Difference between revisions of "Geometry and Topology Seminar 2019-2020"

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== Fall 2010 ==
+
The [[Geometry and Topology]] seminar meets in room '''901 of Van Vleck Hall''' on '''Fridays''' from '''1:20pm - 2:10pm'''.
 +
<br>
 +
For more information, contact Shaosai Huang.
  
The seminar will be held  in room B901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm
+
[[Image:Hawk.jpg|thumb|300px]]
 +
 
 +
 
 +
== Spring 2020 ==
  
 
{| cellpadding="8"
 
{| cellpadding="8"
Line 9: Line 14:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|September 10
+
|Feb. 7
|[http://www.math.wisc.edu/~oh/ Yong-Geun Oh] (UW Madison)
+
|Xiangdong Xie  (Bowling Green University)
|[[#Yong-Geun Oh (UW Madison)|
+
| Minicourse 1: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
''Counting embedded curves in Calabi-Yau threefolds and Gopakumar-Vafa invariants'']]
+
|(Dymarz)
|local
 
|-
 
|September 17
 
|Leva Buhovsky (U of Chicago)
 
|[[#Leva Buhovsky (U of Chicago)|
 
''On the uniqueness of Hofer's geometry'']]
 
|[http://www.math.wisc.edu/~oh/ Yong-Geun]
 
|-
 
|September 24
 
|[http://sites.google.com/site/polterov/home/ Leonid Polterovich] (Tel Aviv U and U of Chicago)
 
|[[#Leonid Polterovich (Tel Aviv U and U of Chicago)|
 
''Poisson brackets and symplectic invariants'']]
 
|[http://www.math.wisc.edu/~oh/ Yong-Geun]
 
|-
 
|October 8
 
|[http://www.math.wisc.edu/~stpaul/ Sean Paul] (UW Madison)
 
|[[#Sean Paul (UW Madison)|
 
''Canonical Kahler metrics and the stability of projective varieties'']]
 
|local
 
|-
 
|October 15
 
|Conan Leung (Chinese U. of Hong Kong)
 
|[[#Conan Leung (Chinese U. of Hong Kong)|
 
''SYZ mirror symmetry for toric manifolds'']]
 
|Honorary fellow, local
 
|-
 
|October 22
 
|[http://www.mathi.uni-heidelberg.de/~banagl/ Markus Banagl] (U. Heidelberg)
 
|[[# Markus Banagl (U. Heidelberg)|
 
''Intersection Space Methods and Their Application to Equivariant Cohomology, String Theory, and Mirror Symmetry'']]
 
|[http://www.math.wisc.edu/~maxim/ Maxim]
 
 
|-
 
|-
|October 29
+
|Feb. 14
|[http://www.math.umn.edu/~zhux0086/ Ke Zhu] (U of Minnesota)
+
|Xiangdong Xie  (Bowling Green University)
|[[#Ke Zhu (U of Minnesota)|
+
| Minicourse 2: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
''Thick-thin decomposition of Floer trajectories and adiabatic gluing'']]
+
|(Dymarz)
|[http://www.math.wisc.edu/~oh/ Yong-Geun]
 
 
|-
 
|-
|November 5
+
|Feb. 21
|[http://www.math.psu.edu/tabachni/ Sergei Tabachnikov] (Penn State)
+
|Xiangdong Xie (Bowling Green University)
|[[#Sergei Tabachnikov (Penn State)|
+
| Minicourse 3: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
''Algebra, geometry, and dynamics of the pentagram map'']]
+
|(Dymarz)
|[http://www.math.wisc.edu/~maribeff/ Gloria]
 
 
|-
 
|-
|November 19
+
|Feb. 28
|Ma Chit (Chinese U. of Hong Kong)
+
|Kuang-Ru Wu (Purdue University)
|[[#Ma Chit (Chinese U. of Hong Kong)|
+
|Griffiths extremality, interpolation of norms, and Kahler quantization
''A growth estimate of lattice points in Gorenstein cones using toric Einstein metrics'']]
+
|(Huang)
|Graduate student, local
 
