(September 8) Brent Doran: “Say X × A¹ = Aⁿ. Please solve for X” ... and related questions — Starting with the problem of the title -- in algebraic geometry, this is known as the Zariski Cancellation Problem -- we will show how it touches upon a number of deep problems in mathematics. Classification of contractible manifolds (and an historic mis-proof of the Poincaré conjecture), non-reductive group actions and quotients, classical invariant theory and Hilbert's 14th problem, the automorphism group of affine space, reinterpretations of singularities, moduli of algebraic vector bundles, and probing the extent to which detailed algebraic geometry can be captured by new techniques from algebraic topology ... all of these emerge naturally, and at heart are rather simple to explain, at least conceptually. Indeed, our basic object of study is quite friendly: free additive group actions on affine space. Pretty examples abound. This draws on joint work with Aravind Asok and with Frances Kirwan.

(September 15) Tom Baird: Moduli spaces of flat bundles over nonorientable surfaces — Moduli spaces of flat bundles over orientable surfaces have been actively studied for many years. They have wide ranging applications in such diverse fields as algebraic geometry, low dimensional topology and mathematical physics.

Moduli spaces of flat bundles over nonorientable surfaces have less scrutiny, and have recently been shown to possess interesting geometric and topological properties. For rank 2 bundles the topology is well understood and I will begin my lecture by describing this case. I will then provide a conjectural description of the higher rank case and detail efforts to prove this conjecture.

(September 22) Reza Seyyedali: Chow stability of ruled manifolds — In 1980, Morrison proved that slope stability of a vector bundle of rank 2 over a compact Riemann surface implies Chow stability of the projectivization of the bundle with respect to certain polarizations. Using very different techniques, namely Donaldson's notion of a "balanced metric" and recent work of Donaldson, Wang, and Phong-Sturm, we show that the statement holds for higher rank vector bundles as well.

(October 6) Sean Lawton: The topology of the moduli of free group representations — Let G be a complex affine reductive group and let K be a maximal compact subgroup. We have recently proved that the moduli space of representations Hom(F,G)//G deformation retracts to the quotient space Hom(F,K)/K for any rank r free group F. If F is replaced by other finitely generated groups the theorem may be false, but not always. In this talk we discuss this theorem and some examples.

(October 27) Florent Schaffhauser: Decomposable representations of surface groups — In this talk, we generalize to arbitrary surface groups and arbitrary compact connected Lie groups the notion of decomposable representation, first introduced by Falbel and Wentworth for unitary representations of the punctured sphere group. We show that such decomposable representations are the elements of the fixed-point set of an anti-symplectic involution defined on the moduli space of representations, forming therefore a Lagrangian submanifold of this moduli space. The existence of decomposable representations is obtained as a corollary of a real convexity theorem for group-valued momentum maps.

(November 3) Jane Gilman: The geometry and algebra of palindromes and PSL(2,C) discreteness criteria — Let G= be a rank two free group. A word W(A,B) in G is primitive if it along with another group element generates the group. It is a palindrome (with respect to A and B) if it reads the same forwards and backwards. It is known that in any rank two free group any primitive element is conjugate either to a palindrome or to the product of two palindromes, but known iteration schemes for all primitive words give only the conjugacy class. In this talk we present a new iteration scheme that gives either the unique palindrome in the conjugacy class or expresses the word as a unique product of two unique palindromes. We then derive a necessary and a sufficient geometric condition for the discreteness of a non-elementary PSL(2,C) representation of G, roughly, that the axes of all palindromes intersect a certain core geodesic in H³ in a compact subinterval. In the case of a geometrically finite representation we describe the pleating locus. This is joint work with L. Keen.

(November 17) Georgios Daskalopoulos: Harmonic maps between singular spaces — We will discuss regularity questions of harmonic maps from a simplicial complex to metric spaces of non-positive curvature. We will also discuss the relation with rigidity questions of group actions on these spaces.

(December 1) Bruno Klingler: Non-Abelian Hodge theory and quaternionic rigidity for complex hyperbolic lattices — Let L be a cocompact lattice in the group SU(n,1), n>1, of biholomorphic isometries of the complex hyperbolic n-space. Can we deform L in the group Sp(n,1) of isometries of the quaternionic hyperbolic n-space ? The main tool for answering this question will be Simpson's non-Abelian Hodge theory .

