Seminars and Colloquia by Series

Renormalization for the almost Mathieu operator and related skew products.

Series
CDSNS Colloquium
Time
Friday, November 1, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hans KochUniv. of Texas, Austin

Considering SL(2,R) skew-product maps over circle rotations,
we prove that a renormalization transformation
associated with the golden mean alpha
has a nontrivial periodic orbit of length 3.
We also present some numerical results,
including evidence that this period 3 describes
scaling properties of the Hofstadter butterfly
near the top of the spectrum at alpha,
and scaling properties of the generalized eigenfunction
for this energy.

New invariants of homology cobordism

Series
School of Mathematics Colloquium
Time
Thursday, October 31, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kristen HendricksRutgers

This is a talk about 3-manifolds and knots. We will begin by reviewing some basic constructions and motivations in low-dimensional topology, and will then introduce the homology cobordism group, the group of 3-manifolds with the same homology as the 3-dimensional sphere up to a reasonable notion of equivalence. We will discuss what is known about the structure of this group and its connection to higher dimensional topology. We will then discuss some existing invariants of the homology cobordism group coming from gauge theory and symplectic geometry, particularly Floer theory. Finally, we will introduce a new invariant of homology cobordism coming from an equivariant version of the computationally-friendly Floer-theoretic 3-manifold invariant Heegaard Floer homology, and use it to construct a new filtration on the homology cobordism group and derive some structural applications. Parts of this talk are joint work with C. Manolescu and I. Zemke; more recent parts of this talk are joint work with J. Hom and T. Lidman.

TBA

Series
Mathematical Biology Seminar
Time
Wednesday, October 30, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Claudia Solis-LemusUniversity of Wisconsin-Madison

TBA

Degenerating Einstein spaces

Series
PDE Seminar
Time
Tuesday, October 29, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ruobing ZhangStony Brook University
In the talk we discuss singularity formation of Einstein metrics as the underlying spaces degenerate or collapse. The usual analytic tools such as uniform Sobolev inequalities and nonlinear a priori estimates are unavailable in this context. We will describe an entirely new way to handle these difficulties, and construct degenerating Ricci-flat metrics with quantitative singularity behaviors.

Tropical covers with an abelian group action

Series
Algebra Seminar
Time
Tuesday, October 29, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dmitry ZakharovCentral Michigan University

Given a topological space X and a group G, a G-cover of X is a covering space X’ --> X together with an invariant G-action on X’ that acts freely and transitively on the fibers. In tropical geometry, we are naturally led to consider more general objects: finite morphisms C’ --> C of tropical curves admitting an invariant G-action on C’, such that the induced action on the fibers is transitive, but not necessarily free. A natural question is to classify all such covers of a given curve C.

I will show that when G is a finite abelian group, a G-cover of a tropical curve C is determined by two data: a stratification S of C by subgroups of G, and an element of a cohomology group H1(C,S) generalizing the first simplicial cohomology group H1(C,G). This classification can be viewed as a tropical version of geometric class field theory. I will discuss the realizability problem for tropical abelian covers, and the relationship between cyclic covers of a curve C and the corresponding torsion subgroup of Jac(C). The realizability problem for prime cyclic covers turns out to be related to the classical nowhere-zero flow problem in graph theory.

This is joint work with Yoav Len and Martin Ulirsch.

Connected Floer homology of covering involutions

Series
Geometry Topology Seminar
Time
Monday, October 28, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skile 006
Speaker
Sungkyung KangChinese University of Hong Kong

Using the covering involution on the double branched cover of S3 branched along a knot, and adapting ideas of Hendricks-Manolescu and Hendricks-Hom-Lidman, we define new knot (concordance) invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots. This is a joint work with A. Alfieri and A. Stipsicz.

Heegaard Floer homology and Seifert manifolds

Series
Geometry Topology Seminar Pre-talk
Time
Monday, October 28, 2019 - 12:45 for 1 hour (actually 50 minutes)
Location
Skile 006
Speaker
Sungkyung KangChinese University of Hong Kong

Heegaard Floer homology gives a powerful invariant of closed 3-manifolds. It is always computable in the purely combinatorial sense, but usually computing it needs a very hard work. However, for certain graph 3-manifolds, its minus-flavored Heegaard Floer homology can be easily computed in terms of lattice homology, due to Nemethi. I plan to give the basic definitions and constructions of Heegaard Floer theory and lattice homology, as well as the isomorphism between those two objects.

Effective bounds for the measure of rotations

Series
CDSNS Colloquium
Time
Monday, October 28, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 05
Speaker
Alex HaroUniv. de Barcelona

A fundamental question in Dynamical Systems is to identify regions of
phase/parameter space satisfying a given property (stability,
linearization, etc).  In this talk, given a family of analytic circle
diffeomorphisms depending on a parameter, we obtain effective (almost
optimal) lower bounds of the Lebesgue measure of the set of parameters
for which that diffeomorphism is conjugate to a rigid rotation.
We estimate this measure using an a-posteriori KAM
scheme that relies on quantitative conditions that
are checkable using computer-assistance. We carefully describe
how the hypotheses in our theorems are reduced to a finite number of
computations, and apply our methodology to the case of the
Arnold family, in the far-from-integrable regime.

This is joint work with Jordi Lluis Figueras and Alejandro Luque.

 

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