 
|-
 
|-
|December 3
+
|Mar. 6
|[http://www.math.northwestern.edu/~zaslow/ Eric Zaslow]  (Northwestern University)
+
|Yuanqi Wang (University of Kansas)
|[[#Eric Zaslow (Northwestern University)|
+
|Moduli space of G2−instantons on 7−dimensional product manifolds
''Ribbon Graphs and Mirror Symmetry'']]
+
|(Huang)
|[http://www.math.wisc.edu/~oh/ Yong-Geun and Conan Leung]
 
 
|-
 
|-
|December 10
+
|Mar. 13 <b>CANCELED</b>
|Wenxuan Lu  (MIT)
+
|Karin Melnick (University of Maryland)
|[[#Wenxuan Lu (MIT)|
+
|A D'Ambra Theorem in conformal Lorentzian geometry
''Instanton Correction, Wall Crossing And Mirror Symmetry Of Hitchin's Moduli
+
|(Dymarz)
Spaces'']]
 
|[http://www.math.wisc.edu/~oh/ Young-Geun and Conan Leung]
 
 
|-
 
|-
 +
|<b>Mar. 25</b> <b>CANCELED</b>
 +
|Joerg Schuermann (University of Muenster, Germany)
 +
|An introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles
 +
|(Maxim)
 
|-
 
|-
 +
|Mar. 27 <b>CANCELED</b>
 +
|David Massey (Northeastern University)
 +
|Extracting easily calculable algebraic data from the vanishing cycle complex
 +
|(Maxim)
 
|-
 
|-
 +
|<b>Apr. 10</b> <b>CANCELED</b>
 +
|Antoine Song (Berkeley)
 +
|TBA
 +
|(Chen)
 
|}
 
|}
  
== Spring 2011 ==
+
== Fall 2019 ==
 
 
The seminar will be held  in room B901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm
 
  
 
{| cellpadding="8"
 
{| cellpadding="8"
Line 90: Line 68:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|January 21
+
|Oct. 4
|Mohammed Abouzaid (Clay Institute & MIT)
+
|Ruobing Zhang (Stony Brook University)
|[[#Mohammed Abouzaid (Clay Institute & MIT)|
+
| Geometric analysis of collapsing Calabi-Yau spaces
''A plethora of exotic Stein manifolds'']]
+
|(Chen)
|[http://www.math.wisc.edu/~oh/ Yong-Geun]
 
 
|-
 
|-
|March 4
 
|[http://www.massey.math.neu.edu/ David Massey] (Northeastern)
 
|[[#David Massey (Northeastern)|
 
''TBA'']]
 
|[http://www.math.wisc.edu/~maxim/ Maxim]
 
 
|-
 
|-
|March 11
+
|Oct. 25
|Danny Calegari (Cal Tech))
+
|Emily Stark (Utah)
|[[#Danny Calegari (Cal Tech)|
+
| Action rigidity for free products of hyperbolic manifold groups
''TBA'']]
+
|(Dymarz)
|[http://www.math.wisc.edu/~oh/ Yong-Geun]
 
 
|-
 
|-
|May 6
+
|Nov. 8
|[http://www.math.neu.edu/~suciu/ Alex Suciu] (Northeastern)
+
|Max Forester (University of Oklahoma)
|[[#Alex Suciu (Northeastern)|
+
|Spectral gaps for stable commutator length in some cubulated groups
''TBA'']]
+
|(Dymarz)
|[http://www.math.wisc.edu/~maxim/ Maxim]
+
|-
 +
|Nov. 22
 +
|Yu Li (Stony Brook University)
 +
|On the structure of Ricci shrinkers
 +
|(Huang)
 
|-
 
|-
 
|}
 
|}
  
== Abstracts ==
+
==Spring Abstracts==
 
 
==Fall 2010==
 
 
 
===Yong-Geun Oh (UW Madison)===
 
''Counting embedded curves in Calabi-Yau threefolds and Gopakumar-Vafa invariants''
 