(December 8) Jonathan Rosenberg: Topological problems connected with T-duality in string theory — T-duality is a conjectured equivalence between physics on two different space-time manifolds, with tori in one space-time replaced by "dual" tori in the other. There is by now considerable evidence for such dualities, both mathematical and physical. In this talk we will focus on the topological aspects of T-duality, and what it says about the structure of torus bundles with a 3-cohomology class (the "H-flux").

(December 10) Francois Labourie: The energy functional on Teichmüller space — We study two classes of linear representations of a surface group: Hitchin and maximal symplectic representations. We relate them to cross ratios and thus deduce that they are displacing which means that their translation lengths are roughly controlled by the translations lengths on the Cayley graph. As a consequence, we show that the mapping class group acts properly on the space of representations and that the energy functional associated to such a representation is proper. This implies the existence of minimal surfaces in the quotient of the associated symmetric spaces, a fact which leads to two consequences: a rigidity result for maximal symplectic representations and a partial result concerning a purely holomorphic description of the Hichin component.

(December 15) Pierre Will: Bending Fuchsian representations of fundamental groups of punctured surfaces in PU(2,1) — Discrete subgroups of PU(2,1) are a higher dimmensional analogue of Fuchsian groups. Contrary to the case of PSL(2,R), very few examples of such groups are known. In this talk, I will present a family of embeddings of the Teichmüller space of a hyperbolic Riemann surface S with punctures into the PU(2,1)-representation variety associated to the fundamental group of S. These embeddings are non-trivial, and have the property that the image of any point of the Teichmüller space is a class of discrete, faithful and type preserving representation of the fundamental group of S in PU(2,1).

(February 2) Olivier Guichard: Connected Components of Maximal Representations — The representations of a surface group into the symplectic group Sp(2n,R) whose characteristic number (Toledo invariant) is maximal are called maximal representations. Their moduli space can be seen as a generalization of the Teichmüller space as for representations into Sp(2,R) one recovers exactly the classical moduli space. Indeed the moduli space of maximal representations shares many properties with the Teichmüller space. However its topology is richer as it has many connected components. We will explain in this talk how one can defined topological invariants for maximal representations and how these invariants help in distinguishing between the different connected components. This is a joint work with Anna Wienhard.

(February 9) Yanir Rubinstein: 1-D Schrödinger map flow — This talk will be aimed at non-specialists. The Schrödinger flow is the Hamiltonian analogue of the Eells-Sampson harmonic map flow. A conjecture of W.-Y. Ding states that the initial value problem for maps from 1-D domains (namely, the real line and the circle) into Kähler manifolds should be globally well-posed in time. We will discuss recent joint work with I. Rodnianski and G. Staffilani on this problem.

(March 9) Young-Heon Kim: Determinants of Laplacians as functions on spaces of Riemannian metrics — The determinant of the Laplacian is a global Riemannian invariant which is defined formally as the product of the nonzero eigenvalues of the Laplacian of a given Riemannian metric. It gives a continuous function on the space of Riemannian metrics. In this talk we are interested in the case of compact surfaces with boundary and will discuss the properness of the determinant function on the moduli space of hyperbolic surfaces with geodesic boundary and on the moduli space of flat surfaces with boundary of constant geodesic curvature. We will also discuss an application to the following isospectral compactness problem: On a given compact surface with boundary, consider the set of all smooth flat metrics having the same Dirichlet Laplacian spectrum. Is it compact in C^∞ topology, so that the surfaces having the same spectrum don't degenerate?

(March 23) Anna Wienhard: Domains of discontinuity for Anosov representations — Let Γ be the fundamental group of a closed surface of genus g ≥ 2 and ρ: Γ → G a representation into a semisimple Lie group. When does there exists a parabolic subgroup P and a non-empty open subset Ω ∈ G/P such that Γ acts properly discontinuously on Ω with compact quotient? A positive answer to this question was known for some special classes of representations, e.g. quasi-Fuchsian representations of Γ into PSL(2, C) or convex representations into PSL(3,R). We construct such domains of discontinuity for a much bigger class of representations, so called Anosov representations, which are characterized by certain dynamical properties. This class includes in particular all "higher Teichmüller spaces". This is joint work with Olivier Guichard.

