Geometry-Topology Archives for Fall 2023 to Spring 2024
Sharp symplectic embeddings via integrable systems and ECH capacities
When: Mon, September 12, 2022 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Brayan Ferreira (IAS) -
Abstract: The question of whether a symplectic manifold embeds into another is central in Symplectic topology. Since the Gromov nonsqueezing theorem, it is known that this is a different problem from volume preserving embeddings. Embedded contact homology (ECH) has been shown to be very useful to obtain obstructions for symplectic embeddings in dimension 4. In this talk we will recall some results in dimension 4 due to Mcduff-Schlenk, Hutchings, Frenkel-Müller, Cristofaro-Gardiner, and Ramos. Furthermore, we will explain some recent results about embeddings into the disk cotangent bundle of the round sphere and of some spheres of revolution. This is a joint work with Vinicius Ramos and Alejandro Vicente.
A generalization of Geroch's conjecture
When: Mon, September 19, 2022 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Florian Johne (Columbia) -
Abstract:
Closed manifolds with topology N = M x S^1 do not admit metrics of
positive Ricci curvature by the theorem of Bonnet-Myers, while the the
resolution of the Geroch conjecture implies that the torus T^n does
not admit a metric of positive scalar curvature. In this talk we
explain a non-existence result for metrics of positive m-intermediate
curvature (a notion of curvature reducing to Ricci curvature for m =
1, and scalar curvature for m = n-1) on closed manifolds
with topology N^n = M^{n-m} x T^m for n <= 7. Our proof uses
minimization of weighted areas, the associated stability inequality,
and delicate estimates on the second fundamental form. This is joint
work with Simon Brendle and Sven Hirsch.
Pleated surfaces in PSL_d(C)
When: Mon, October 3, 2022 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Sara Maloni (Virginia) -
Abstract:
Pleated surfaces are an important tool introduced by Thurston to study hyperbolic 3-manifolds and can be described as piecewise totally geodesic surfaces, bent along a geodesic lamination lambda. Bonahon generalized this notion to representations of surface groups in PSL_2(C) and described a holomorphic parametrization of the resulting open charts of the character variety in term of shear-bend cocycles.
In this talk I will discuss joint work with Martone, Mazzoli and Zhang, where we generalize this theory to representations in PSL_d(C). In particular, I will discuss the notion of d-pleated surfaces, and their holomorphic parametrization.
Compact spaces of surface group representations in genus zero
When: Mon, October 10, 2022 - 3:00pmWhere: 3206
Speaker: Arnaud Maret (Heidelberg) -
Abstract. This talk is about a special kind of representations of the fundamental group of a punctured sphere into Hermitian Lie groups. Even if the target groups are never compact, the moduli space of representations is compact. I will explain how to parametrize these representations in the most basic case. The coordinates we describe are analogous to Fenchel-Nielsen coordinates for Teichmüller space.
Higher dimensional examples of a smooth closing lemma
When: Mon, October 17, 2022 - 3:00pmWhere: Kirwan Hall 0411
Speaker: Shira Tanny (IAS) -
Abstract: Given a flow on a manifold, an old question is whether it is possible to perturb it in order to create a periodic orbit passing through a given region. Statements of this kind are called "closing lemmas". An interesting relation between this problem and pseudo-holomorphic curves was found by Irie in dimension 3. I will discuss such a relation in general dimensions, including a proof of a conjecture of Irie regarding certain flows on the boundaries of ellipsoids. All symplectic preliminaries will be explained. This is a joint work with Julian Chaidez, Ipsita Datta and Rohil Prasad.
Affine structures and the twisted cubic cone
When: Mon, October 24, 2022 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Bill Goldman (UMD) -
Abstract: In 1950 Kuiper classified complete affine structures on 2-tori.
Later Baues showed that the deformation space (the analog of Teichmuller space),
is homeomorphic to R^2, and in particular is Hausdorff.