 
 
Gopakumar-Vafa BPS invariant is some integer counting invariant of the cohomology
 
of D-brane moduli spaces in string theory. In relation to the Gromov-Witten theory,
 
it is expected that the invariant would coincide with the `number' of embedded
 
(pseudo)holomorphic curves (Gopakumar-Vafa conjecture). In this talk, we will explain the speaker's recent
 
result that the latter integer invariants can be defined for a generic choice of
 
compatible almost complex structures. We will also discuss the corresponding
 
wall-crossing phenomena and some open questions towards a complete solution to
 
the Gopakumar-Vafa conjecture.
 
 
 
===Leva Buhovsky (U of Chicago)===
 
''On the uniqueness of Hofer's geometry''
 
 
 
In this talk we address the question whether Hofer's metric is unique among the Finsler-type bi-invariant metrics on the group of Hamiltonian diffeomorphisms. The talk is based on a recent joint work with Yaron Ostrover.
 
 
 
===Leonid Polterovich (Tel Aviv U and U of Chicago)===
 
''Poisson brackets and symplectic invariants''
 
  
We discuss new invariants associated to collections of closed subsets
+
===Xiangdong Xie===
of a symplectic manifold. These invariants are defined
 
through an elementary variational problem involving Poisson brackets.
 
The proof of non-triviality of these invariants requires methods of modern
 
symplectic topology (Floer theory). We present applications
 
to approximation theory on symplectic manifolds and to Hamiltonian dynamics.
 
The talk is based on a work in progress with Lev Buhovsky and Michael Entov.
 
  
===Sean Paul (UW Madison)===
+
The quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces has played
''Canonical Kahler metrics and the stability of projective varieties"
+
an important role in various  rigidity questions in geometry and group theory.
 +
In these talks I  shall give an introduction to this topic.  In the first talk I will introduce Gromov hyperbolic spaces, define their ideal boundary,  and  discuss their basic properties.  In the second and third talks I will define the visual metrics on the ideal boundary, explain the connection between quasiisometries of  Gromov hyperbolic space and quasiconformal maps on  their ideal boundary, and indicate  how the quasiconformal structure on the ideal boundary can be used to deduce rigidity.
  
I will give a survey of my own work on this problem, the basic reference is:
+
===Kuang-Ru Wu===
http://arxiv.org/pdf/0811.2548v3
 
  
===Conan Leung (Chinese U. of Hong Kong)===
+
Following Kobayashi, we consider Griffiths negative complex Finsler bundles, naturally leading us to introduce Griffiths extremal Finsler metrics. As we point out, this notion is closely related to the theory of interpolation of norms, and is characterized by an equation of complex Monge– Ampere type, whose corresponding Dirichlet problem we solve. As applications, we prove that Griffiths extremal Finsler metrics quantize solutions to a natural PDE in Kahler geometry, related to the construction of flat maps for the Mabuchi metric. This is joint work with Tamas Darvas.
''SYZ mirror symmetry for toric manifolds''
 
  
===Markus Banagl (U. Heidelberg)===
+
===Yuanqi Wang===
''Intersection Space Methods and Their Application to Equivariant Cohomology, String Theory, and Mirror Symmetry.''
+
$G_{2}-$instantons are 7-dimensional analogues of flat connections in dimension 3. It is part of Donaldson-Thomas’ program to generalize the fruitful gauge theory in dimensions 2,3,4 to dimensions 6,7,8. The moduli space of  $G_{2}-$instantons, with virtual dimension $0$, is  expected to have interesting  geometric structure and yield enumerative invariant for the underlying $7-$dimensional manifold.  
  
Using homotopy theoretic methods, we shall associate to certain classes of
+
In this talk, in some reasonable special cases and a fairly complete manner, we will describe the relation between the moduli space of $G_{2}-$instantons and an algebraic geometry moduli on a Calabi-Yau 3-fold.
singular spaces generalized geometric Poincaré complexes called intersection
 
spaces. Their cohomology is generally not isomorphic to intersection
 
cohomology.
 
