Nov 18, 2019
Monday

09:15 AM  09:30 AM


Welcome

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
 Supplements



09:30 AM  10:30 AM


ArakelovMilnor inequalities and maximal variations of Hodge structure
Oscar GarciaPrada (Consejo Superior de Investigaciones Científicas (CSIC))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We consider the moduli space of GHiggs bundles over a compact Riemann surface X, where G is a real semisimple Lie group. By the nonabelian Hodge correspondence this is homeomorphic to the moduli space of representations of the fundamental group of X in G. We are interested in the fixed point subvarieties under the action of C*, obtained by rescaling the Higgs field. The fixed points correspond to variations of Hodge structure. They also correspond to critical subvarieties of a Morse function on the moduli space of Higgs bundles, known as the Hitchin functional. We show that one can define in this context an invariant that generalizes the Toledo invariant in the case where G is of Hermitian type. Moreover, there are bounds on this invariant similar to the Milnor–Wood inequalities of the Hermitian case. These bounds also generalize the Arakelov inequalities of classical Hodge bundles arising from families of varieties over a compact Riemann surface. We study the case where this invariant is maximal, and show that there is a rigidity phenomenon, relating to Fuchsian representations and higher Teichmüller spaces (Joint work with O. Biquard, B. Collier and D. Toledo).
 Supplements



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Novikov’s problem: from physics of metals to modern dynamical systems
Alexandra Skripchenko (Higher School of Economics)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will discuss a famous problem posed by S. P. Novikov in 1982 in connection with the conductivity theory of monocrystals: what one can say about different types of asymptotic behavior of plane sections of a triply periodic surface? Apparently the most efficient approach to this problem is closely related to the techniques used in Teichmuller dynamics. I will try to explain historical perspectives and recent developments as well as challenging questions that still remain open.
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Geometric Recursion
Jorgen Ellegaard Andersen (Syddansk Universitet (University of Southern Denmark))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We shall review the geometric recursion and its relation to topological recursion. In particular, we shall consider the target theory of continuous functions on Teichmüller spaces and we shall exhibit a number of classes of mapping class group invariant functions, which satisfies the geometric recursion. Many of these classes of functions are integrable over moduli spaces and we prove that these averages over moduli spaces satisfies topological recursion. The talk will end with some future perspectives of applications of geometric recursion. The construction of geometric recursion and the results relating it to topological recursion is joint work with Borot and Orantin.
 Supplements



03:00 PM  04:00 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



04:00 PM  05:00 PM


The saddle connection complex
Anja Randecker (University of Toronto)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
A halftranslation surface is given by a quadratic differential on a closed Riemann surface. It can be visualized by polygons in the plane whose edges are identified via translations and reflectiontranslations. The geometric properties of a halftranslation surface can be captured in the saddle connection complex. The vertices of this complex are saddle connections (i.e. geodesic segments that connect a zero or pole to a zero or pole) and the simplices are formed by saddle connections that are nonintersecting. I will explain properties of the saddle connection complex, in particular the following rigidity result: Every simplicial isomorphism between the saddle connection complexes of two halftranslation surfaces is induced by an affine diffeomorphism between the underlying surfaces. The talk is based on joint work with Valentina Disarlo and Robert Tang.
 Supplements



05:15 PM  06:15 PM


Screening of Secrets of the Surface: The Mathematical Vision of Maryam Mirzakhani

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
The Mathematical Sciences Research Institute (MSRI) and George Csicsery have started production of a onehour documentary film about a remarkable mathematician whose contributions were recognized with a Fields Medal just a few years before her untimely death.
The biographical film is about Maryam Mirzakhani, a brilliant woman, and Muslim immigrant to the United States who became a superstar in her field. The story of her life will be complemented with sections about Mirzakhani’s mathematical contributions, as explained by colleagues and illustrated with animated sequences. Throughout, we will look for clues about the sources of Mirzakhani’s insights and creativity.
 Supplements




