The history of the Hilbert-Waring problem dates back to Diophantus of Alexandria and his famous book Arithmetica. It states that given a power $k\geq 2$, every positive integer $n$ can be written as a sum of $\mu_{k}$ $k$-powers, where $\mu_{k}$ depends on $k$; that is, $n=x_1^k+...+x_{\mu_k}^k$. I will discuss a short solution on a variation of this problem.
Abstract: Milnor fibers are invariants that arise in the study of hypersurface singularities. A major open conjecture predicts that for hyperplane arrangements, the Betti numbers of the Milnor fiber depend only on the combinatorics of the arrangement. I will discuss how tropical geometry can be used to study related invariants, the virtual Hodge numbers of a hyperplane arrangement’s Milnor fiber. This talk is based on joint work with Max Kutler.
Abstract: The algebra of conformal blocks is a commutative ring built from the spaces of conformal blocks of the Wess-Zumino-Novikov-Witten model of conformal field theory attached to a smooth projective curve and a simple Lie algebra. For algebraic geometers, these rings emerge naturally as the total coordinate rings of the moduli of principal bundles on the curve.