Geometry-Topology Archives for Fall 2022 to Spring 2023
Organizational meeting
When: Mon, August 30, 2021 - 3:00pmWhere: Kirwan 3206
for the Geometry-Topology seminar
Positivity and connected components of character varieties
When: Mon, September 13, 2021 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Brian Collier (UCR) - https://sites.google.com/view/brian-collier/home
Abstract: In this talk, we will define a special class of nilpotent elements of complex semisimple Lie algebra and describe how a Slodowy slice construction defines connected components of moduli spaces of G-Higgs bundles on a compact Riemann surface for specific real groups G. Through the nonabelian Hodge correspondence, these components define connected components of the G-character variety with many nice features. We will then describe how these nice features, together with recent work of Guichard-Labourie-Wienhard on positive representations imply that such components are higher Teichmuller spaces, that is, they consist entirely of discrete and faithful representations. In the remaining time, we will describe what remains to be proven to finish the classification of higher Teichmuller spaces.
A zero surgery obstruction from involutive Heegaard Floer homology
When: Mon, September 20, 2021 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Peter Johnson (University of Virginia) - https://peterkj1.github.io/
Abstract: A fundamental result in 3-manifold topology due to Lickorish and Wallace says that every closed oriented connected 3-manifold can be realized as surgery on a link in the 3-sphere. One may therefore ask: which 3-manifolds can be obtained by surgery on a link with a single component, i.e. a knot, in the 3-sphere? More specifically, one can ask: which 3-manifolds are obtained by zero surgery on a knot in the 3-sphere? In this talk, we give a brief outline of some known results to this question in the context of small Seifert fibered spaces. We then sketch a new method, using involutive Heegaard Floer homology, to show that certain 3-manifolds cannot be obtained by zero surgery on a knot in the three sphere. In particular, we produce a new infinite family of weight 1 irreducible small Seifert fibered spaces with first homology Z which cannot be obtained by zero surgery on a knot in the 3-sphere, extending a result of Hedden, Kim, Mark and Park.
The Simplicity Conjecture
When: Mon, September 27, 2021 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Daniel Cristofaro-Gardiner (UMD) -
Abstract: In the 60s and 70s, there was a flurry of activity concerning the question of whether or not various subgroups of homeomorphism groups of manifolds are simple, with beautiful contributions by Fathi, Kirby, Mather, Thurston, and many others. A funnily stubborn case that remained open was the case of area-preserving homeomorphisms of surfaces. For example, for balls of dimension at least 3, the relevant group was shown to be simple by work of Fathi from the 1970s, but the answer in the two-dimensional case was not known. I will explain recent joint work proving that the group of compactly supported area preserving homeomorphisms of the two-disc is in fact not a simple group, which answers the ``Simplicity Conjecture” in the affirmative. Our proof uses a new tool for studying area-preserving surface homeomorphisms, called periodic Floer homology (PFH) spectral invariants; these recover the classical Calabi invariant of monotone twists. Time permitting, I will also briefly mention a generalization of our result to compact surfaces of any genus.
The spheres of sol
When: Mon, October 4, 2021 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Richard Schwartz (Brown University) - https://www.math.brown.edu/reschwar/
Abstract: This is joint work with Matei Coiculescu.
Sol is probably the weirdest of the 8 Thurston geometries.
In this talk I will give an exact characterization of when
a geodesic segment in Sol is distance minimizing and
then explain why the result implies that the metric
spheres in Sol are topological spheres, smooth away
from at most 4 arcs. One form of the characterization
is a homogenous function on R^3 involving the
arithmetic-geometric mean of Gauss. I'll illustrate the
talk with some computer demos.
Contact type hypersurfaces in R^4
When: Mon, October 11, 2021 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Tom Mark (UVA)
Abstract: Given a closed, orientable 3-manifold M, it is a subtle and often difficult problem to determine whether M may be smoothly embedded in R^4. Even among integer homology spheres, and restricting to special classes such as Seifert manifolds, the problem is open in general, with positive answers for some such manifolds and negative answers in other cases. However, recent work shows that if suitable geometric conditions are imposed then there is a uniform answer for an important class of 3-manifolds called Brieskorn homology spheres: no such 3-manifold admits an embedding as a hypersurface "of contact type" in R^4, which is to say as the boundary of a region that is convex from the point of view of symplectic geometry. I'll describe further context and background for this result, which is joint work with Bülent Tosun, give some highlights of the proof, and indicate connections with complex geometry and further potential directions.