Nov 19, 2019
Tuesday

09:30 AM  10:30 AM


The Euler characteristic of moduli spaces of meromorphic differentials
Martin Moeller (Johann Wolfgang GoetheUniversität Frankfurt)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We compute the cotangent bundle to the moduli spaces of meromorphic differentials. This relies on the fine structure at the boundary of the smooth compactification by multiscale differentials. As a consequence we obtain a canonical bundle formula and an algorithm to compute Euler characteristics.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Spectral networks and stability conditions
Fabian Haiden (University of Oxford)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
Spectral networks are certain labeled graphs drawn on a Riemann surface which generalize the saddle connections and closed loops for the flat metric of a quadratic differential. They were introduced by physicists Gaiotto, Moore, and Neitzke as a way of computing BPS spectra. A conjectural mathematical theory was suggested by Kontsevich, in which spectral networks are supports of semistable objects for a stability condition in the sense of Bridgeland. This is joint work in progress with Katzarkov, Kontsevich, Pandit, and Simpson. An example in which everything can be described explicitly is the A2xA5 category, where a recursive procedure constructs all spectral networks which have a "spider web" like shape.
 Supplements



12:00 PM  01:15 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



01:15 PM  02:15 PM


K3 surfaces: dynamics and moduli spaces
Simion Filip (University of Chicago)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
A compact complex surface that has a nowhere vanishing holomorphic 2form, and which is simplyconnected, is called a K3 surface. The geometry and dynamics of K3s is rich: they admit Ricciflat metrics and have homogeneous parameter spaces, analogous to Teichmuller and moduli spaces of Riemann surfaces. Additionally, K3s often admit interesting automorphisms about which many questions remain open. I will first provide the necessary background, following an analogy with Riemann surface equipped with a holomorphic 1form. Then I will discuss some results regarding the interaction of automorphisms and Ricciflat metrics on K3s. Joint work with Valentino Tosatti.
 Supplements



02:30 PM  03:30 PM


The realization problem for twisted quadratic differentials (dilation surfaces)
Jane Wang (Indiana University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Twisted quadratic differentials, also known as dilation surfaces, are geometric structures that are in a way a generalization of translation surfaces. We can define a dilation surface either as a quadratic differential twisted by some real holonomy or as a collection of polygons with sides identified by translations and dilations by nonzero real factors. This small generalization is enough to introduce interesting new dynamical behaviors on dilation surfaces that do not occur for translation surfaces. In this talk, we will introduce dilation surfaces and discuss some of the new and interesting dynamical behaviors that can occur on them. We will then formulate the realization problem, which asks which mapping class group elements can be realized as affine automorphisms of a dilation surfaces, and discuss challenges and progress toward resolving this problem.
 Supplements



03:30 PM  04:00 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



04:00 PM  05:00 PM


Cyclic harmonic bundles on noncompact surfaces
Qiongling Li (Chern Institute of Mathematics)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
For a cyclic Higgs bundle over a noncompact Riemann surface, we study the space of Todalike harmonic metrics depending on the asymptotic behavior of the deteminant of the Higgs field. We will attach the problem in both the HitchinKobayashi method and the partial differential equation method. This is a joint work with Takuro Mochizuki.
 Supplements



05:00 PM  07:00 PM


Reception

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements




Nov 21, 2019
Thursday

09:30 AM  10:30 AM


Meanders and holomorphic differentials
Vincent Delecroix (Université de Bordeaux I)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
A meander is a (homotopy class of) a pair of curves on the sphere with transverse intersections. The meandric number M_n is the count of "rooted" meanders with 2n intersections. One has M_1=1, M_2=2, M_3=8 and M_4=42. Understanding the asymptotic growth rate of M_n remains an open problem in mathematics/statistical physics. In this talk we will consider the refinement M_{n,k} that counts meanders with 2n intersections and whose complement has exactly k bigons. In this settings, we can describe the asymptotics as n tends to infinity for each fixed k.
Our solution relies on associating a quadratic differential to a meander (a so called square tiled surface) and relates the counting of M_{n,k} to the MasurVeech volumes of quadratic differentials on the sphere with k punctures.
This is a joint work with E. Goujard, P. Zograf and A. Zorich.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Higher Teichmüller spaces and Anosov representations
Anna Wienhard (RuprechtKarlsUniversität Heidelberg)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Hitchin components, positive representations in the sense of Fock and Goncharov, and maximal representations are families of higher Teichmüller spaces and important examples of Anosov representations, which provide a large class of discrete subgroups in higher rank Lie groups. In this talk I will discuss some recent results and open open questions.
 Supplements