In this talk, we shall concentrate on the applications of the new
 
cohomology theory to the equivariant real cohomology of isometric actions of
 
torsionfree discrete groups, to type II string theory and D-branes, and to
 
the relation of the new theory to classical intersection cohomology under
 
mirror symmetry.
 
  
===Ke Zhu (U of Minnesota)===
+
===Karin Melnick===
''Thick-thin decomposition of Floer trajectories and adiabatic gluing''
 
  
Let f be a generic Morse function on a symplectic manifold M.
+
D'Ambra proved in 1988 that the isometry group of a compact, simply connected, real-analytic Lorentzian manifold must be compact. I will discuss my recent theorem that the conformal group of such a manifold must also be compact, and how it relates to the Lorentzian Lichnerowicz Conjecture.
For Floer trajectories of Hamiltonian \e f, as \e goes to 0 Oh proved that  
 
they converge to “pearl complex” consisiting of J-holomorphic spheres
 
and joining gradient segments of f. The J-holomorphic spheres come from the  
 
“thick” part of Floer trajectories and the gradient segments come from
 
the “thin” part. Similar “thick-thin” compactification result has
 
also been obtained by Mundet-Tian in twisted holomorphic map setting. In
 
this talk, we prove the reverse gluing result in the simplest setting: we
 
glue from disk-flow-dsik configurations to nearby Floer trajectories of
 
Hamitonians K_{\e} for sufficiently small \e and also show the  
 
surjectivity. (Most part of the Hamiltonian K_{\e} is \ef). We will discuss
 
the application to PSS isomorphism. This is a joint work with Yong-Geun Oh.
 
  
===Sergei Tabachnikov (Penn State)===
+
===Joerg Schuermann===
''Algebra, geometry, and dynamics of the pentagram map''
 
  
Introduced by R. Schwartz almost 20  years ago, the pentagram map acts on plane n-gons, considered up to projective equivalence, by drawing the diagonals that connect second-nearest vertices and taking the new n-gon formed by their intersections. I shall survey recent work on the pentagram map, in particular, I shall demonstrate  that the dynamics of the pentagram map  is completely integrable. I shall also explain that the pentagram map is a discretization of the Boussinesq equation, a well known completely integrable partial differential equation. A surprising relation between the spaces of polygons and combinatorial objects called the 2-frieze patterns (generalizing the frieze patterns of Coxeter) will be described. Eight new(?) configuration theorems of projective geometry will be demonstrated. The talk is illustrated by computer animation.
+
We give an introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles, based on stratified Morse theory for constructible functions. The corresponding local index of an isolated critical point (in a stratified sense) of a one-form depends on the constructible function, specializing for different choices to well known indices like the radial, GSV or Euler obstruction index.
  
===Ma Chit (Chinese U. of Hong Kong)===
+
===David Massey===
''A growth estimate of lattice points in Gorenstein cones using toric Einstein metrics''
 
  
Using the existence of Einstein metrics on toric Kahler and Sasaki manifolds, a lower bound estimate on the growth of lattice points is obtained for Gorenstein cones. This talk is based on a joint work with Conan Leung.   
+
Given a complex analytic function on an open subset U  of C<sup>n+1</sup>, one may consider the complex of sheaves of vanishing cycles along f of the constant sheaf Z<sub>U</sub>. This complex encodes on the cohomological level the reduced cohomology of the Milnor fibers of f at each of f<sup>-1</sup>(0). The question is: how does one calculate (ideally, by hand) any useful numbers about this vanishing cycle complex? One answer is to look at the Lê numbers of f. We will discuss the precise relationship between these objects/numbers.
  
===Eric Zaslow (Northwestern University)===
+
===Antoine Song===
''Ribbon Graphs and Mirror Symmetry''
 
  
I will define, for each ribbon graph, a dg category,
+
TBA
and explain the conjectural relation to mirror symmetry.
 