Finite conjugacy classes and split exact cochain complexes
When: Mon, October 18, 2021 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Christian Rosendal (UMD) -
Abstract: We will present the theory behind and new results on the cohomology of super-reflexive Banach G-modules X, where G is a countable discrete group. In particular, we shall show how the cohomology is controlled by the FC-centre of G, that is, the subgroup of elements having finite conjugacy classes. For example, using purely cohomological tools, we show that when X is an isometric super-reflexive Banach G-module so that X has no almost invariant unit vectors under the action of the FC-centre, then the associated cochain complex is split exact. Further connections to the work of Bader-Furman-Gelander-Monod, Nowak, and Bader-Rosendal-Sauer will be presented.
Coarse embeddings and homological filling functions
When: Mon, October 25, 2021 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Mark Pengitore (University of Virginia) - https://pengitore.weebly.com/
Abstract: In this talk, we will relate homological filling functions and the existence of coarse embeddings. In particular, we will demonstrate that a coarse embedding of a group into a group of geometric dimension 2 induces an inequality on homological Dehn functions in dimension 2. As an application of this, we are able to show that if a finitely presented group coarsely embeds into a hyperbolic group of geometric dimension 2, then it is hyperbolic. Another application is a characterization of subgroups of groups with quadratic Dehn function. If there is enough time, we will talk about various higher dimensional generalizations of our main result.
Building homogeneous parabolic geometries out of curvature
When: Mon, November 1, 2021 - 3:00pmWhere:
Speaker: Jacob Erickson (UMD)
When studying parabolic geometries, such as conformal, projective, and CR geometries, it can be useful to have examples with many automorphisms despite nontrivial curvature. In this talk, we will explore a recent result that allows us to construct globally homogeneous parabolic geometries from a choice of curvature, significantly extending a previous local result due to Kruglikov and The.
Mixed Hodge Structures on Character Varieties of Nilpotent Groups
When: Mon, November 8, 2021 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Sean Lawton (GMU) -
Abstract: Let R be the connected component of the identity of the variety of representations of a finitely generated nilpotent group N into a connected reductive complex affine algebraic group G. We determine the mixed Hodge structure on the representation variety R and on the character variety R//G. See https://arxiv.org/abs/2110.07060. This work is in collaboration with C. Florentino and J. Silva.
Conformal groups of compact Lorentzian manifolds
When: Mon, November 15, 2021 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Karin Melnick (UMD) -
Abstract: The Lorentzian Lichnerowicz Conjecture is a Lorentzian analogue of the Ferrand-Obata Theorem on conformal transformation groups of Riemannian manifolds. I will discuss my verification of the conjecture in dimension three, for real-analytic metrics, in recent joint work with C. Frances.
The Hitchin connection for parabolic G-bundles
When: Mon, November 22, 2021 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Richard Wentworth (UMD) -
For a simple and simply connected complex group G, I will discuss some elements of the proof of the existence of a flat projective connection on the bundle of nonabelian theta functions on the moduli space of semistable parabolic G-bundles over families of smooth projective curves with marked points. Under the isomorphism with the bundle of conformal blocks, this connection is equivalent to the one constructed by conformal field theory. This is joint work with Indranil Biswas and Swarnava Mukhopadhyay.
A para-hyperKähler structure on the space of GHMC AdS-manifolds
When: Mon, November 29, 2021 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Filippo Mazzoli (UVA)
Abstract: A celebrated result by Donaldson asserts that the space of almost-Fuchsian manifolds admits a natural hyperKähler structure invariant under the action of the mapping class group, extending the Weil-Petersson Kähler structure of Teichmüller space. In this talk we will discuss the occurrence of a similar phenomenon for the deformation space of globally hyperbolic Anti-de Sitter 3-manifolds. In particular we will see how such space carries a para-hyperKähler structure, where a pseudo-Riemannian metric and 3 symplectic structures coexist with a integrable complex structure and two para-complex structures, satisfying the relations of para-quaternionic numbers. This project is a joint work with Andrea Seppi (Université Grenoble Alpes) and Andrea Tamburelli (Rice University).
