Tuesday, February 6, 2018
Public Access

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February 2018
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11:00am [11:45am]Rajiv Garg
Commutative Algebra Seminar Speaker: Rajiv Garg Date & Time: 6th February, 11:45am-13:00pm Venue: Room 215 Title: Boij-S\ddot{\text{o}}derberg Theory over Standard Graded Rings Abstract: In 2009, Eisenbud and Schreyer prove that extremal rays of Betti cone over a polynomial ring are spanned by Betti diagrams of pure Cohen-Macaulay S-modules, where S={\sf k}[X_1,\dots, X_n]. In this talk, we discuss Boij-S\ddot{\text{o}}derberg theory for standard graded {\sf k}-algebras. We note the obstacles in using their techniques in the general situation and identify classes of rings where we can prove some of these results.

3:00pm [3:00pm]Pranav Pandit
Speaker: Pranav Pandit Date & Time: 6 February, 3pm Venue: Room 215 Title: Categorical Kähler Geometry: from derived categories to dynamical systems Abstract:Mirror symmetry is a phenomenon predicted by string theory in physics. It allows one to translate questions in symplectic geometry to questions in complex geometry, and vice versa. The homological mirror symmetry program interprets mirror symmetry within the unifying categorical framework of derived noncommutative geometry. After introducing these ideas, I will describe an approach to a theory of Kähler metrics in derived noncommutative geometry. We will see how this leads to (i) a non-Archimedean categorical analogue of the Donaldson-Uhlenbeck-Yau theorem, inspired by symplectic geometry, and (ii) the discovery of a refinement of the Harder-Narasimhan filtration which controls the asymptotic behavior of certain geometric flows. This talk is based on joint work with Fabian Haiden, Ludmil Katzarkov, and Maxim Kontsevich.