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Mean curvature flow, the gradient flow of the area functional, is the most natural geometric heat flow for embedded hypersurfaces. Being non linear, the flow develops singularities, at which it stops being smooth. One fundamental, often delicate, question for such non linear flows is that of backwards uniqueness. In this talk I will discuss recent backwards uniqueness results, obtained jointly with Josh Daniels-Holgate, which can address some singularities. I will also compare these results to (commonly more robust) forward uniqueness results, and also to the situation in other equations.   

I will discuss joint work with Terry Song on the calculation of the virtual Hodge numbers (i.e. Hodge—Deligne polynomial, Hodge-Euler characteristic, etc.) of the moduli space of degree-d maps to projective space from smooth n-pointed curves of genus g. Up to Brill—Noether loci in genus g>= 3, I will show how to reduce the calculation to the corresponding invariants of M_g,n. This reduction implies a strong stability statement for the virtual Hodge numbers as functions of the degree d, and suggests homological stability properties of the moduli space generalizing known statements in genus zero.  As an intermediate result, I will outline an analogous calculation for the universal Jacobian over M_g,n. As I will discuss, the theory of symmetric functions is fundamental to our approach.