Positive scalar curvature on manifolds with boundary
When: Mon, November 29, 2021 - 4:00pmWhere: Kirwan Hall 3206
Speaker: Jonathan Rosenberg (UMD) -
Abstract: Since work of Gromov and Lawson around 1980, we have known (under favorable circumstances)
necessary and sufficient conditions for a closed manifold to admit a Riemannian metric of
positive scalar curvature, but not much was known about analogous results for manifolds
with boundary (and suitable boundary conditions). In joint work with Shmuel Weinberger of the
University of Chicago, we give necessary and sufficient conditions in many cases for
compact manifolds with non-empty boundary to admit:
(a) a positive scalar curvature metric which is a product metric in a neighborhood of the boundary
or
(b) a positive scalar curvature metric with positive mean curvature on the boundary.
Rigidity of Uniform Roe Algebras
When: Mon, December 6, 2021 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Bruno Braga (UVA) -
Abstract: Given a uniformly locally finite metric space $X$ (also known as a metric space with bounded geometry), one can define a C*-algebra of bounded operators on $\ell_2(X)$ called the uniform Roe algebra of $X$, denoted by $C*_u(X)$. The quintessential examples of uniformly locally finite metric spaces are the Cayley graphs of finitely generated groups endowed with the shortest path metric. Uniform Roe algebras were introduced by John Roe in the context of index theory of elliptical operators on noncompact manifolds and they capture many of the large-scale (i.e., coarse) geometric properties of $X$. The rigidity problem for uniform Roe algebras is the problem of whether such algebras remember all the large-scale aspects of their base metric spaces. Precisely, if $C*_u(X)$ and $C*_u(Y)$ are isomorphic as C*-algebras, does it follow that X and Y are coarsely equivalent? This problem has recently been solved by myself together with F. Baudier, I. Farah, A. Khukhro, A. Vignati, and R. Willett. In this talk, I will give an overview of this rigidity problem and explain the main steps involved in its solution.
Seminar cancelled (to be rescheduled to spring)
When: Mon, December 13, 2021 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Sara Maloni (UVA) -
Contact manifolds with exactly two Reeb orbits
When: Mon, January 31, 2022 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Dan Cristofaro-Gardiner (UMD) -
Abstract: The influential "Weinstein conjecture" asserts that for every contact form on a closed manifold, the associated Reeb vector field has at least one closed orbit. For contact three-manifolds, this was proved by Taubes around 2007. In fact, in dimension three, it was later shown that there are always at least two distinct Reeb orbits; moreover, examples with precisely two orbits exist. I will discuss recent joint work aimed at better understanding the case of two orbits. We show that if there are exactly two orbits, then the manifold is a lens space, and the orbits are the core circles of a genus one Heegaard splitting. We also obtain further information about the Reeb dynamics. For example, the Reeb flow has a disk-like global surface of section and so its dynamics are described by a pseudorotation; and, in the case of the three-sphere, the product of the periods of the orbits is equal to the contact volume of the manifold.
Towards an extension of Taubes' Gromov invariant to Calabi--Yau 3-folds
When: Mon, February 7, 2022 - 3:00pmWhere: Online. Meeting ID: 591 226 6463. The password is the first 6 Fibonacci numbers, starting with 1,1.
Speaker: Mohan Swaminathan (Princeton) - https://sites.google.com/view/mohanswaminathan/home
Abstract: I will describe the construction of a virtual count of
embedded pseudo-holomorphic curves of a given genus in a Calabi--Yau
3-fold lying in two times a primitive homology class. The result is an
integer-valued symplectic deformation invariant which can be viewed
as an analogue of Taubes' Gromov invariant (which is defined for
symplectic 4-manifolds). The construction depends on a detailed
bifurcation analysis of the moduli space of embedded curves along a
generic path of almost complex structures and is partly motivated by
Wendl's resolution of Bryan--Pandharipande's super-rigidity
conjecture. This is based on joint work with Shaoyun Bai.