I will being by reviewing how T-duality relates
 
coherent sheaves on toric varieties to constructible sheaves
 
on a vector space, then use this relation to glue
 
toric varieties together.  In one-dimension, the
 
category of sheaves on such gluings has a
 
description in terms of ribbon graphs.
 
These categories are conjecturally
 
related to the Fukaya category of a noncompact
 
hypersurface mirror to the variety with toric
 
components.
 
  
I will use very basic examples.
+
==Fall Abstracts==
This work is joint with Nicolo' Sibilla
 
and David Treumann.
 
  
 +
===Ruobing Zhang===
  
===Wenxuan Lu (MIT)===
+
This talk centers on the degenerations of Calabi-Yau metrics. We will focus on the interactions between algebraic degenerations and metric convergence with highly singular behaviors in the collapsing case. As the complex structures degenerate, the collapsing Calabi-Yau metrics may exhibit various wild geometric properties with highly non-algebraic features.
''Instanton Correction, Wall Crossing And Mirror Symmetry Of Hitchin's Moduli
 
Spaces''
 
  
We study two instanton correction problems of Hitchin's moduli spaces along with
+
First, as motivating examples, we will describe our recent results on the new collapsing mechanisms of K3 surfaces. Next, we will switch to higher dimensions and we will exhibit some entirely new constructions of degenerating Calabi-Yau metrics which are expected to work in broader contexts. Complex structures degeneration will be accurately characterized by the bubbling and singularity analysis in a geometric manner.
their wall crossing formulas. The hyperkahler metric of a Hitchin's moduli space
 
can be put into an instanton-corrected form according to physicists Gaiotto,
 
Moore and Neitzke. The problem boils down to the construction of a set of
 
special coordinates which can be constructed as Fock-Goncharov coordinates
 
associated with foliations of quadratic differentials on a Riemann surface. A
 
wall crossing formula of Kontsevich and Soibelman arises both as a crucial
 
consistency condition and an effective computational tool. On the other hand
 
Gross and Siebert have succeeded in determining instanton corrections of
 
complex structures of Calabi-Yau varieties in the context of mirror symmetry
 
from a singular affine structure with additional data.   We will show that the
 
two instanton correction problems are equivalent in an appropriate sense. This
 
is a nontrivial statement of mirror symmetry of Hitchin's moduli spaces which
 
till now has been mostly studied in the framework of geometric Langlands
 
duality.  This result provides examples of Calabi-Yau varieties where the
 
instanton correction (in the sense of mirror symmetry) of  metrics and complex
 
structures can be determined.
 
  
==Spring 2011==
+
===Emily Stark===
  
===Mohammed Abouzaid (Clay Institute & MIT)===
+
The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold group is not action rigid for all n at least three. In contrast, we show that free products of closed hyperbolic manifold groups are action rigid. Consequently, we obtain the first examples of Gromov hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. This is joint work with Daniel Woodhouse.
''A plethora of exotic Stein manifolds''
 
  
In real dimensions greater than 4, I will explain how a smooth
+
===Max Forester===
manifold underlying an affine variety admits uncountably many distinct
 
(Wein)stein structures, of which countably many have finite type,
 
and which are distinguished by their symplectic cohomology groups.
 
Starting with a Lefschetz fibration on such a variety, I shall per-
 
form an explicit sequence of appropriate surgeries, keeping track of
 
the changes to the Fukaya category and hence, by understanding
 
open-closed maps, obtain descriptions of symplectic cohomology af-
 
ter surgery. (joint work with P. Seidel)
 
  
===David Massey (Northeastern)===
+
I will discuss stable commutator length (scl) in groups, and some gap theorems for the scl spectrum. Such results say that for various groups, scl of an element is always either zero or is larger than some uniform constant. I will discuss the cases of right-angled Artin groups and certain right-angled Coxeter groups. This is joint work with Pallavi Dani, Ignat Soroko, and Jing Tao.
''TBA''
 
  
===Danny Calegari (Cal Tech)===
+
===Yu Li===
''TBA''
+
We develop a structure theory for non-collapsed Ricci shrinkers without any curvature condition. As an application, we show that any Ricci shrinker whose second eigenvalue of the curvature operator is positive must be a quotient of sphere.
  