12:00 PM  01:15 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



01:15 PM  02:15 PM


Aspects of the cohomology ring of the moduli space of torsionfree sheaves on an irreducible nodal curve
Inder Kaur (Institute of Pure and Applied Mathematics (IMPA))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Let M (2, L) denote the moduli space of stable vector bundles of rank 2 and determinant L of odd degree, on a smooth curve of genus g ≥ 2. Owing to the work of Mumford, Newstead, Thaddeus, King, Kirwan and several others, questions such as the generators and relations, higher rank Torellitype theorems as well as the Hodge conjecture for the cohomology ring of M (2, L) are well understood. In this talk I will survey some of these aspects for the smooth case and discuss analogous results for the case when the underlying curve is irreducible, nodal. This is joint work with S. Basu and A. Dan.
 Supplements



02:30 PM  03:30 PM


Degeneration of the Period Matrix and ShimuraTeichmuller Curves
Chaya Norton (University of Michigan)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In joint work with David Aulicino, we show there are no ShimuraTeichmuller curves in genus 5. Together with Moeller's results this completes the classification of Teichmuller curves with completely degenerate KontsevichZorich spectrum. In this talk I hope to highlight the main new technique  a parametric jump problem  with the hope it can be useful to answer other questions.
 Supplements



03:30 PM  04:00 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



04:00 PM  05:00 PM


Maximal surfaces in pseudohyperbolic spaces of rank 2
François Labourie (Universite de Nice Sophia Antipolis)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In this joint work with Jérémy Toulisse and Mike Wolf, we prove the existence (and discuss the uniqueness) of maximal surfaces in the pseudohyperbolic space $H_{2,n}$
of signature (2,n) with a prescribed quasisymmetric boundary at infinity.
No knowledge of Lorentzian geometry will be assumed. We will mainly discuss the notion of quasisymmetric curves in the boundary at infinity of $H_{2,n}$, boundary known as the EInstein universe
 Supplements




Nov 22, 2019
Friday

09:30 AM  10:30 AM


Holomorphic differentials and the geometry of dgcategories
Pranav Pandit (International Centre for Theoretical Sciences)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The DonaldsonUhlenbeckYau theorem relates the stability of a holomorphic bundle on a Kähler manifold to the existence of a good Hermitian metric on the bundle. There is an analogous conjectural picture in symplectic geometry, where objects in Fukaya categories play the role of holomorphic bundles, and holomorphic differentials play the role of Kähler forms. I will outline a categorytheoretic framework for studying these phenomena, and discuss some applications of this framework. This is based on joint work with Fabian Haiden, Ludmil Katzarkov, and Maxim Kontsevich.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


On typepreserving representations of thrice punctured projective plane group
Sara Maloni (University of Virginia)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
In the first part of the talk we will discuss famous topological and dynamical questions and conjectures about character varieties and the associated action of the mapping class group. In the second part of the talk we will discuss joint work with F. Palesi and T. Yang about typepreserving representations of the fundamental group of the threeholed projective plane N into PGL(2, R). First, we prove Kashaev’s conjecture on the number of connected components with nonmaximal euler class. Second, we discuss Bowditch's question about which of these components have the following property: there is a a simple closed curve sent to a nonhyperbolic element. Finally, we show, in most cases, that the action of the pure mapping class group Mod(N) on these nonmaximal components is ergodic, proving Goldman conjecture. Time permitting we will discuss a work in progress with Palesi where we expand these results to all five surfaces (orientable and nonorientable) of characteristic 2.
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:00 PM  04:00 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