(April 13) Feng Luo: Variational principles on polyhedral surfaces — We will discuss some of the recent work on variational principles associated to polyhedral surfaces. These include the work of Colin de Verdiere, Rivin, and others. The relationship 2-dimensional variational principles, the cosine law, the 3-dimensional hyperbolic geometry and the Schlaefli formula will be addressed in details.

(April 20) Melissa Macasieb: Character varieties of a family of 2-bridge knot complements — To every hyperbolic finite volume 3-manifold M, one can associate a pair of related algebraic varieties X(M) and Y(M), the SL(2,C)- and PSL(2,C)-character varieties of M. These varieties carry much topological information about M, but are in general difficult to compute. If M has one cusp, then both these varieties have dimension one. In this talk, I will also show how to obtain explicit equations for the character varieties associated to a family of hyperbolic two-bridge knots K(m,n) and discuss consequences of these results related to the existence of nonintegral points on these curves. This is joint work with Kate Petersen and Ronald van Luijk.

(April 30) Rebecca Goldin: Full Orbifold K-theory of Abelian Symplectic Quotients — In this talk, we will describe the full orbifold K-theory of an orbifold X that occurs as a symplectic reduction of a manifold by an Abelian Lie group. We introduce the *inertial K-theory* of a Hamiltonian T-space M (where T is an Abelian Lie group) and show that it surjects onto the full orbifold K-theory of the symplectic quotient M//T of M at a regular value. This research essentially involves three ingredients, which we will discuss in detail: (0) The construction of this particular class of orbifold via symplectic reduction (1) The fact that equivariant K-theory of M maps surjectivity onto the K-theory of M//T (2) The introduction of a fancy product on the inertial K-theory of M, so that it surjects onto the full orbifold K-theory of M//T. These ideas are base on a similar (though rational) story in cohomology which we will also discuss. This is joint work with T. Holm, M. Harada, and T. Kimura.

(May 4) Samuel Grushevsky: String scattering amplitudes and modular forms — The (super)string measure is one of the central quantities involved in the formulation of string theory. Various expressions for the bosonic string measure were proposed in 1980s. The question of finding the superstring measure is much harder than for the bosonic measure, as the supermoduli are present in the theory. D'Hoker and Phong started the modern program of computing the superstring measure from factorization constraints (restrictions to lower genera). In this talk we discuss the recently proposed ansatze for the superstring measure, using their ideas, and a detailed study of the moduli spaces of Riemann surfaces, and appropriate modular forms. We will show that the proposed ansatz for the superstring measure satisfies some physical constraints, and discuss further open questions. The knowledge of string theory will not be required to understand this talk.

(May 11) Kelly Delp: Marked Length Spectra and Strictly Convex Projective Orbifolds — A convex real projective structure on an orbifold gives a (non-Riemannian) metric which assigns a length to every element of the fundamental group. We show that two distinct structures assign the same length to every element if and only if the two structures are dual, and that a structure is self dual if and only if it is a hyperbolic structure. This work extends (and slightly corrects) Inkang Kim's result for manifolds. This is joint work with Daryl Cooper.

(June 19) Hisham Sati: Higher Topological Structures and their Geometry — "Higher Topological Structures" are new kinds of structures on manifolds which have appeared recently in the physics literature. They can be viewed has "higher" versions of spin and spin^c structures. We will discuss what these structures are, why they are interesting, and what geometry is associated with them.

(July 15) Andy Sanders: Introduction to Geometrization of 3-manifolds via Ricci flow — In this first of a series of talks I will introduce the Ricci flow equation, discuss some of its basic properties, and explain the strategy for solving Thurston's Geometrization conjecture for 3-manifolds via the Ricci flow, following a manuscript of Besson et al. Later talks will focus on several important components of the argument. The goal is to provide a reasonable understanding of the steps required to implement the Ricci flow approach.