A Weyl law for PFH spectral invariants
When: Mon, February 14, 2022 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Rohil Prasad (Princeton) -
Abstract: Hutchings' periodic Floer homology (PFH) is a Floer-theoretic
invariant associated to an area-preserving diffeomorphism of a closed,
oriented surface. It has a set of associated quantitative invariants,
called "PFH spectral invariants", which encode information about periodic
orbits of this diffeomorphism. The main topic of this talk is a "Weyl law"
for PFH spectral invariants, which relates the asymptotics of PFH spectral
invariants to the Calabi invariant of Hamiltonian surface diffeomorphisms.
We will state the Weyl law and discuss a bit of its proof, which relies on
a quantitative analysis of the Lee-Taubes isomorphism of PFH and monopole
Floer homology. Time permitting, we will discuss some applications to the
dynamics of area-preserving surface diffeomorphisms. This talk is based on
joint work with Dan Cristofaro-Gardiner and Boyu Zhang.
Lagrangian Cobordisms and Enriched Knot Diagrams
When: Mon, February 21, 2022 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Ipsita Datta (Institute for Advanced Study) - https://www.math.ias.edu/~ipsi/
Abstract: We present new obstructions to the existence of Lagrangian cobordisms in \R^4. The obstructions arise from studying moduli spaces of holomorphic disks with corners and boundaries on immersed objects called Lagrangian tangles. The obstructions boil down to area relations and sign conditions on disks bound by knot diagrams of the boundaries of the Lagrangian. We present examples of pairs of knots that cannot be Lagrangian cobordant and knots that cannot bound Lagrangian disks.
Combinatorics of graphs via right-angled Artin groups
When: Mon, February 28, 2022 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Thomas Koberda (University of Virginia) -
Abstract: Right-angled Artin groups are a rich class of finitely presented groups that are determined by a finite graph, and which play a central role in combinatorial and geometric group theory. Conversely, a right-angled Artin group uniquely determines a graph, and so the combinatorics of the underlying graph and the algebraic structure of the right-angled Artin group should be intimately related. In this talk, I will survey some of the dictionary of known results in this area: namely, I will investigate the way in which algebraic properties of right-angled Artin groups are reflected in the combinatorics of graphs, and the way in which the combinatorics of graphs are reflected in the group theory of right-angled Artin groups. I will finish with recent work joint with R. Flores and D. Kahrobaei, in which we characterize planarity of graphs via the algebraic structure of right-angled Artin groups.
TBA
When: Sun, March 6, 2022 - 3:00pmWhere: Zoom
Speaker: Yusheng Luo (Stony Brook) -
Generalized soap bubbles and the topology of manifolds with positive scalar curvature
When: Mon, March 7, 2022 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Chao Li (NYU) -
Abstract: It has been a classical conjecture that closed aspherical manifolds do not admit any Riemannian metric of positive scalar curvature. I will review some history of this question, and present a solution in dimensions 4 and 5. The proof is based on analyzing generalizes soap bubbles - surfaces that are stable solutions to the prescribed mean curvature problem. If time permits, I will also discuss some other applications of the key idea. This talk is based on joint work with O. Chodosh.
Cantor sets of infinite staircases of ellipsoid embedding functions of Hirzebruch surfaces
When: Mon, March 28, 2022 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Morgan Weiler (Cornell) -
Abstract: Symplectic embeddings generalize coordinate changes in physical systems, and the question of whether one symplectic manifold embeds into another is quite delicate. In 2012, McDuff and Schlenk found that as a four-dimensional ellipsoid's eccentricity varies, the size of the smallest 4-ball it symplectically embeds into (i.e. the value of its ellipsoid embedding function) is governed by a recursive pattern of rigidity and flexibility called an "infinite staircase." Since then, several authors have studied infinite staircases. We will present two new types of infinite staircases and will discuss an almost complete description of which Hirzebruch surfaces (one-point blowups of the projective plane) have ellipsoid embedding functions containing infinite staircases. The set of such Hirzebruch surfaces is modeled on the Cantor set. This partially answers a conjecture of Cristofaro-Gardiner--Holm--Mandini--Pires in the case of Hirzebruch surfaces.
