Geometry-Topology Archives for Academic Year 2018

Geometric recursion

When: Tue, September 4, 2018 - 3:15pm
Where: Kirwan Hall 3206
Title: Geometric recursion

Speaker: Gaetan Borot (Max Planck Institute for Mathematics, Bonn) -

Abstract: Assume one is given for each topological bordered oriented surface, a 'natural' topological vector space which is a representation of the mapping class group. With some extra data and assumptions, I will explain how to construct mapping class group invariants in these spaces for surfaces of arbitrary topology, by induction on the Euler characteristic. We call "geometric recursion" this construction. For instance, one can consider the space of continuous functions over the Teichmuller space of bordered surfaces, with topology of convergence on every compact. The geometric recursion then yield continuous functions on the moduli spaces out of initial data for pairs of pants and tori with one boundary. Integrating these functions on the moduli space of bordered Riemann surfaces for fixed boundary lengths with respect to the Weil-Petersson volume form, satisfies a topological recursion. The fact that the constant function 1 can be obtained via the geometric recursion for well-chosen initial data is the content of Mirzakhani's generalization of McShane identity. The topological recursion is in this case Mirzakhani's recursion for the Weil-Petersson volumes. I will present a generalization of Mirzakhani's identity, which expresses linear statistics of the hyperbolic length of simple closed curves as a result the geometric recursion for a twist of the initial data.

Based on joint work with Jorgen Ellegaard Andersen and Nicolas Orantin.

Conformal limits and the Bialynicki-Birula stratifications of lambda-connections

When: Mon, September 10, 2018 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Brian Collier (UMD) -
Abstract: Both the Higgs bundle moduli space and the moduli space of holomorphic connections have a natural stratification induced by a C* action. In both of these stratifications, each stratum is a holomorphic fibration over a connected component of complex variations of Hodge structure. While the nonabelian Hodge correspondence provides a homeomorphism between Higgs bundles and holomorphic connections, this homeomorphism does not preserve the respective strata. The closed strata on the Higgs bundle side is the image of the Hitchin section and the closed strata in the space of flat connections is the space of opers. In this talk we will show how many of the relationships between opers and the Hitchin section extend to general strata. This is based on joint work with Richard Wentworth.

Moduli spaces of semistable sheaves; an alternative construction method

When: Mon, September 17, 2018 - 3:15pm
Where: CHE 2110
Speaker: Matei Toma (l'Universite de Lorraine) -
Abstract: For a compact Kaehler manifold (X,\omega) the Kobayashi-Hitchin correspondence gives homeomorphisms between moduli spaces of irreducible Hermitian-Yang-Mills connections and moduli spaces of stable vector bundles on X. When X is projective and \omega is a rational class, modular compactifications of the latter spaces have been constructed in Algebraic Geometry by putting appropriate classes of semistable sheaves at the boundary. These compactifications appear as global quotients. In this talk we present an alternative construction method concentrating on "local quotients" and which also covers the case when \omega is an arbitrary Kaehler class. This is the subject of joint recent work with Daniel Greb. Essential use is made of the notion introduced by Jarod Alper of a good moduli space of an algebraic stack. Besides solving a wall-crossing issue appearing in the context of projective manifolds, this alternative method is likely to extend to the general case of Kaehler manifolds.

How to reconstruct a metric by its unparameterized geodesics

When: Mon, September 24, 2018 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Vladamir Matveev (Friedrich-Schiller-Universität Jena ) -
Abstract: We discuss whether it is possible to reconstruct an affine connection, a (pseudo)-Riemannian metric or a Finsler metric
by its unparameterized geodesics, and how to do it effectively. We explain why this problem is interesting for
general relativity. We show how to understand whether all curves from a sufficiently
big family are unparameterized geodesics of a certain affine connection, and how
to reconstruct algorithmically a generic 4-dimensional metric by its unparameterized
geodesics. I will also explain how this theory helped to solve two problems
explicitly formulated by Sophus Lie in 1882. This portion of results is joint with R. Bryant, A. Bolsinov, V. Kiosak, G. Manno, G. Pucacco. At the end of my talk, I will explain that the so-called chains in the CR-geometry are geodesics of a so-called Kropina Finsler metric. I will show that sufficiently many geodesics determine the Kropina Finsler metric, which reproves and generalizes the famouse result of Jih-Hsin Cheng, 1988, that chains dermine the CR structure. This correspondence between chains and Kropina geodesics allows us to use the methods of metric geometry to study chains, we employ it to re-prove the result of H. Jacobowitz, 1985, that locally any two points of a strictly pseudoconvex CR manifolds can be joined by a chain, and generalize it to a global setting. This portion of results is joint with J.-H. Cheng, T. Marugame, R. Montgomery.

