Aug 19, 2019
Monday

09:15 AM  09:30 AM


Introduction to MSRI

 Location
 SLMath: Eisenbud Auditorium
 Video


 Abstract
 
 Supplements



09:30 AM  10:30 AM


Introduction to holomorphic differentials, I
Michael Wolf (Rice University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
The first of a series of lectures highlighting topics involving holomorphic differentials. Abelian differentials and their periods and flat cone metrics. Teichmuller space, its cotangent and tangent spaces. The covector for energy.
 Supplements



10:30 AM  11:00 AM


Coffee Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


An introduction to flat surfaces  I
Elise Goujard (Université de Bordeaux I)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Holomorphic differentials equip naturally Riemann surfaces with a flat structure. Motivated by the study of billiards in polygons, I will review the geometry and dynamics of moduli spaces of flat surfaces: their statification, the GL(2,R) action, and the Teichmüller flow, and then focus on several invariants such as MasurVeech volumes of strata, SiegelVeech constants (related to the counting of closed geodesics), Lyapunov exponents, and their interconnection.
 Supplements



12:00 PM  02:15 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:15 PM  03:15 PM


An introduction to Higgs bundles  I
Qiongling Li (Chern Institute of Mathematics)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
I will review the basic concepts of Higgs bundles and the moduli space of Higgs bundles, and then focus the nonAbelian Hodge correspondence: relation between the moduli space of Higgs bundles with harmonic maps and the character variety. I will also explain some examples: rank two Higgs bundles, cyclic Higgs bundles, variation of Hodge structure and so on.
 Supplements



03:15 PM  03:45 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:45 PM  04:45 PM


Geometry and Physics of BPS States  I
Du Pei (Harvard University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Over the course of three lectures, we will explore the wonderful world of BPS states. We will start with their physics origin and then focus on applications to mathematics, especially to the study of moduli spaces of Higgs bundles.
 Supplements




Aug 20, 2019
Tuesday

09:30 AM  10:30 AM


Introduction to holomorphic differentials, II
Michael Wolf (Rice University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
The middle lecture of a series highlighting topics involving holomorphic differentials. The covector for periods. The metric on the space of metrics and the WeilPetersson metric. The Thurston metric and a hint about the pressure metric. Teichmuller maps and metric. Measured foliations and laminations. The HubbardMasur theorem.
 Supplements



10:30 AM  11:00 AM


Coffee Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


An introduction to flat surfaces  II
Elise Goujard (Université de Bordeaux I)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Holomorphic differentials equip naturally Riemann surfaces with a flat structure. Motivated by the study of billiards in polygons, I will review the geometry and dynamics of moduli spaces of flat surfaces: their statification, the GL(2,R) action, and the Teichmüller flow, and then focus on several invariants such as MasurVeech volumes of strata, SiegelVeech constants (related to the counting of closed geodesics), Lyapunov exponents, and their interconnection.
 Supplements



12:00 PM  02:15 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:15 PM  03:15 PM


An introduction to Higgs bundles  II
Qiongling Li (Chern Institute of Mathematics)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
I will review the basic concepts of Higgs bundles and the moduli space of Higgs bundles, and then focus the nonAbelian Hodge correspondence: relation between the moduli space of Higgs bundles with harmonic maps and the character variety. I will also explain some examples: rank two Higgs bundles, cyclic Higgs bundles, variation of Hodge structure and so on.
 Supplements



03:15 PM  03:45 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:45 PM  04:45 PM


Geometry and Physics of BPS States  II
Du Pei (Harvard University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Over the course of three lectures, we will explore the wonderful world of BPS states. We will start with their physics origin and then focus on applications to mathematics, especially to the study of moduli spaces of Higgs bundles.
 Supplements



04:45 PM  06:20 PM


Reception

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements




Aug 21, 2019
Wednesday

09:30 AM  10:30 AM


Introduction to holomorphic differentials, III
Michael Wolf (Rice University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
The final lecture of a series highlighting topics involving holomorphic differentials. Apology for inefficiency, and selection of topics from previous lectures that I didn't get to. Complex projective structures, Thurston's Riemannian metric, and mention of opers. ODE's with irregular singular points and Stokes phenomena, classical and contemporary. Convex RP^3 structures and Pick differentials.
 Supplements



