Seminars and Colloquia by Series

Julia sets with Ahlfors-regular conformal dimension one by InSung Park

Series
Geometry Topology Seminar
Time
Monday, February 22, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
InSung ParkIndiana University Bloomington

Please Note: Office hours will be held 3-4pm EST.

Complex dynamics is the study of dynamical systems defined by iterating rational maps on the Riemann sphere. For a rational map f, the Julia set Jf  is a beautiful fractal defined as the repeller of the dynamics of f. Fractal invariants of Julia sets, such as Hausdorff dimensions, have information about the complexity of the dynamics of rational maps. Ahlfors-regular conformal dimension, abbreviated by ARconfdim, is the infimum of the Hausdorff dimension in a quasi-symmetric class of Ahlfors-regular metric spaces. The ARconfdim is an important quantity especially in geometric group theory because a natural metric, called a visual metric, on the boundary of any Gromov hyperbolic group is determined up to quasi-symmetry. In the spirit of Sullivan's dictionary, we can use ARconfdim to understand the dynamics of rational maps as well. In this talk, we show that the Julia set of a post-critically finite hyperbolic rational map f has ARconfdim 1 if and only if there is an f-invariant graph G containing the post-critical set such that the dynamics restricted to G has topological entropy zero.  

Braids, quasimorphisms, and slice-Bennequin inequalities.

Series
Geometry Topology Seminar
Time
Monday, February 8, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Peter FellerETH Zurich

The writhe of a braid (=#pos crossing - #neg crossings) and the fractional Dehn twist coefficient of a braid (a rational number that measures "how much the braid twists") are the two most prominent examples of what is known as a quasimorphism (a map that fails to be a group homomorphism by at most a bounded amount) from Artin's braid group on n-strands to the reals.
We consider characterizing properties for such quasimorphisms and talk about relations to the study of knot concordance. For the latter, we consider inequalities for quasimorphism modelled after the so-called slice-Bennequin inequality:
writhe(B) ≤ 2g_4(K) - 1 + n for all n-stranded braids B with closure a knot K.
Based on work in progress.

Symmetric knots and the equivariant 4-ball genus

Series
Geometry Topology Seminar
Time
Monday, February 1, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Ahmad IssaUniversity of British Columbia

Given a knot K in the 3-sphere, the 4-genus of K is the minimal genus of an orientable surface embedded in the 4-ball with boundary K. If the knot K has a symmetry (e.g. K is periodic or strongly invertible), one can define the equivariant 4-genus by only minimising the genus over those surfaces in the 4-ball which respect the symmetry of the knot. I'll discuss some work with Keegan Boyle trying to understanding the equivariant 4-genus.

The asymptotic dimension of big mapping class groups

Series
Geometry Topology Seminar
Time
Monday, January 25, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Yvon VerberneGeorgia Institute of Technology

Please Note: Dan Margalit is inviting you to a scheduled Zoom meeting. https://zoom.us/j/94410378648?pwd=TVV6UDd0SnU3SnAveHA1NWxYcmlTdz09 Meeting ID: 944 1037 8648 Passcode: gojackets

In 2010, Bestvina-Bromberg-Fujiwara proved that the mapping class group of a finite type surface has finite asymptotic dimension. In contrast, we will show the mapping class group of an infinite-type surface has infinite asymptotic dimension if it contains an essential shift. This work is joint with Curtis Grant and Kasra Rafi.

Identifying Dehn Functions of Bestvina--Brady Groups From Their Defining Graphs

Series
Geometry Topology Seminar
Time
Monday, January 11, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Yu-Chan ChangEmory University

Please Note: https://zoom.us/j/8833025617?pwd=R1FvQWp1MVlRSTVBdFZNejE3ZURmUT09 Meeting ID: 883 302 5617

Bestvina--Brady groups are subgroups of right-angled Artin groups, and their Dehn functions are bounded above by quartic functions. There are examples of Bestvina--Brady groups whose Dehn functions are linear, quadratic, cubic, and quartic. In this talk, I will give a class of Bestvina--Brady groups that have polynomial Dehn functions, and we can identify the Dehn functions by the defining graphs of those Bestvina--Brady groups. 

Taut foliations and Dehn surgery along positive braid knots

Series
Geometry Topology Seminar
Time
Monday, November 30, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
online
Speaker
Siddhi KrishnaGeorgia Tech

The L-space conjecture has been in the news a lot lately. It predicts a surprising relationship between the algebraic, geometric, and Floer-homological properties of a 3--manifold Y. In particular, it predicts exactly which 3-manifolds admit a ``taut foliation". In this talk, I'll discuss some of my past and forthcoming work investigating these connections. In particular, I'll discuss a strategy for building taut foliations manifolds obtained by Dehn surgery along knots realized as closures of ``positive braids". As an application, I will show how taut foliations can be used to obstruct positivity for cable knots. All are welcome; no background in foliation or Floer homology theories will be assumed.

https://bccte.zoom.us/j/91883463721

Meeting ID: 918 8346 3721

 

A Combinatorial Description of the knot concordance invariant epsilon

Series
Geometry Topology Seminar
Time
Monday, November 9, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Hakan DogaUniversity of Buffalo

Computing, understanding the behavior of concordance invariants obtained from knot Floer homology theories is quite central to the study of the concordance group and low-dimensional topology in general. In this talk, I will describe the method that allows us to compute the concordance invariant epsilon using combinatorial knot Floer homology and talk about some computational results. This is a joint work with S. Dey.

Knots and Links in overtwisted contact structures

Series
Geometry Topology Seminar
Time
Monday, November 2, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
on line
Speaker
Rima ChatterjeeLSU

Please Note: Knots/links associated to overtwisted contact structures have been less explored. There are two types of knots/links in overtwisted contact manifolds, namely loose and non-loose. In this talk, I will start with an overview of these knots and then discuss some of my recent work involving these knots and links. Specifically, I will talk about a coarse classification result of loose, null-homologous Legendrian and transverse links . Next relating them with open book decompositions, I will show that coarse equivalence class of loose null-homologous Legendrian links has support genus zero. I will end with some interesting open questions.

Embedding closed hyperbolic 3-manifolds in small volume hyperbolic 4-manifolds

Series
Geometry Topology Seminar
Time
Monday, October 26, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michelle ChuUniversity of Illinois at Chicago

The smallest volume cusped hyperbolic 3-manifolds, the figure-eight knot complement and its sister, contain many immersed but no embedded closed totally geodesic surfaces. In this talk we discuss the existence or lack thereof of codimension-1 closed embedded totally geodesic submanifolds in minimal volume cusped hyperbolic 4-manifolds. This talk is based on joint work with Alan Reid.

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