Organizers: Caterina Consani, Chi Li, Yueqiao Wu
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September 15: Daniil Serebrennikov (JHU) Finiteness, Boundedness, and Constructibility for Calabi-Yau Varieties
Abstract: Grothendieck’s foundational work shows that many geometric properties of fibers in a projective family, for example smoothness, define constructible subsets of the base. However, one natural property is absent from this list: given a fiber, is the locus of points whose fibers are isomorphic to it constructible? Surprisingly, this question is closely related to the Minimal Model Program and the Kawamata-Morrison cone conjecture, a longstanding central problem in birational geometry. In this talk, I will explain the relationship between the cone conjecture and this constructibility problem and show how it yields a partial answer to the Kawamata-Matsuki conjecture on the finiteness of isomorphism classes of Calabi-Yau varieties within a fixed birational class.
September 22: Eric Jovinelly (Brown) Curves in Singular Fano Varieties
Abstract: One way to study a variety is through the curves it contains. This paradigm has shaped our understanding of smooth Fano varieties, but far less is known in the singular setting. This talk will discuss several aspects of the theory of curves on singular Fano varieties, including the existence of free curves and Mori's Bend-and-Break. This covers joint work with Osamu Fujino, Brian Lehmann, and Eric Riedl.
September 29: Travis Mandel (Oklahoma) Valuative independence and theta reciprocity
Abstract: A discrete valuation of a linear combination of terms is greater than or equal to the minimum of the valuations of the terms. A fundamental fact in toric geometry says that if the terms are monomials and the valuation corresponds to a toric boundary divisor, then this inequality will actually be an equality. We call this valuative independence. I will discuss joint work with M.-W. Cheung, T. Magee, and G. Muller in which we show that theta functions constructed from a positive scattering diagram (e.g., the Gross-Hacking-Keel-Kontsevich theta functions on cluster varieties) also satisfy valuative independence. Applications include finding theta function bases for global sections of line bundles, linear independence of theta functions with specialized coefficients, and a general gluing lemma for theta functions from moduli of local systems on marked surfaces. We then prove theta reciprocity, a symmetry property between valuations and theta functions on mirror pairs: roughly, val_v(theta_u)=val_u(theta_v).
October 6: Yueqiao Wu (JHU)
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October 13: Shubhodip Mondal (Purdue)
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October 20: Junyan Zhao (UMD)
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October 27: Harold Blum (Georgia Tech)
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November 3:
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November 10: Guido Bosco (Princeton)
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November 17:
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