Geometry and Topology Seminar 2019-2020: Difference between revisions

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== Spring 2011 ==
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 901 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)
|-
|-
|January 21
|Feb. 7
|Mohammed Abouzaid (Clay Institute & MIT)
|Xiangdong Xie  (Bowling Green University)
|[[#Mohammed Abouzaid (Clay Institute & MIT)|
| Minicourse 1: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
''A plethora of exotic Stein manifolds'']]
|(Dymarz)
|[http://www.math.wisc.edu/~oh/ Yong-Geun]
|-
|-
|February 4
|Feb. 14
|[http://www.math.wisc.edu/~maxim/ Laurentiu Maxim] (UW-Madison)
|Xiangdong Xie  (Bowling Green University)
|[[#Laurentiu Maxim (UW-Madison)|
| Minicourse 2: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
''Intersection Space Homology and Hypersurface Singularities'']]
|(Dymarz)
|local
|-
|-
|February 11
|Feb. 21
|[http://www.math.wisc.edu/~rkent/ Richard Kent] (UW-Madison)
|Xiangdong Xie  (Bowling Green University)
|[[#Richard Kent (UW-Madison)|
| Minicourse 3: Quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces
''Mapping class groups through profinite spectacles'']]
|(Dymarz)
|local
|-
|-
|February 18
|Feb. 28
|[http://www.math.wisc.edu/~jeffv/ Jeff Viaclovsky] (UW-Madison)
|Kuang-Ru Wu (Purdue University)
|[[#Jeff Viaclovsky (UW-Madison)|
|Griffiths extremality, interpolation of norms, and Kahler quantization
''Rigidity and stability of Einstein metrics for quadratic curvature functionals'']]
|(Huang)
|local
|-
|-
|March 4
|Mar. 6
|[http://www.massey.math.neu.edu/ David Massey] (Northeastern)
|Yuanqi Wang (University of Kansas)
|[[#David Massey (Northeastern)|
|Moduli space of G2−instantons on 7−dimensional product manifolds
''Lê Numbers and the Topology of Non-isolated Hypersurface Singularities'']]
|(Huang)
|[http://www.math.wisc.edu/~maxim/ Maxim]
|-
|-
|March 11
|Mar. 13 <b>CANCELED</b>
|Danny Calegari (Cal Tech)
|Karin Melnick (University of Maryland)
|[[#Danny Calegari (Cal Tech)|
|A D'Ambra Theorem in conformal Lorentzian geometry
''Random rigidity in the free group'']]
|(Dymarz)
|[http://www.math.wisc.edu/~oh/ Yong-Geun]
|-
|-
|'''March 23, Wed'''
|<b>Mar. 25</b> <b>CANCELED</b>
|Joerg Schuermann (University of Muenster, Germany)
|Joerg Schuermann (University of Muenster, Germany)
|[[#Joerg Schuermann (University of Muenster, Germany)|
|An introduction to Poincare-Hopf theorems for singular spaces via characteristic cycles
''Generating series for invariants of symmetric products'']]
|(Maxim)
|[http://www.math.wisc.edu/~maxim/ Maxim]
|-
|-
|April 8
|Mar. 27 <b>CANCELED</b>
|[http://www.iazd.uni-hannover.de/~dancohen/ Ishai Dan-Cohen] (U. Hannover)
|David Massey (Northeastern University)
|[[#Ishai Dan-Cohen (U. Hannover)|
|Extracting easily calculable algebraic data from the vanishing cycle complex
''Moduli of unipotent representations'']]
|(Maxim)
|[http://www.math.wisc.edu/~ellenber/ Jordan]
|-
|-
|April 15
|<b>Apr. 10</b> <b>CANCELED</b>
|[http://euclid.colorado.edu/~gwilkin/ Graeme Wilkin] (U of Colorado-Boulder)
|Antoine Song (Berkeley)
|[[#Graeme Wilkin (U of Colorado-Boulder)|
|TBA
''Moment map flows and the Hecke correspondence for quivers'']]
|(Chen)
|[http://www.math.wisc.edu/~mehrotra/ Sukhendu]
|}
 
== Fall 2019 ==
 
{| cellpadding="8"
!align="left" | date
!align="left" | speaker
!align="left" | title
!align="left" | host(s)
|-
|Oct. 4
|Ruobing Zhang (Stony Brook University)
| Geometric analysis of collapsing Calabi-Yau spaces
|(Chen)
|-
|-
|April 22
|[http://www.math.wisc.edu/~oh/ Yong-Geun Oh] (UW-Madison)
|[[#Yong-Geun Oh (UW-Madison)|
''Floer homology and continuous Hamiltonian dynamics'']]
|local
|-
|-
|April 29
|Oct. 25
|Steven Simon (Courant Institute, NYU)
|Emily Stark (Utah)
|[[#Steven Simon (Courant)|
| Action rigidity for free products of hyperbolic manifold groups
''Equivariant and Orthogonal Ham Sandwich Theorems'']]
|(Dymarz)
|[http://www.math.wisc.edu/~maxim/ Maxim]
|-
|-
|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
''Betti numbers of abelian covers'']]
|(Dymarz)
|[http://www.math.wisc.edu/~maxim/ Maxim]
|-
|-
|May 13
|Nov. 22
|[http://www.math.wustl.edu/~apelayo/ Alvaro Pelayo] (IAS)
|Yu Li (Stony Brook University)
|[[#Alvaro Pelayo (IAS)|
|On the structure of Ricci shrinkers
''Symplectic Dynamics of integrable Hamiltonian systems'']]
|(Huang)
|[http://www.math.wisc.edu/~oh/ Yong-Geun]
|-
|-
|}
|}


