Apr 13, 2015
Monday

09:15 AM  09:30 AM


Welcome

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
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09:30 AM  10:30 AM


Convergence of quasifuchsian hyperbolic 3manifolds
Richard Canary (University of Michigan)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Thurston's Double Limit Theorem provided a criterion ensuring convergence, up to subsequence, of a sequence of quasifuchsian representations. This criterion was the key step in his proof that 3manifolds which fiber over the circle are geometrizable. In this talk, we describe a complete characterization of when a sequence of quasifuchsian representations has a convergent subsequence. Moreover, we will see that the asymptotic behavior of the conformal structures determines the ending laminations and parabolic loci of the algebraic limit and how the algebraic limit ``wraps'' inside the geometric limit. (The results described are joint work with Jeff Brock, Ken Bromberg, Cyril Lecuire and Yair Minsky.)
 Supplements



10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


The (asymptotic) location of eigenvalues of a representation in the Hitchin component
Andrés Sambarino (Université de Paris VII (Denis Diderot) et Université de Paris VI (Pierre et Marie Curie))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The Hitchin component is a (special) connected component of the space of homomorphisms of a surface group into $\textrm{PSL}(d,\mathbb{R}).$ This component is a higher rank analogue of the Teichmuller space of the surface. The purpose of the talk is to show that the critical exponent of a Hitchin representation has a rigid upper bound. This is a joint work with Rafael Potrie.
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


A collar lemma for Hitchin representations
Tengren Zhang (University of Michigan)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
There is a wellknown result known as the collar lemma for hyperbolic surfaces. It has the following consequence: if two closed curves, a and b, on a closed orientable hyperbolizable surface have nonzero geometric intersection number, then there is an explicit lower bound for the length of a in terms of the length of b, which holds for any hyperbolic structure one can choose on the surface. Furthermore, this lower bound for the length of a grows logarithmically as we shrink the length of b. By slightly weakening this lower bound, we generalize this statement to hold for all Hitchin representations instead of just hyperbolic structures. This is joint work with GyeSeon Lee.
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Maximal representations of complex hyperbolic lattices
Maria Beatrice Pozzetti (RuprechtKarlsUniversität Heidelberg)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
There are natural incidence structures on the boundary of the complex hyperbolic space and on some suitable boundary S associated to the group PU(m,n). Such structures have striking rigidity properties: I will prove that a (measurable) map from the boundary of the complex hyperbolic space to S that preserves these incidence structures needs to be algebraic. This implies that, if G is a lattice in SU(1,p) and n is greater than m, there exist Zariski dense maximal representations of G in SU(m,n) only if (m,n) is equal to (1,p). In particular the restriction to G of the diagonal embedding of SU(1,p) in SU(m,pm+k) is locally rigid.
 Supplements



Apr 14, 2015
Tuesday

09:30 AM  10:30 AM


The action of a pseudoAnosov on the Hitchin component
Francis Bonahon (University of Southern California)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The action of a pseudoAnosov on the Hitchin component
Abstract: The Hitchin component is a preferred component of the space of homomorphisms from a surface group to PSL_n(R). I will describe a parametrization of this component that is welladapted to a general geodesic lamination on the surface. This can be used to provide a very explicit description of the action on the Hitchin component of a pseudoAnosov homeomorphism of the surface. This is joint work with Guillaume Dreyer.
 Supplements



10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Anosov representations and proper actions
Fanny Kassel (Institut des Hautes Études Scientifiques (IHES))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Anosov representations of word hyperbolic groups into reductive Lie groups provide a generalization of convex cocompact representations to higher real rank. I will explain how these representations can be used to construct properly discontinuous actions on homogeneous spaces. For a rankone simple group G, this construction covers all proper actions on G, by left and right multiplication, of quasiisometrically embedded discrete subgroups of G×G; in particular, such actions remain proper after small deformations, and we can describe them explicitly. This is joint work with F. Guéritaud, O. Guichard, and A. Wienhard.
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Coordinates for representation varieties of 3manifold groups
Christian Zickert (University of Maryland)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We discuss coordinates for representation varieties of 3manifold groups. These coordinates are 3dimensional analogues of the FockGoncharov coordinates on higher Teichmuller spaces.
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Buildings, spectral networks, and the RiemannHilbert correspondence at infinity
Pranav Pandit (International Centre for Theoretical Sciences)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will describe joint work with Katzarkov, Noll, and Simpson, which introduces the notion of a versal harmonic map to a building associated with a given spectral cover of a Riemann surface, generalizing to higher rank the leaf space of the foliation defined by a quadratic differential. A motivating goal is to develop a geometric framework for studying spectral networks that affords a new perspective on their role in the theory of Bridgeland stability structures and the WKB theory of differential equations depending on a small parameter. This talk will focus on the WKB aspect: I will discuss the sense in which the asymptotic behavior of the RiemannHilbert correspondence is governed by versal harmonic maps to buildings.
 Supplements


