We will indicate a proof of M. Artin's theorem on solutions of analytic equations. Some examples of how this theorem can be used to prove surprising results will be indicated.

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We'll make a precise statement of Green's conjecture. We'll then discuss commutative algebraic aspects of canonical curves: their Hilbert series, Castelnuovo-Mumford regularity, Gorenstein property.

Abstract: In this talk, our focus will be on certain classical results due to Ingham, Levinson and Paley-Wiener which find optimal decay of the Fourier transform of nonzero functions vanishing on `large sets'. We will talk about these theorems in details and their generalisations on the $n$- dimensional Euclidean space, the $n$-dimensional torus and certain non-commutative Lie groups.

Abstract: In this talk we present a derandomized form of the famous martingale inequality of Kazuoki Azuma (1967), and the bounded differences inequality of Colin McDiarmid (1988) based on it. We further show how to embed limited independence in the concentration bounds of Angluin-Valiant, motivated by work of Svante Janson (2003) on sums of partially dependent random variables for the Chernov-Hoeffding inequality. We then demonstrate that the derandomized McDiarmid-inequality is an easy applicable and elegant frame work for derandomization in presence of dependent random variables. As an example we choose the randomized algorithm for the vertex cover (or hitting set) problem in hypergraphs due to Mourad El Ouali, Helena Fohlin and Anand Srivastav (2016), which for hypergraphs with bounded vertex degree gives the presently best approximation bounds. This is joint work with Mayank.

Abstract. We will indicate a proof of M. Artin's theorem on solutions of analytic equations. Some examples of how this theorem can be used to prove surprising results will be indicated.

Abstract: The Riemann-Roch theorem is fundamental to algebraic geometry. In 2006, Baker and Norine discovered an analogue of the Riemann-Roch theorem for graphs. In fact, this theorem is not a mere analogue but has concrete relations with its algebro-geometric counterpart. Since its conception this topic has been explored in different directions, two significant directions are i. Connections to topics in discrete geometry and commutative algebra ii. As a tool to studying linear series on algebraic curves. We will provide a glimpse of these developments. Topics in commutative algebra such as Alexander duality and minimal free resolutions will make an appearance. This talk is based on joint work with i. Omid Amini, ii. Bernd Sturmfels, ii. Frank-Olaf Schreyer and John Wilmes. I gave a talk on this topic during my visit here in August 2016, I hope to report on some progress since then.