== Abstracts ==
==Spring Abstracts==
 
===Mohammed Abouzaid (Clay Institute & MIT)===
''A plethora of exotic Stein manifolds''
 
In real dimensions greater than 4, I will explain how a smooth
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)
 
===Laurentiu Maxim (UW-Madison)===
''Intersection Space Homology and Hypersurface Singularities''
 
A recent homotopy-theoretic procedure due to Banagl assigns to a certain singular space a cell complex, its intersection space, whose rational cohomology possesses Poincare duality. This yields a new cohomology theory for singular spaces, which has a richer internal algebraic structure than intersection cohomology (e.g., it has cup products), and which addresses certain questions in type II string theory related to massless D-branes arising during a Calabi-Yau conifold transition.
 
While intersection cohomology is stable under small resolutions, in recent joint work with Markus Banagl we proved that the new theory is often stable under smooth deformations of hypersurface singularities. When this is the case, we showed that the rational cohomology of the intersection space can be endowed with a mixed Hodge structure compatible with Deligne's mixed Hodge structure on the ordinary cohomology of the singular hypersurface.
 
===Richard Kent (UW-Madison)===
''Mapping class groups through profinite spectacles''


It is a theorem of Bass, Lazard, and Serre, and, independently,
===Xiangdong Xie===
Mennicke, that the special linear group SL(n,Z) enjoys the congruence
subgroup property when n is at least 3.  This property is most quickly
described by saying that the profinite completion of the special
linear group injects into the special linear group of the profinite
completion of Z.  There is a natural analog of this property for
mapping class groups of surfaces.  Namely, one may ask if the
profinite completion of the mapping class group embeds in the outer
automorphism group of the profinite completion of the surface group.


M. Boggi has a program to establish this property for mapping class
The quasiconformal structure on the ideal boundary of Gromov hyperbolic spaces has played
groups.  I'll discuss some partial results, and what remains to be
an important role in various  rigidity questions in geometry and group theory.
done.
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.


===Jeff Viaclovsky (UW-Madison)===
===Kuang-Ru Wu===
''Rigidity and stability of Einstein metrics for quadratic curvature functionals''


===David Massey (Northeastern)===
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.
''Lê Numbers and the Topology of Non-isolated Hypersurface Singularities''


The results of Milnor from his now-classic 1968 work "Singular Points of Complex Hypersurfaces" are particularly strong when the singular points are isolated. One of the most striking subsequent results in this area, was the 1976 result of Lê and Ramanujam, in which the h-Cobordism Theorem was used to prove that constant Milnor number implies constant topological-type, for families of isolated hypersurfaces.
===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, I will discuss the Lê cycles and Lê numbers of a singular hypersurface, and the results which seem to indicate that they are the "correct" generalization of the Milnor number for non-isolated hypersurface singularities.
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.


===Danny Calegari (Cal Tech)===
===Karin Melnick===
''Random rigidity in the free group''


We prove a rigidity theorem for the geometry of the unit ball in the stable commutator length norm spanned by k random elements of the commutator subgroup of a free group of fixed big length n; such unit balls are C^0 close to regular octahedra. A heuristic argument suggests that the same is true in all hyperbolic groups. This is joint work with Alden Walker.
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 (Muenster)===
===Joerg Schuermann===
''Generating series for invariants of symmetric products"


We explain new formulae for the generating series of
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.
Hodge theoretical invariants for symmetric products
of complex quasi-projective varieties and mixed Hodge module
complexes. These invariants include the corresponding Hodge
polynomial as well as Hirzebruch characteristic classes,
including those accociated to middle intersection cohomology.
This is joint work with L. Maxim, M. Saito, S. Cappell,
J. Shaneson and S. Yokura.


===Ishai Dan-Cohen (U. Hannover)===
===David Massey===
''Moduli of unipotent representations''


Let $G$ be a unipotent group over a field of characteristic zero. The moduli problem posed by all representations of a fixed dimension $n$ is badly behaved. We set out to define an appropriate nondegenracy condition, and to construct a quasi-projective variety parametrinzing isomorphism classes of nondegenerate representations. In my thesis I defined an invariant $w$ of $G$, its \textit{width}, and a nondegeneracy condition appropriate for representations of dimension $n \le w+1$. Unfortunately, the width is bounded by the depth. But for groups $G$, unipotent of depth $\le 2$, a different nondegeneracy condition gives rise to a quasi projective moduli space for \textit{all} $n$.
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.


