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January 15: Roberto Svaldi (EPFL)
SPECIAL TIME: 10:30-11am pretalk, 11am-12pm main talk Title: Applications of birational geometry to holomorphic foliations, Part 2 Abstract: This will be the continuation to Calum's talk. The plan, building on what Calum explained, is to discuss some recent work building towards the birational classification of holomorphic foliations on projective varieties (particularly 3folds) in the spirit of the Minimal Model program. We will explain some applications of these ideas to the study of the dynamics and geometry of foliations and foliation singularities. Features works of C.Spicer and P. Cascini and joint work with C. Spicer. |
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January 22: Yalong Cao (IPMU)
Title: Gopakumar-Vafa type invariants for Calabi-Yau 4-folds Abstract: Gopakumar-Vafa type invariants on Calabi-Yau 4-folds (which are non-trivial only for genus zero and one) are defined by Klemm-Pandharipande from Gromov-Witten theory, and their integrality is conjectured. In this talk, I will explain how to give a sheaf theoretical interpretation of them using counting invariants on moduli spaces of one dimensional stable sheaves. Based on joint works with D. Maulik and Y. Toda. |
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January 29: Rahul Pandharipande (ETH)
SPECIAL TIME: 11:30am-12pm pretalk, 12pm-1pm main talk Title: K3 surfaces: curves, sheaves, and moduli Abstract: I will talk about some results and open questions related to the moduli of maps of curves to K3 surfaces, sheaves on K3 surfaces, and moduli of K3 surfaces themselves. |
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February 5: Kisun Lee (UCSD)
SPECIAL NOTE: No pretalk Title: Finding and certifying numerical roots of systems of equations Abstract: Numerical algebraic geometry studies methods to approach problems in algebraic geometry numerically. Especially, finding roots of systems of equations using theory in algebraic geometry involves symbolic algorithm which requires expensive computations. However, numerical techniques often provides faster methods to tackle these problems. We establish numerical techniques to approximate roots of systems of equations and ways to certify its correctness. As techniques for approximating roots of systems of equations, homotopy continuation method will be introduced. Since numerical approaches rely on heuristic method, we study how to certify numerical roots of systems of equations. Krawczyk method from interval arithmetic and Smale's alpha theory will be used as main paradigms for certification. Furthermore, as an approach for multiple roots, we establish the local separation bound of a multiple root. For a regular quadratic multiple zero, we give their local separation bound and study how to certify an approximation of such multiple roots. |
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February 12: Ljudmila Kamenova (Stony Brook University)
SPECIAL TIME: 11:30am-12pm pretalk, 12pm-1pm main talk Title: Algebraic non-hyperbolicity of hyperkahler manifolds Abstract: A projective manifold is algebraically hyperbolic if the degree of any curve is bounded from above by its genus times a constant, which is independent from the curve. This is a property which follows from Kobayashi hyperbolicity. We prove that hyperkahler manifolds are not algebraically hyperbolic when the Picard rank is at least 3, or if the Picard rank is 2 and the SYZ conjecture on existence of Lagrangian fibrations is true. We also prove that if the automorphism group of a hyperkahler manifold is infinite, then it is algebraically non-hyperbolic. These results are joint with Misha Verbitsky. |
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February 19: Ruijie Yang (Stony Brook University)
Title: Decomposition theorem for semisimple local systems Abstract:In complex algebraic geometry, the decomposition theorem asserts that semisimple geometric objects remain semisimple after taking direct images under proper algebraic maps. This was conjectured by Kashiwara and is proved by Mochizuki and Sabbah in a series of long papers via harmonic analysis and D-modules. In this talk, I would like to explain a simpler proof in the case of semisimple local systems using a more geometric approach. As a byproduct, we recover a weak form of Saito's decomposition theorem for variations of Hodge structures. Joint work in progress with Chuanhao Wei. |
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February 26: Mircea Mustaţă (University of Michigan)
SPECIAL TIME: 11:30am-12pm pretalk, 12pm-1pm main talk Title: The minimal exponent of hypersurface singularities Abstract:I will introduce and discuss an invariant of hypersurface singularities, Saito's minimal exponent (a.k.a. Arnold exponent in the case of isolated singularities). This can be considered as a refinement of the log canonical threshold, which is interesting in the case of rational singularities. I will focus on recent work on this invariant and remaining open problems. |
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March 5: Antonella Grassi (Università di Bologna and University of Pennsylvania)
SPECIAL TIME: 9:30-10am pretalk, 10-11am main talk Title: Kodaira's birational classification of singular elliptic fibers (and threefolds with Q-factorial and non Q-factorial terminal singularities) Abstract: Kodaira classified the singular elliptic fibers occurring on relatively minimal elliptic surfaces (over C). I will explain a birational Kodaira's classifications for higher dimensional elliptic fibrations. (Based on work in collaboration with T. Weigand) |
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March 12: Daniel Huybrechts (Bonn)
SPECIAL TIME: 9:30-10am pretalk, 10-11am main talk Title: Brilliant families of K3 surfaces Abstract: We explain how Hodge theory unifies three a priori very different types of deformations of K3 surfaces: twistor spaces, Brauer (or Tate-Shafarevich) families and Dwork families. All three share the property of transporting complex multiplication from one fibre in the Noether-Lefschetz locus to any other. This phenomenon is at the moment observed in all three cases but geometrically only explained for Brauer families. The motivation comes from the Hodge conjecture for squares of K3 surfaces which is still open. |
Organizers: Elham Izadi, James McKernan and Dragos Oprea
This seminar is supported in part by grants from the NSF. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Past quarters: Fall 2013, Winter 2014, Spring 2014, Fall 2014, Fall 2017, Winter 2018, Spring 2018, Fall 2018, Winter 2019, Fall 2019, Winter 2020, Spring 2020, Fall 2020.
The design of this webpage is copied shamelessly from the MIT Number Theory seminar site. Contact Samir Canning at srcannin@ucsd.edu about problems with the website or posters.