04:30 PM  06:20 PM


Reception

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements




Apr 15, 2015
Wednesday

09:30 AM  10:30 AM


Three proofs from dynamics of rigidity of surface group actions
Kathryn Mann (University of California, Berkeley)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In previous talks (not a prerequisite!), I've described examples of actions of a surface group G on the circle that are totally rigid  they are essentially isolated points in the representation space Hom(G, Homeo+(S^1))/~. These examples are interesting from many perspectives, ranging from foliation theory to the classification of connected components of representation spaces.
In this talk, I will illustrate three separate approaches to prove rigidity of these actions, including my original proof. Each one uses fundamentally different techniques, but all have a common dynamical flavor:
1. Structural stability of Anosov foliations (Ghys/Bowden, under extra hypotheses)
2. Rotation number "trace coordinates" on the representation space (Mann)
3. New "pingpong" lemmas for surface groups (Matsumoto)
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Entropy of pseudoAnosovs which fix homology
Ian Agol (University of California, Berkeley)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We show that the entropy of a pseudoAnosov which fixes a kdimensional subspace of homology of a genus g surface is comparable to (k+1)/g. This answers a question of Ellenberg. Key use is made of a recent inequality of KojimaMcShane giving an upper bound on volume in terms of dilatation
 Supplements



Apr 16, 2015
Thursday

09:30 AM  10:30 AM


Non archimedean representations of surface groups in PGL(3) and A2Euclidean buildings
Anne Parreau (Université de Grenoble I (Joseph Fourier))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In this talk we study FockGoncharov generalized shearing parameters for representations of a punctured surface group in PGL(3,K) for an ultrametric field K, acting on the associated (real) Euclidean building. The main motivation is to understand degenerations of real convex projective structures on surfaces. We will explain how to interpret these parameters in the Euclidean building, using a geometric classification of triples of ideal chambers. Under simple open conditions on the parameters, we construct a nice invariant subspace and an associated finite A2complex encoding the marked length spectrum. This allows to describe degenerations of convex projective strucures in an open cone of parameters.
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Degeneration of real projective structures on open surfaces
Daniele Alessandrini (RuprechtKarlsUniversität Heidelberg)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
This talk will be about the theory of compactification of the character variety.
Methods coming from algebraic geometry and tropical geometry can be
used to construct and to study some of these compactifications. With
these methods it is often possible to get some results by direct
computations. We will mostly discuss the special case of deformation
spaces of convex real projective structures on open surfaces, a subset
of the character variety of the free group in SL(3,R).
This is joint work with S. Choi.
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Asymptotic Behavior of Certain Families of Higgs bundles in Hitchin Components
Qiongling Li (Chern Institute of Mathematics)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Hitchin component for $SL(n,R)$ is the component in the space of surface group representations into $SL(n,R)$ which can deform to Fuchsian locus. The Hitchin component is in correspondence with the moduli space of $SL(n,R)$Higgs bundles. I will introduce recent work with Brian Collier on asymptotic behaviors of families in Hitchin component in terms of certain families of Higgs bundles. Namely, given a family of Higgs bundles by scaling Higgs field by $t$, we analyze the asymptotic behavior of the corresponding representations as $t$ goes to $\infty$ in two special cases.
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


A transfer principle: from periods to isoperiodic foliations
Bertrand Deroin (École Normale Supérieure)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Schiffer variations produce isoperiodic deformations on the moduli space of abelian differentials on algebraic curves of genus g, and define there an interesting algebraic foliations. I'll explain that the dynamics of this latter can be understood through the dynamics of the lattice Sp(2g,Z) on the homogeneous space Sp(2g,R)/Sp(2g2,R). This transfer principle is based on a topological property of the period map (its fibers are connected). The fact that this property holds answers a question of McMullen, and generalizes a previous work of Simpson. This is a joined work with Calsamiglia and Francaviglia
 Supplements



Apr 17, 2015
Friday

09:30 AM  10:30 AM


Affine sieve and expansion in linear groups
Alireza Salehi Golsefidy (University of California, San Diego)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
 
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Dynamics and geometry of the Weil Petersson metric
Ursula Hamenstädt (Rheinische FriedrichWilhelmsUniversität Bonn)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We exhibit some dynamical properties of the Weil Petersson geodesic flow. As an application, we obtain geometric information on random hyperbolic 3manifolds which fibre over the circle.
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Counting closed geodesics on a hyperbolic surface
Maryam Mirzakhani (Stanford University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We discuss the asymptotic behavior of the number of closed geodesics of a given combinatorial type on a hyperbolic surface (the closed curve can be disconnected or have self intersections).
We will see that this question is closely related to the distribution of lengths of closed curves on a random pants decomposition of a hyperbolic surface. We use the ergodicity of the earthquake flow to study this problem.
More generally, we consider the action of mapping class group on the space of geodesic currents, and discuss the growth of the orbit of an arbitrary point. We discuss both topological and geometric versions of this problem.
 Supplements


