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-MÃ¼ller, 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.
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.
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.
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.
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.
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 HÃ¶lder regularity is the key to make this work. Time permitting, I will mention some applications.
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.
When: Mon, November 21, 2022 - 3:00pm Where: Kirwan Hall 3206
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.
When: Mon, December 5, 2022 - 2:00pm Where: Kirwan Hall 1308
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.
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.
When: Mon, December 5, 2022 - 3:00pm Where: Kirwan Hall 3206
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 areÂ a prioriÂ orbi-spaces in general. This is based on joint work with Guangbo Xu.
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 Labourieâ€™s conjecture. I will explain our construction, which starts from minimal surfaces in R^3, and what new questions this raises.
Abstract: In 1980s, Thurstonâ€™s formulated the geometrization conjecture for 3-manifolds, and proved the hyperbolization theorem. The keys to Thurstonâ€™s proof are two bounded results for certain deformation spaces of Kleinian groups. In early 1990s, motivated by Thurstonâ€™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 Thurstonâ€™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
Abstract: The Nielsenâ€“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!
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. â€¨â€¨
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.
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.
Abstract:Â 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 TeichmÃ¼ller 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.
Abstract: Each element of the mapping class group has one of three types: periodic, pseudo-Anosov, or reducible. Â In joint work with Strenner, Taylor, and Yurttas, we give an algorithm to determine which, and also to determine some associated data. Â Our algorithm has polynomial complexity with respect to word length in the mapping class group. Â A polynomial time algorithm for a closely related problem was previously given by Bell and Webb. Â 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. Â This talk is meant to be accessible to a wide audience of mathematicians with an interest in topology.
Abstract: Lusztigâ€™s total positivity in split real Lie groups is a key ingredient in Fock and Goncharovâ€™s theory of higher TeichmÃ¼ller 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 TeichmÃ¼ller spaces. In this talk, I will give an overview of these ideas, with a goal of understanding theÂ non-commutativeÂ cluster atlases which underly these spaces. Based on joint work with Anna Wienhard, Merik Niemeyer, and Zack Greenberg.
4176 Campus Drive - William E. Kirwan Hall
College Park, MD 20742-4015
P: 301.405.5047 | F: 301.314.0827