Geometry-Topology Archives for Academic Year 2019

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.

The Total Curvature and Betti Numbers of Complex Projective Manifolds

When: Mon, December 3, 2018 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Joseph Hoisington (Smith College) -
Abstract: We will show that the Betti numbers of a complex projective manifold can be bounded above in terms of its total curvature, and we will characterize the complex projective manifolds whose total curvature is minimal. These results extend a classical family of theorems proved by Chern and Lashof to complex projective space.

On the geometric analysis of BPS monopoles

When: Mon, February 11, 2019 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Akos Nagy (Duke University) -
Abstract: While the 3-dimensional BPS equations have long been studied by mathematicians and physicists as well, many conjectured analytic properties of solutions have not yet been rigorously proved. In particular, the asymptotic behavior of finite energy solution, which is commonly used in the literature, has only been proved for SU(2)-monopoles by Jaffe and Taubes. The issue becomes even more complicated, and less explored, when the solutions have nonmaximal symmetry breaking. In this talk, I will give a detailed introduction to the problem, and report on our contribution to the theory. This is a joint project with Benoit Charbonneau and Gonçalo Oliveira.

A structured spectrum of Clifford modules representing K-theory

When: Mon, February 18, 2019 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Peter Ulrickson (The Catholic University of America) -
Abstract: Atiyah, Bott, and Shapiro revealed a close connection between K-theory and Clifford algebras. I will describe an E-infinity ring spectrum representing topological K-theory which is made from Clifford modules, inspired by the Atiyah-Bott-Shapiro construction. I will further describe a notion of Clifford system, and sketch a way to produce a spectrum from an abelian category, generalizing the construction of topological K-theory. This ongoing project is joint with Dmitri Pavlov.

Polylogarithms, motivic cohomology, and quiver mutations

When: Mon, February 25, 2019 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Christian Zickert (UMD) -
Abstract: We discuss the polylogarithm function and its role in motivic cohomology. Much of the talk will be a survey of Goncharov's Bloch complexes that conjecturally compute the rational motivic cohomology of a field. Goncharov's work uses a variant of polylogarithm which is a continuous real valued function on CP^1. We shall introduce a complex valued polylogarithm defined on the universal cover of C-{0,1}, and lift Goncharov's Bloch complexes to a complex which we speculate computes the integral motivic cohomology. If time permits, we shall discuss methods for finding functional relations from quivers.


When: Mon, March 11, 2019 - 3:15pm
Where: Kirwan Hall 1313

Counting square-tiled surfaces with prescribed real and imaginary foliations

When: Mon, March 25, 2019 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Francisco Arana Herrera (Stanford University) -
Abstract: Let X be a closed, connected, hyperbolic surface of genus 2. Is it more likely for a simple closed geodesic on X to be separating or non-separating? How much more likely? In her thesis, Mirzakhani gave very precise answers to these questions. One can ask analogous questions for square-tiled surfaces of genus 2 with one horizontal cylinder. Is it more likely for such a square-tiled surface to have separating or non-separating horizontal core curve? How much more likely? Recently, Delecroix, Goujard, Zograf, and Zorich gave very precise answers to these questions. Surprisingly enough, their answers were exactly the same as the ones in Mirzakhani’s work. In this talk we explore the connections between these counting problems, showing they are related by more than just an accidental coincidence.

Polylogarithms and cluster algebras

When: Mon, April 1, 2019 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Christian Zickert (UMD) -
Abstract: We discuss some rather mysterious relationships between cluster algebras and polylogarithms.

Hyperholomorphic line bundles and energy functionals

When: Mon, April 8, 2019 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Florian Beck (University of Hamburg) -
Abstract: Hyperkähler (HK) manifolds have attracted a lot of attention both by mathematicians and physicists due to their rich and intricate structure. Most prominently, they come equipped with three complex structures satisfying the quaternionic relations. Even though an HK manifold M is a differential-geometric object, the HK structure can be encoded in one complex-analytic object, the twistor space Z of M. By construction, it fibers over the complex line and the sections of this fibration provide the link to the HK structure on M.
In this talk, we pursue and extend this complex-analytic approach for HK manifolds that admit an appropriate circle action. For each such HK manifold M, Haydys constructed a hyperholomorphic line bundle, i.e. it is holomorphic with respect to each of the three complex structures on M. In turn, this induces a holomorphic line bundle L on Z. By employing L, we define an energy functional on the space of sections of Z. We show that it is a natural complex-analytic extension of the moment map of the circle action on M. This gives new insights on the space of sections of Z if M is the moduli space of solutions to Hitchin's self-duality equations.

