ball quotients. Next, we present our constructions of new symplectic and non-symplectic exotic 4-manifolds with non-negative signatures that have the smallest Euler characteristics in the so-called ‘arctic region’ on the geography chart.

 

More precisely, we prove that there exist infinite families of irreducible symplectic and infinite families of irreducible non-symplectic, exotic 4-manifolds that have the smallest Euler characteristics among the all known simply connected 4-manifolds with nonnegative signatures and with more than one smooth structures. This is a joint work with A. Akhmedov and S.-K. Yeung.

Quasi-morphisms on Surface Diffeomorphism groups

Series
Geometry Topology Seminar
Time
Monday, March 28, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Jonathan Bowden

We discuss the problem of constructing quasi-morphisms on the group of diffeomorphisms of a surface that are isotopic to the identity, thereby resolving a problem of Burago-Ivanov-Polterovich from the mid 2000’s. This is achieved by considering a new kind of curve graph, in analogy to the classical curve graph first studied by Harvey in the 70’s, on which the full diffeomorphism group acts isometrically. Joint work with S. Hensel and R. Webb. 

The Grand Arc Graph -- A "curve graph" for infinite-type surfaces

Series
Geometry Topology Seminar
Time
Monday, March 14, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Assaf Bar-NatanUniversity of Toronto

In this talk, I will be defining the grand arc graph for infinite-type surfaces. This simplicial graph is motivated by the works of Fanoni-Ghaswala-McLeay, Bavard, and Bavard-Walker to define an infinite-type analogue of the curve graph. As in these earlier works, the grand arc graph is connected, (oftentimes) infinite-diameter, and (sometimes) delta hyperbolic. Moreover, the mapping class group acts on it by isometries, and the action is continuous on the visible boundary. If there's time, this talk will degenerate into open speculation about what the boundary looks like and what we can do with it.

An invariant for families of contact structures in monopole Floer homology

Series
Geometry Topology Seminar
Time
Monday, March 7, 2022 - 14:00 for
Location
Skiles 006
Speaker
Juan Muñoz-EchánizColumbia University

The contact invariant, introduced by Kronheimer and Mrowka,
is an element in the monopole Floer homology of a 3-manifold which is
canonically attached to a contact structure. I will describe an
application of monopole Floer homology and the contact invariant to
study the topology of spaces of contact structures and
contactomorphisms on 3-manifolds. The main new tool is a version of
the contact invariant for families of contact structures.
 

Finite-order mapping classes of del Pezzo surfaces

Series
Geometry Topology Seminar
Time
Monday, February 28, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Seraphina LeeUniversity of Chicago

Let $M$ be the underlying smooth $4$-manifold of a degree $d$ del Pezzo surface. In this talk, we will discuss two related results about finite subgroups of the mapping class group $\text{Mod}(M) := \pi_0(\text{Homeo}^+(M))$. A motivating question for both results is the Nielsen realization problem for $M$: which finite subgroups $G$ of $\text{Mod}(M)$ have lifts to $\text{Diff}^+(M) \leq \text{Homeo}^+(M)$ under the quotient map $\pi: \text{Homeo}^+(M) \to \text{Mod}(M)$? For del Pezzo surfaces $M$ of degree $d \geq 7$, we will give a complete classification of such finite subgroups. Furthermore, we will give a classification of, and a structure theorem for, all involutions in $\text{Mod}(M)$ for all del Pezzo surfaces $M$. This yields a positive solution to the Nielsen realization problem for involutions on $M$ and a connection to Bertini's classification of birational involutions of $\mathbb{CP}^2$ (up to conjugation by birational automorphisms of $\mathbb{CP}^2$).

Symplectic Lefschetz fibrations, rational blowdowns, and small exotic 4-manifolds

Series
Geometry Topology Seminar
Time
Monday, February 21, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jonathan SimoneGeorgia Tech

Fiber sums and the rational blowdown have been very useful tools in constructing smooth, closed, oriented 4-manifolds. Applying these tools to genus g>1 Lefschetz fibrations with clustered nodal fibers, we will construct symplectic Lefschetz fibrations realizing all the lattice points in the symplectic geography plane below the Noether line, providing a symplectic extension of classical works populating the complex geography plane with holomorphic Lefschetz fibrations. Moreover, Lefschetz fibrations with certain clustered nodal fibers provide rational blowdown configurations that yield new constructions of small symplectic exotic 4-manifolds. We will present an example of a construction of a minimal symplectic exotic CP^2#-5CP^2 through this procedure applied to a genus-3 fibration. This work is joint with Inanc Baykur and Mustafa Korkmaz.

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