This talk is based in part on my thesis, and in part on joint work with Anton Geraschenko, part of which was covered by his recent talk in the number theory seminar here in Madison.
===Antoine Song===


===Graeme Wilkin (U of Colorado-Boulder)===
TBA
''Moment map flows and the Hecke correspondence for quivers''


Quiver varieties are a fundamental part of Nakajima's work in
==Fall Abstracts==
Geometric Representation Theory, but some of their basic topological
invariants (such as the cohomology ring) are not yet well-understood. In
the first part of the talk I will give the definition of a quiver variety
and describe some examples, before giving an overview (again with
examples) of some of Nakajima's constructions, one of which is the Hecke
correspondence for quivers. In the second part of the talk I will explain
a new theorem that gives an analytic description of the Hecke
correspondence in terms of the gradient flow of an energy functional.
This is related to an ongoing program to use Morse theory to study the
cohomology of quiver varieties, and, if time permits, then I will state
some conjectures in this direction.


===Yong-Geun Oh (UW-Madison)===
===Ruobing Zhang===
''Floer homology and continuous Hamiltonian dynamics''


Alexander isotopy on the n-disc exists in almost all the known categories
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.
of existing topology; e.g., diffeomorphism, homeomorphism, symplectic diffeomorphism
and symplectic homeomorphism, measure-preserving homeomorphism and others.  
In this talk, we will explain our recent result  that Alexander isotopy exists in the category
of Hamiltonian homeomorphisms which  were introduced  by Mueller and the speaker a
few years ago. As a consequence, this implies that the group
of area preserving homeomorphisms of the 2-disc (compactly supported in the interior)
is not simple. The proof uses chain-level Floer homology theory in full throttle.
We will try to give some overview of the proof in this talk.  


===Steven Simon (Courant Institute, NYU)===
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.
''Equivariant and Orthogonal Ham Sandwich Theorems''


This talk will present two generalizations of the Ham Sandwich Theorem, which states that under very broad conditions, any n finite measures on R^n can be bisected by a single hyperplane. Giving the theorem a S^0 interpretation, we provide equivariant analogues for the finite subgroups of the spheres S^1 and S^3. Secondly, we ask for the maximum number of pairwise orthogonal hyperplanes which can bisect a generic set of m measures on R^n, m<n. An ``orthogonal" Ham Sandwich Theorem is found in the form of a lower bound for this number which is defined in terms of the span of real projective space. The two generalizations are unified by finding orthogonal versions of the equivariant results.
===Emily Stark===


===Alex Suciu (Northeastern)===
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.
''Betti numbers of abelian covers''


The regular covers of a connected, finite cell
===Max Forester===
complex <i>X</i>, with group of deck transformations
a fixed abelian group <i>A</i> admit a natural parameter
space, which in the case of free abelian covers of
rank <i>r</i> is simply the Grassmannian of <i>r</i>-planes
in <i>H</i><sup>1</sup>(<i>X</i>, <b>Q</b>). 
The Betti numbers of such covers are determined by the
jump loci for homology with coefficients in rank 1
local systems on <i>X</i>, and the way these loci intersect
with certain algebraic subgroups in the group of characters 
of the fundamental group of <i>X</i>.  Under favorable
circumstances, the finiteness of those Betti numbers is
controlled by the jump loci of the cohomology ring of <i>X</i>. 
In this talk, I will discuss this circle of ideas, and give
some examples from geometry, topology, and group theory
where such computations play a role.


===Alvaro Pelayo (IAS)===
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.
''Symplectic Dynamics of integrable Hamiltonian systems''


I will start with a review the basic notions of Hamiltonian/symplectic vector field and of Hamiltonian/symplectic
===Yu Li===
group action, and the classical structure theorems of Kostant, Atiyah,
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.
Guillemin-Sternberg and Delzant on Hamiltonian torus actions.
Then I will state a structure theorem for general symplectic torus
actions, and give an idea of its proof. In the second part of the talk
I will introduce new symplectic invariants of completely integrable
Hamiltonian systems in low dimensions, and explain how these invariants
determine, up to isomorphisms, the so called "semitoric systems".
Semitoric systems are Hamiltonian systems which lie somewhere between the more
rigid toric systems and the usually complicated general integrable
systems. Semitoric systems form a fundamental class of integrable systems,
commonly found in simple physical models such as the coupled
spin-oscillator, the Lagrange top and the spherical pendulum. Parts of
this talk are based on joint work with with Johannes J. Duistermaat and
San Vu Ngoc.


[[Fall-2010-Geometry-Topology]]
== Archive of past Geometry seminars ==
2018-2019  [[Geometry_and_Topology_Seminar_2018-2019]]
<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 18:56, 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

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