Isovariant homotopy theory

When: Mon, April 15, 2019 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Sarah Yeakel (UMD) -
Abstract: Fixed point theory studies the extent to which fixed points of a self-map of a space are intrinsic. Variations of fixed point problems have solutions which use tools from stable and equivariant homotopy theory. In this talk, I will discuss progress towards a homotopy theory for an isovariant category of spaces with a finite group action. In particular, we will discuss an isovariant analogue of Elmendorf's Theorem.

McShane identities for higher Teichmuller theory and the Goncharov-Shen potential

When: Mon, April 22, 2019 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Sun Zhe (University of Luxembourg) -
Abstract: Goncharov and Shen introduce a family of mapping class group invariant regular functions on the moduli space A(G, S) of decorated twisted G-local systems. They use them to parametrize certain canonical bases and to formulate an explicit homological mirror symmetry picture. We observe these functions as generalized horocycle lengths, called Goncharov--Shen potential, to describe a family of Mcshane-type identities for simple root length on positive surface group representations into PGL(n,R). It looks very similar to Mirzakhani's generalized McShane identity for bordered surfaces which is used to again prove Witten-Kontsevich theorem, except new parameters appear. We find nice properties for these new parameters. We find some applications for these identities. This is joint work with Yi Huang.

Exotic real projective Dehn surgery space

When: Mon, April 29, 2019 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Jeffrey Danciger (University of Texas) -
Abstract: We study properly convex real projective structures on closed 3-manifolds. A hyperbolic structure is one special example, and in some cases the hyperbolic structure may be deformed non-trivially as a convex projective structure. However, such deformations seem to be exceedingly rare. By contrast, we show that many closed hyperbolic manifolds admit a second convex projective structure not obtained through deformation. We find these examples through a theory of properly convex projective Dehn filling, generalizing Thurston’s picture of hyperbolic Dehn surgery space. Joint work with Sam Ballas, Gye-Seon Lee, and Ludovic Marquis.

Counting lariats

When: Tue, April 30, 2019 - 3:15pm
Where: 2300.0
Speaker: Scott Wolpert (UMD) -
Abstract: On a hyperbolic surface a lariat is a bi infinite simple geodesic with both ends at a common cusp. The reduced length of a lariat is defined by choosing a horocycle for truncation. We discuss adapting Mirzakhani’s method from “Growth of the number of simple closed geodesics …” to count by length the lariats at a given cusp. The method combines counting and estimating the integral lattice points in MGL, the space of measured geodesic laminations; Masur’s ergodicity for the mapping class group acting on MGL and integration over the moduli space to evaluate constants.
The overall result is that lariats counted by length have the same growth rate as simple closed geodesics and as in Mirzakhani’s result, the leading constant in the count is given by characteristic classes on the moduli space. The original method is adapted by counting cosets in the mapping class group. The approach is applied to count pants decompositions with controlled ratios for lengths.

Twisted Equivariant K-theory of Compact Lie Groups

When: Mon, May 6, 2019 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Valentin Zakharevich (Johns Hopkins) -
Abstract: In this talk I will discuss a computation of twisted equivariant K-theory of a compact Lie group G acting on itself by conjugation where the group G is not connected. The computation is motivated by the connection between the Verlinde ring of the Chern-Simons theory based on G and twisted equivariant K-theory as developed by Freed, Hopkins and Teleman. When G is simply connected this connection is well understood. When G is not connected, the Chern-Simons theory based on G is not a priory well defined mathematically. On the formal level, it should correspond to gauging of a finite symmetry action on the modular tensor category corresponding to the Chern-Simons theory of the component of the identity of G, as developed by Barkeshli, Bonderson, Cheng, and Wang. Comparing the the K-theory ring with the Verlinda ring obtained by gauging a symmetry suggest the isomorphism between the twisted equivariant K-theory of G and the Verlinde ring of G persists for non-connected compact Lie groups.

Dynamics on geodesic currents and atoroidal subgroups of Out(F_N)

When: Mon, May 13, 2019 - 3:15pm
Where: Kirwan Hall 3206
Speaker: Caglar Uyanik (Yale University) -
Abstract: Geodesic currents on surfaces are measure theoretic generalizations of closed curves on surfaces and they play an important role in the study of the Teichmuller spaces. I will talk about their analogs in the setting of free groups, and try to illustrate how the dynamics and geometry of the Out(F_N) action reflects on the algebraic structure of Out(F_N).