===Alex Suciu (Northeastern)===
+
== Archive of past Geometry seminars ==
''TBA''
+
2018-2019  [[Geometry_and_Topology_Seminar_2018-2019]]
[[Fall-2010-Geometry-Topology]]
+
<br><br>
 +
2017-2018 [[Geometry_and_Topology_Seminar_2017-2018]]
 +
<br><br>
 +
2016-2017  [[Geometry_and_Topology_Seminar_2016-2017]]
 +
<br><br>
 +
2015-2016: [[Geometry_and_Topology_Seminar_2015-2016]]
 +
<br><br>
 +
2014-2015: [[Geometry_and_Topology_Seminar_2014-2015]]
 +
<br><br>
 +
2013-2014: [[Geometry_and_Topology_Seminar_2013-2014]]
 +
<br><br>
 +
2012-2013: [[Geometry_and_Topology_Seminar_2012-2013]]
 +
<br><br>
 +
2011-2012: [[Geometry_and_Topology_Seminar_2011-2012]]
 +
<br><br>
 +
2010: [[Fall-2010-Geometry-Topology]]

Latest revision as of 13:56, 3 September 2020

The Geometry and Topology seminar meets in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm.
For more information, contact Shaosai Huang.

Hawk.jpg


Spring 2020

date speaker title host(s)
Feb. 7 Xiangdong Xie (Bowling Green University) Minicourse 1: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces (Dymarz)
Feb. 14 Xiangdong Xie (Bowling Green University) Minicourse 2: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces (Dymarz)
Feb. 21 Xiangdong Xie (Bowling Green University) Minicourse 3: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces (Dymarz)
Feb. 28 Kuang-Ru Wu (Purdue University) Griffiths extremality, interpolation of norms, and Kahler quantization (Huang)
Mar. 6 Yuanqi Wang (University of Kansas) Moduli space of G2−instantons on 7−dimensional product manifolds (Huang)
Mar. 13 CANCELED Karin Melnick (University of Maryland) A D'Ambra Theorem in conformal Lorentzian geometry (Dymarz)
Mar. 25 CANCELED Joerg Schuermann (University of Muenster, Germany) An introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles (Maxim)
Mar. 27 CANCELED David Massey (Northeastern University) Extracting easily calculable algebraic data from the vanishing cycle complex (Maxim)
Apr. 10 CANCELED Antoine Song (Berkeley) TBA (Chen)

Fall 2019

date speaker title host(s)
Oct. 4 Ruobing Zhang (Stony Brook University) Geometric analysis of collapsing Calabi-Yau spaces (Chen)
Oct. 25 Emily Stark (Utah) Action rigidity for free products of hyperbolic manifold groups (Dymarz)
Nov. 8 Max Forester (University of Oklahoma) Spectral gaps for stable commutator length in some cubulated groups (Dymarz)
Nov. 22 Yu Li (Stony Brook University) On the structure of Ricci shrinkers (Huang)

Spring Abstracts

Xiangdong Xie

The quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces has played an important role in various rigidity questions in geometry and group theory. In these talks I shall give an introduction to this topic. In the first talk I will introduce Gromov hyperbolic spaces, define their ideal boundary, and discuss their basic properties. In the second and third talks I will define the visual metrics on the ideal boundary, explain the connection between quasiisometries of Gromov hyperbolic space and quasiconformal maps on their ideal boundary, and indicate how the quasiconformal structure on the ideal boundary can be used to deduce rigidity.

Kuang-Ru Wu

Following Kobayashi, we consider Griffiths negative complex Finsler bundles, naturally leading us to introduce Griffiths extremal Finsler metrics. As we point out, this notion is closely related to the theory of interpolation of norms, and is characterized by an equation of complex Monge– Ampere type, whose corresponding Dirichlet problem we solve. As applications, we prove that Griffiths extremal Finsler metrics quantize solutions to a natural PDE in Kahler geometry, related to the construction of flat maps for the Mabuchi metric. This is joint work with Tamas Darvas.