10:30 AM  11:00 AM


Coffee Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


An introduction to flat surfaces  III
Elise Goujard (Université de Bordeaux I)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Holomorphic differentials equip naturally Riemann surfaces with a flat structure. Motivated by the study of billiards in polygons, I will review the geometry and dynamics of moduli spaces of flat surfaces: their statification, the GL(2,R) action, and the Teichmüller flow, and then focus on several invariants such as MasurVeech volumes of strata, SiegelVeech constants (related to the counting of closed geodesics), Lyapunov exponents, and their interconnection.
 Supplements




Aug 22, 2019
Thursday

09:30 AM  10:30 AM


Quadratic differentials, WKB analysis, and cluster coordinates
Dylan Allegretti (University of Sheffield)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
The WKB method was originally introduced by Wentzel, Kramers, and Brillouin in 1926 as a way of finding approximate solutions of the Schrodinger equation in the semiclassical limit in quantum mechanics. The modern theory of WKB analysis is a refinement of this method which is deeply related to the theory of quadratic differentials and the associated spectral networks on Riemann surfaces. In this talk, I will review the notion of a Voros symbol from WKB analysis. Voros symbols are nonconvergent formal series whose Borel sums define analytic functions under certain conditions. Recently, Iwaki and Nakanishi observed that the wallcrossing behavior of Voros symbols is governed by cluster transformations. I will present an extension of their result, which says that in fact the Borel sums of Voros symbols arise naturally as cluster coordinates on certain moduli spaces.
 Supplements



10:30 AM  11:00 AM


Coffee Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


The asymptotic geometry of the Hitchin moduli space
Laura Fredrickson (University of Oregon)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Hitchin's equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichmuller theory, and the geometric Langlands correspondence. The Hitchin moduli space carries a natural hyperkahler metric. A conjectural description of its asymptotic structure appears in the work of physicists GaiottoMooreNeitzke and there has been a lot of progress on this recently. I will discuss some recent results.
 Supplements



12:00 PM  02:15 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:15 PM  03:15 PM


An introduction to Higgs bundles  III
Qiongling Li (Chern Institute of Mathematics)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
I will review the basic concepts of Higgs bundles and the moduli space of Higgs bundles, and then focus the nonAbelian Hodge correspondence: relation between the moduli space of Higgs bundles with harmonic maps and the character variety. I will also explain some examples: rank two Higgs bundles, cyclic Higgs bundles, variation of Hodge structure and so on.
 Supplements



03:15 PM  03:45 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:45 PM  04:45 PM


Geometry and Physics of BPS States  III
Du Pei (Harvard University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Over the course of three lectures, we will explore the wonderful world of BPS states. We will start with their physics origin and then focus on applications to mathematics, especially to the study of moduli spaces of Higgs bundles.
 Supplements




Aug 23, 2019
Friday

09:30 AM  10:30 AM


Maximal Representations, real Spectrum, and harmonic Maps
Marc Burger (ETH Zürich)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
The character variety of maximal Sp(2n,R)representations of the fundamental group G of a surface S is homeomorphic to a real semi algebraic set X and as such admits a compactification by the set of closed points in the real spectrum of the coordinate ring of X. This compactification has very good topological properties and the mapping class group acts on it with virtually abelian stabilizers. Its ideal points are represented by certain Sp(2n,F) representations of G over various real closed fields F. Characterizing these representations and understanding their geometric properties is an ongoing theme of our research. In this talk we focus on the Rbuilding B(n,F) associated to Sp(2n,F) and will explain why the corresponding Gaction is proper in the sense of Korevaar and Schoen, implying the existence of an equivariant harmonic map for every complex structure h on S. We also explain that if the systole of the Gaction on B(n,F) is strictly positive, the energy functional, as a function of h, is proper on Teichmueller space. Such actions with strictly positive systole form an open set in the real spectrum boundary, that is nonempty as soon as n is at least 2. joint work with A.Iozzi, A.Parreau and B. Pozzetti.
 Supplements



10:30 AM  11:00 AM


Coffee Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Magical nilpotents and higher Teichmuller spaces
Brian Collier (University of California, Riverside)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
In this talk, we will present a special class of nilpotent elements of a complex semisimple Lie algebra. For such a nilpotent element, we will describe how Higgs bundles can be used to construct components of the character variety of a closed surface of genus at least 2. Moreover, such components are deformation spaces of Teichmuller space. The classification of this class of nilpotent elements turns out to be equivalent to Guichard, and Wienhard's notion of Thetapositivity, and so, this construction should describe all Higher Teichmuller spaces.
 Supplements



12:00 PM  02:15 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