The mapping group action is the usual LINEAR action of GL(2,Z) on R^2
whose quotient (the analog of the Riemann moduli space) is thus an intractable
non-Hausdorff mess. Last year Deligne observed the deformation space
naturally identifies with the cone on the twisted cubic curve in real projective $3$-space.
In my talk I will describe several perspectives on this striking result.
Transversal Hölder cohomology for flat Wieler solenoids
When: Mon, October 31, 2022 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Rodrigo Trevino (UMD) -
Abstract: de Rham regularization is a tool to find an isomorphism between the smooth de Rham cohomology of a smooth manifold and the de Rham cohomology using forms with coefficients in some Banach space of functions of finite regularity. Flat Wieler solenoids are inverse limit spaces with the local product structure of a Euclidean space and a Cantor set. In this talk I will discuss how one can obtain a regularization-type of result for these spaces and how understanding transversal Hölder regularity is the key to make this work. Time permitting, I will mention some applications.
Counting closed curves in hyperbolic surfaces
When: Mon, November 7, 2022 - 3:00pmWhere: https://umd.zoom.us/j/5665210745
Speaker: Pouya Honaryar (Toronto) -
Fixing a hyperbolic surface X of genus $g > 1$, Mirzakhani proved that the number of simple closed curves (that is, closed curves without self-intersection) of length at most $L$ in X grows like a constant time $L^{6g - 6}$. Later, Eskin-Mirzakhani-Mohammadi obtained a power-saving error term for this count. In this talk, I will discuss a closely related counting problem, also considered by Mirzakhani, and discuss the recent advancements made by Arana-Herrera in obtaining an error term for this problem. Then I describe a work in progress that builds on and generalizes his result, and time permitting, I will mention the new ingredients used in the proof.
Convergence of the gradient flow of renormalized volume
When: Mon, November 21, 2022 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Frenco Pallette (Yale) -In this talk, we will study the gradient flow of renormalized volume with respect to the Weil-Peterson metric for (relatively) acylindrical convex cocompact hyperbolic 3-manifolds. We will see how, given any starting metric, the flow converges towards the metric M_geod with totally geodesic convex core boundary. This is based on joint work with Martin Bridgeman and Ken Bromberg.
Title: Constructing proper affine actions via higher strip deformations
When: Mon, December 5, 2022 - 2:00pmWhere: Kirwan Hall 1308
Speaker: Neza Korenjak (UT Austin) - Abstract: Free groups are negatively curved, but they still admit properly discontinuous actions on affine space, which is flat. This allows us to utilize hyperbolic geometry to study affine actions. In this talk, we will generalize an approach of Danciger-Gueritaud-Kassel for constructing proper affine actions in three dimensions. We use a hyperbolic surface to construct higher strip deformations, which can be used to define proper actions of Fuchsian free groups on affine (4n-1)-space for any n.
Eigenvalue asymmetry for convex real projective surfaces
When: Mon, December 5, 2022 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Jeff Danciger (UT Austin) -abstract: A convex real projective surface is one obtained as the quotient of a properly convex open set in the projective plane by a discrete subgroup of SL(3,R), called the holonomy group, that preserves this convex set. The most basic examples are hyperbolic surfaces, for which the convex set is an ellipse, and the holonomy group is conjugate into SO(2,1). In this case, the eigenvalues of elements of the holonomy group are symmetric. More generally, the asymmetry of the eigenvalues of the holonomy group is a natural measure of how far a convex real projective surface is from being hyperbolic. We study the problem of determining which elements (and more generally geodesic currents) may have maximal eigenvalue asymmetry. We will present some limited initial results that we hope may be suggestive of a bigger picture. Joint work with Florian Stecker.