Almost Complex Structures on Homotopy Complex Projective Spaces
When: Mon, April 4, 2022 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Keith Mills (UMD) -
Abstract: We completely answer the question of when a homotopy $\CP^n$, a smooth closed manifold with the oriented homotopy type of $\CP^n$, admits an almost complex structure for $3 \leq n \leq 6$. For $3 \leq n \leq 5$ all homotopy $\CP^n$s admit almost complex structures, while for $n=6$ there exist homotopy $\CP^n$s that do not. Our methods provide a new proof of a result of Libgober and Wood on the classification of almost complex structures on homotopy $\CP^4$s.
The Ruelle Invariant And Convexity In Higher Dimensions
When: Mon, April 11, 2022 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Julian Chaidez (Princeton/IAS) -
Abstract: I will explain how to construct the Ruelle invariant of a symplectic cocycle over an arbitrary measure preserving flow. I will provide examples and computations in the case of Hamiltonian flows and Reeb flows (in particular, for toric domains). As an application of this invariant, I will construct toric examples of dynamically convex domains that are not symplectomorphic to convex ones. This is joint work with Oliver Edtmair and extends our previous work in dimension three.
Splitting sequences for endperiodic maps
When: Mon, April 18, 2022 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Michael Landry (Wash. U.) -
Abstract: Given a pseudo-Anosov diffeomorphism f of a surface, Agol used ideas of Hamenstadt to construct a periodic train track splitting sequence which is an invariant of the homotopy class of f. There is a strong, precise sense in which this invariant detects a fibered face of the Thurston norm ball of the mapping torus of f, an object of general interest in low-dimensional topology. In this talk we will discuss a similar picture: given an endperiodic automorphism F we construct a train track splitting sequence using Handel-Miller theory. This invariant is unique up to a natural equivalence relation and detects a "Handel-Miller foliation cone" of the compactified mapping torus of F. Such a cone is analogous to the cone over a fibered face, but in the more general setting of sutured manifolds. Our construction also completes half of the proof of an unwritten theorem of Gabai-Mosher. This is joint work in preparation with Chi Cheuk Tsang.
Localization and flexibilization in symplectic geometry
When: Mon, April 25, 2022 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Oleg Lazarev (UMass) -
Abstract: Localization is an important construction in algebra and topology that allows one to study global phenomena a single prime at a time. Flexibilization is an operation in symplectic topology introduced by Cieliebak and Eliashberg that makes any two symplectic manifolds that are diffeomorphic (plus a bit of tangent bundle data) become symplectomorphic. In this talk, I will explain that it is fruitful to view flexibilization as a localization (at the 'prime' zero ). Building on work of Abouzaid and Seidel, l will also give examples of new localization functors of symplectic manifolds (up to stabilization and subcriticals) that interpolate between flexible and rigid symplectic geometry and can be viewed as symplectic analogs of topological localization of Sullivan, Quillen, and Bousfield. This talk is based on joint work with Z. Sylvan and H. Tanaka.
TBA
When: Sun, May 1, 2022 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Dan Margalit (Gatech) -
Geometric quantization of (some) Poisson manifolds
When: Mon, May 2, 2022 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Jonathan Weitsman (Northeastern) -
Abstract: We review geometric quantization in symplectic geometry, and then study recent proposals, joint with Guillemin and Miranda, on extending such constructions to the Poisson case.
Slice knots in definite 4-manifolds
When: Mon, May 9, 2022 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Alexandra Kjuchukova (Notre Dame) -
Abstract: Let $K\subset S^3$ be a knot and let $X$ be a closed smooth four-manifold. Does $K$ bound a smooth null-homologous disk properly embedded in $X$ minus an open ball? If so, we say $K$ is smoothly H-slice in $X$. The classification of H-slice knots in a 4-manifold $X$ can help detect exotic smooth structures on $X$. I will describe new tools to compute the $\mathbb{CP}^2$ slicing number of a knot $K$, which is the smallest $m$ such that $K$ is smoothly H-slice in $\#^m\mathbb{CP}^2$. I will also discuss the analogous question in the topological category, and give examples where the smooth $\mathbb{CP}^2$ slicing number is strictly larger than its topological counterpart. This talk is based on arXiv:2112.14596.