(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 - n 1980, I. 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.