TBA
When: Mon, January 30, 2023 - 3:00pmWhere: Kirwan Hall 3206
Speaker: David Futer (Temple) -
Title: Mazur and Jester 4-manifolds
When: Mon, February 6, 2023 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Jack Calcut (Oberlin U) -
Abstract: Mazur and Poénaru constructed the first compact, contractible manifolds distinct from disks. More recently, Sparks modified Mazur's construction and defined Jester manifolds. Sparks used Jester manifolds to produce compact, contractible 4-manifolds distinct from the 4-disk that split as the union of two 4-disks meeting in a 4-disk. We present several very different proofs that all Mazur and Jester manifolds are not 4-disks. We discuss the problem of distinguishing these 4-manifolds from one another. And, we present pertinent questions on knots in S^1xS^2 and hyperbolic triangle groups. This is joint work with Alexandra Du.
Arnold conjecture over integers
When: Mon, February 20, 2023 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Shaoyun Bai (Simons Center for Geometry and Physics) - https://web.math.princeton.edu/~shaoyunb/
Abstract: We show that for any compact symplectic manifold, the number of fixed points of any non-degenerate Hamiltonian diffeomorphism is bounded from below by a version of integral Betti number which takes account of torsions of all characteristics. The proof relies on recent advances on defining integer-valued counts from moduli spaces of J-holomorphic curves, which are a priori orbi-spaces in general. This is based on joint work with Guangbo Xu.
Non-uniqueness of minimal surfaces in locally symmetric spaces
When: Mon, February 27, 2023 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Peter Smillie (Mathematical Institute, Heidelberg University) - https://www.mathi.uni-heidelberg.de/~psmillie/
Abstract: If G is a split real Lie group of rank 2, for instance SL(3,R), and S is a closed surface of genus at least 2, then Labourie showed that every Hitchin representation of pi_1(S) into G admits a unique equivariant minimal surface. As Labourie pointed out, this lets you parametrise the space of Hitchin representations by the total space of a vector bundle over the Teichmuller space of S. He conjectured that uniqueness should hold more generally, at least for all SL(n,R).
In joint work with Nathaniel Sagman, we show that for any split G of rank at least 3, and for any S, there is a Hitchin representation with two distinct equivariant minimal surfaces, disproving Labourie’s conjecture. I will explain our construction, which starts from minimal surfaces in R^3, and what new questions this raises.
higher complex structures
When: Mon, March 6, 2023 - 2:00pmWhere: Kirwan Hall 1311
Speaker: Alexander Thomas (Heidelberg) -
Boundedness problems in conformal dynamics
When: Mon, March 6, 2023 - 3:00pmWhere: Zoom:https://umd.zoom.us/s/5665210745
Speaker: Yusheng Luo (Stony Brook) -
https://umd.zoom.us/s/5665210745
Abstract:
In 1980s, Thurston’s formulated the geometrization conjecture for 3-manifolds, and proved the hyperbolization theorem. The keys to Thurston’s proof are two bounded results for certain deformation spaces of Kleinian groups. In early 1990s, motivated by Thurston’s boundedness theorem and the Sullivan dictionary, McMullen conjectured that certain hyperbolic components of rational maps are bounded.
In this talk, I will start with a historical discussion on a general strategy of the proof of Thurston’s boundedness theorem. I will then explain how a similar strategy could work for rational maps, and discuss some recent breakthrough towards McMullen's boundedness conjecture
Canonical for free group automorphisms
When: Mon, March 13, 2023 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Jean Pierre Mutanguha (Princeton) -
Abstract: The Nielsen–Thurston theory of surface homeomorphisms can be thought of as a surface analogue to the Jordan canonical form. I will discuss my progress in developing a similar canonical form for free group automorphisms. (Un)Fortunately, free group automorphisms can have arbitrarily complicated behaviour. This is a significant barrier to translating arguments that worked for surfaces into the free group setting; nevertheless, the overall ideas/strategies do translate!
Classifying plane curves and symplectic 4-manifolds using braid groups: The symplectic isotopy conjecture in CP^2
When: Mon, April 3, 2023 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Amitesh Datta (Princeton University) - https://amiteshdatta.wixsite.com/amitesh-datta/
Abstract: The question of which symplectic 4-manifolds are complex projective surfaces reduces in principle (via branched covering constructions) to the question of which symplectic curves in the complex projective plane CP^2 are isotopic to algebraic curves - the latter is known as the symplectic isotopy problem.