Regularity of limit curves of Anosov representations

When: Tue, September 25, 2018 - 3:15pm
Where: Kirwan Hall 2400
Speaker: Tengren Zhang (National University of Singapore) -
Abstract: Anosov representations are representations of a hyperbolic group to a non-compact semisimple Lie group that are “geometrically well-behaved”. In the case when the target Lie group is PGL(d,R), these representations admit a limit set in d-1 dimensional projective space that is homeomorphic to the boundary of the group. Under some irreducibility conditions, we give necessary and sufficient conditions for when this limit set is a C^{1,a} sub manifold. This is joint work with A. Zimmer.

Branes on Higgs bundle moduli and torsion line bundles

When: Tue, October 9, 2018 - 3:15pm
Where: Kirwan Hall 0103
Speaker: Andre Oliveira (Centro de Matematica da Universidade do Porto) -
Abstract: We study the fixed loci on the moduli space M of GL(n,C)-Higgs bundles (over a curve) for the action of tensorization by a line bundle of order n. This loci is hyperholomorphic and can be equipped with a hyperholomorphic sheaf, hence is a BBB-brane on M. Such brane is expected to be dual, via mirror symmetry, to a BAA-brane on M, i.e. to a complex Lagrangian subvariety equipped with a flat bundle. We find this BAA-brane and show that it can described via certain Hecke modifications. Finally we prove the duality statement via explicit Fourier-Mukai transform. It is noteworthy that these branes lie over the singular locus of the Hitchin fibration. Joint work together with E. Franco, P. Gothen and A. Peon-Nieto.

Positive scalar curvature on pseudomanifolds

When: Mon, October 29, 2018 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Jonathan Rosenberg (UMD) -
Abstract: This talk will describe joint work with Boris Botvinnik about the classification of simply connected manifolds of positive scalar curvature M, with a distinguished codimension 2 submanifold N, such that the metric on a tubular neighborhood of N has a natural specific form, and subject to a spin condition. This involves several interesting questions in algebraic topology and geometry of complex line bundles.

Torsors on semistable curves and the problem of degenerations

When: Mon, November 12, 2018 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Balaji Vikraman (Chennai Mathematical Institute ) -
Abstract: Let G be an almost simple, simply connected algebraic group G over the
field of complex numbers. In this talk I answer two basic questions in the classification
of G-torsors on curves. The first one is to construct a flat degeneration of the moduli
stack G-torsors on a smooth projective curve when the curve degenerates to an irreducible
nodal curve. Torsors for a generalization of the classical Bruhat-Tits group schemes to
two-dimensional regular local rings and an application of the geometric formulation of the
McKay correspondence provide the key tools. The second question is to give an intrinsic
definition of (semi)stability for a G-torsor on an irreducible nodal curve. The absence of
obvious analogues of torsion-free sheaves in the setting of G-torsors makes the question
interesting. This also leads to the construction of a proper separated coarse space for
G-torsors on an irreducible nodal curve.

Spectral Rigidity of q-differential Metrics

When: Mon, November 19, 2018 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Marissa Loving (University of Illinois (UIUC) ) -
Abstract: When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask which curves’ lengths do we really need to know? It is a classical result of Fricke that a hyperbolic metric on a surface is determined by its marked simple length spectrum. More recently, Duchin–Leininger–Rafi proved that a flat metric induced by a unit-norm quadratic differential is also determined by its marked simple length spectrum. In this talk, I will describe a generalization of the notion of simple curves to that of q-simple curves, for any positive integer q, and show that the lengths of q-simple curves suffice to determine a non-positively curved Euclidean cone metric induced by a q-differential metric.


When: Mon, December 3, 2018 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Joseph Hoisington (Smith College) -
Abstract: TBA