Yuanqi Wang

$G_{2}-$instantons are 7-dimensional analogues of flat connections in dimension 3. It is part of Donaldson-Thomas’ program to generalize the fruitful gauge theory in dimensions 2,3,4 to dimensions 6,7,8. The moduli space of $G_{2}-$instantons, with virtual dimension $0$, is expected to have interesting geometric structure and yield enumerative invariant for the underlying $7-$dimensional manifold.

In this talk, in some reasonable special cases and a fairly complete manner, we will describe the relation between the moduli space of $G_{2}-$instantons and an algebraic geometry moduli on a Calabi-Yau 3-fold.

Karin Melnick

D'Ambra proved in 1988 that the isometry group of a compact, simply connected, real-analytic Lorentzian manifold must be compact. I will discuss my recent theorem that the conformal group of such a manifold must also be compact, and how it relates to the Lorentzian Lichnerowicz Conjecture.

Joerg Schuermann

We give an introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles, based on stratified Morse theory for constructible functions. The corresponding local index of an isolated critical point (in a stratified sense) of a one-form depends on the constructible function, specializing for different choices to well known indices like the radial, GSV or Euler obstruction index.

David Massey

Given a complex analytic function on an open subset U of Cn+1, one may consider the complex of sheaves of vanishing cycles along f of the constant sheaf ZU. This complex encodes on the cohomological level the reduced cohomology of the Milnor fibers of f at each of f-1(0). The question is: how does one calculate (ideally, by hand) any useful numbers about this vanishing cycle complex? One answer is to look at the Lê numbers of f. We will discuss the precise relationship between these objects/numbers.

Antoine Song

TBA

Fall Abstracts

Ruobing Zhang

This talk centers on the degenerations of Calabi-Yau metrics. We will focus on the interactions between algebraic degenerations and metric convergence with highly singular behaviors in the collapsing case. As the complex structures degenerate, the collapsing Calabi-Yau metrics may exhibit various wild geometric properties with highly non-algebraic features.

First, as motivating examples, we will describe our recent results on the new collapsing mechanisms of K3 surfaces. Next, we will switch to higher dimensions and we will exhibit some entirely new constructions of degenerating Calabi-Yau metrics which are expected to work in broader contexts. Complex structures degeneration will be accurately characterized by the bubbling and singularity analysis in a geometric manner.

Emily Stark

The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold group is not action rigid for all n at least three. In contrast, we show that free products of closed hyperbolic manifold groups are action rigid. Consequently, we obtain the first examples of Gromov hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. This is joint work with Daniel Woodhouse.

Max Forester

I will discuss stable commutator length (scl) in groups, and some gap theorems for the scl spectrum. Such results say that for various groups, scl of an element is always either zero or is larger than some uniform constant. I will discuss the cases of right-angled Artin groups and certain right-angled Coxeter groups. This is joint work with Pallavi Dani, Ignat Soroko, and Jing Tao.

Yu Li

We develop a structure theory for non-collapsed Ricci shrinkers without any curvature condition. As an application, we show that any Ricci shrinker whose second eigenvalue of the curvature operator is positive must be a quotient of sphere.

Archive of past Geometry seminars

2018-2019 Geometry_and_Topology_Seminar_2018-2019

2017-2018 Geometry_and_Topology_Seminar_2017-2018

2016-2017 Geometry_and_Topology_Seminar_2016-2017

2015-2016: Geometry_and_Topology_Seminar_2015-2016

2014-2015: Geometry_and_Topology_Seminar_2014-2015

2013-2014: Geometry_and_Topology_Seminar_2013-2014

2012-2013: Geometry_and_Topology_Seminar_2012-2013

2011-2012: Geometry_and_Topology_Seminar_2011-2012

2010: Fall-2010-Geometry-Topology