The longstanding symplectic isotopy conjecture posits that every smooth symplectic curve in CP^2 is isotopic to an algebraic curve. In this talk, I will describe a new algebraic theory I have developed on the braid groups in order to prove that all degree three symplectic curves in CP^2 with only A_n-singularities (an A_n-singularity is locally modelled by w^2 = z^n and includes nodes and cusps) are isotopic to algebraic curves. The proof is independent of Gromov's theory of pseudoholomorphic curves, and the theory also addresses the symplectic isotopy conjecture in full generality in upcoming work.
I will review the necessary background from scratch, and along the way, we will discuss beautiful ideas from algebraic geometry, symplectic geometry, monodromy theory and geometric group theory and how they unite in the study of plane curves and 4-manifolds.
BAA branes on the Hitchin moduli space from solutions to the extended Bogomolny equations
When: Mon, April 10, 2023 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Panagiotis Dimakis (Stanford University) -
Abstract: BAA branes are complex Lagrangian submanifolds of the Hitchin space. Recently, there has been interest in these objects due to their appearance in mirror symmetry conjectures and due to their intimate connection with the geometry of the Hitchin space. In this talk I will introduce the above notions. Then I will introduce the extended Bogomolny equations and explain how their solutions lead to holomorphic data associated with a Riemann surface. I will show that the moduli of these holomorphic data is a BAA brane. Some of the BAA branes obtained this way are known but some are new.
Extension of homeomorphisms and vector fields of the circle: From Anti-de Sitter to Minkowski geometry
When: Mon, April 24, 2023 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Farid Diaf (Grenoble University) -
Abstract: In 1990, Mess gave a proof of Thurston's earthquake theorem using the Anti-de Sitter geometry. Since then, several of Mess's ideas have been used to investigate the correspondence between surfaces in 3-dimensional Anti de Sitter space and Teichmüller theory.
In this spirit, we investigate the problem of the existence of vector fields giving infinitesimal earthquakes on the hyperbolic plane, using the so-called Half-pipe geometry which is the dual of Minkowski geometry in a suitable sense. In particular, we recover Gardiner's theorem, which states that any Zygmund vector field on the circle can be represented as an infinitesimal earthquake. Our findings suggest a connection between vector fields on the hyperbolic plane and surfaces in 3-dimensional Half-pipe space, which may be suggestive of a bigger picture.
Fast Nielsen--Thurston Classification
When: Mon, May 1, 2023 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Dan Margalit (Gatech) -
Abstract: Each element of the mapping class group has one of three types: periodic, pseudo-Anosov, or reducible. In joint work with Strenner, Taylor, and Yurttas, we give an algorithm to determine which, and also to determine some associated data. Our algorithm has polynomial complexity with respect to word length in the mapping class group. A polynomial time algorithm for a closely related problem was previously given by Bell and Webb. In this talk we will explain the piecewise linear action of the mapping class group on the space of measured foliations, and how we use piecewise-linear algebra to determine the type of a mapping class. This talk is meant to be accessible to a wide audience of mathematicians with an interest in topology.
Positivity, Teichmüller Theory and Generalisations
When: Mon, May 22, 2023 - 3:00pmWhere: Kirwan Hall 3206
Speaker: Dani Kaufman (University of Copenhagen) -
Abstract: Lusztig’s total positivity in split real Lie groups is a key ingredient in Fock and Goncharov’s theory of higher Teichmüller spaces. Recently, Guichard and Wienhard have given a generalisation of total positivity to non-split real Lie groups, called Theta-Positivity which may be used to construct new higher Teichmüller spaces. In this talk, I will give an overview of these ideas, with a goal of understanding the non-commutative cluster atlases which underly these spaces. Based on joint work with Anna Wienhard, Merik Niemeyer, and Zack Greenberg.