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October 16: Martijn Kool (Utrecht)
SPECIAL TIME: 11-11:30am pretalk, 11:30am-12:30pm main talk Title: Virtual Segre and Verlinde numbers of projective surfaces Abstract: Recently, Marian-Oprea-Pandharipande proved Lehn's conjecture for Segre numbers associated to Hilbert schemes of points on surfaces. They also provided a conjectural correspondence between Segre and Verlinde numbers. For surfaces with holomorphic 2-form, we propose conjectural generalizations of their results to moduli spaces of stable sheaves of any rank. We provide several verifications by using Mochizuki's formula. Joint work with Göttsche. |
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October 23: Ben Davison (University of Edinburgh)
SPECIAL TIME: 10:30-11am pretalk, 11am-12pm main talk Title: Finite-dimensional Jacobi algebras, flopping curves, and BPS invariants Abstract: Floppable curves in threefolds are a fundamental feature of the minimal model programme in 3 dimensions. Not only do they provide the stepping stones between different minimal models, but their local geometry itself turns out to be tremendously rich. The list of examples starts with Atiyah's flop, then through the infinite class in Reid's "Pagoda" into the zoo of (-3,1) curves. Associated to these curves are a number of enumerative invariants, amongst them the Gopakumar-Vafa invariants, which can be reinterpreted as modified counts of coherent sheaves on the threefold containing the curve. The contraction algebra, controlling the deformation theory of the structure sheaf of the curve, turns out to be a strictly more refined invariant. This is a finite-dimensional Jacobi algebra introduced by Donovan and Wemyss. In this talk I will explain how the Brown-Wemyss conjecture that all finite-dimensional Jacobi algebras arise as contraction algebras implies a strong positivity statement for BPS invariants of these algebras (while defining what this all means), and explain how to prove this positivity statement, while also explaining how the classical Gopakumar-Vafa invariants of floppable curves can be "categorified" using cohomological DT theory. |
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October 30: Noah Arbesfeld (Imperial College London)
SPECIAL TIME: 10:30-11am pretalk, 11am-12pm main talk Title: Donaldson-Thomas theory and the Hilbert scheme of points on a surface Abstract: One approach to computing integrals over Hilbert schemes of points on surfaces (and other moduli spaces of sheaves on surfaces) is to reduce to the special case when the surface in question is C^2. I'll explain how to use the (K-theoretic) Donaldson-Thomas theory of threefolds to deduce identities for holomorphic Euler characteristics of tautological bundles over the Hilbert scheme of points on C^2. I'll also explain how these identities control the behavior of such Euler characteristics over Hilbert schemes of points on general surfaces. |
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November 6: Jeongseok Oh (Korea Institute for Advanced Study)
SPECIAL TIME: 4-4:30pm pretalk, 4:30-5:30pm main talk Title: Counting sheaves on Calabi-Yau 4-folds Abstract: We define a localised Euler class for isotropic sections, and isotropic cones, in SO(N) bundles. We use this to give an algebraic definition of Borisov-Joyce sheaf counting invariants on Calabi-Yau 4-folds. When a torus acts, we prove a localisation result. This talk is based on the joint work with Richard. P. Thomas. |
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November 13: Calum Spicer (King's College London)
SPECIAL TIME: 10:30-11am pretalk, 11am-12pm main talk Title: Applications of birational geometry to holomorphic foliations Abstract: A foliation on an algebraic variety is a partition of the variety into "parallel" disjoint immersed complex submanifolds. This turns out to be a very useful notion and holomorphic foliations have played a central role in several recent developments in the study of the geometry of projective varieties. This is the first part of a two talks series (with Roberto Svaldi) in which we will explain some recent work building towards the birational classification of holomorphic foliations on projective varieties 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 joint work with P. Cascini and R. Svaldi. |
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November 20: Alex Perry (University of Michigan)
Title: Kuznetsov's Fano threefold conjecture via K3 categories Abstract: Kuznetsov conjectured the existence of a correspondence between different types of Fano threefolds which identifies a distinguished semiorthogonal component of the derived category on each side. I will explain joint work with Arend Bayer which resolves one of the outstanding cases of this conjecture. This relies on the study of the Hodge theory of certain K3 categories associated to the semiorthogonal components. |
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November 25: Ming Zhang (University of British Columbia)
SPECIAL ENUMERATIVE GEOMETRY SEMINAR: Main Talk 1-2:30pm Title: The Verlinde/Grassmannian Correspondence Abstract: In the 90s', Witten gave a physical derivation of an isomorphism between the Verlinde algebra of GL(n) of level l and the quantum cohomology ring of the Grassmannian Gr(n,n+l). In the joint work arXiv:1811.01377 with Yongbin Ruan, we proposed a K-theoretic generalization of Witten's work by relating the GL_n Verlinde numbers to the level l quantum K-invariants of the Grassmannian Gr(n,n+l), and refer to it as the Verlinde/Grassmannian correspondence. The correspondence was formulated precisely in the aforementioned paper, and we proved the rank 2 case (n=2) there. In this talk, I will first explain the background of this correspondence and its interpretation in physics. Then I will discuss the main idea of the proof for arbitrary rank. A new technical ingredient is the virtual non abelian localization formula developed by Daniel Halpern-Leistner. |
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December 4: Rohini Ramadas (Brown University)
Title: Dynamics on the moduli space M_{0,n} Abstract: A rational function f(z) in one variable determines a self-map of P^1. A rational function is called post-critically finite (PCF) if every critical point is either pre-periodic or periodic. PCF rational functions have been studied for their special dynamics, and their special distribution within the moduli space of all rational maps. By works of W. Thurston and S. Koch, every PCF map (with a well-understood class of exceptions) arises as an isolated fixed point of an algebraic dynamical system on the moduli space M_{0,n} of point-configurations on P^1; these dynamical systems are called Hurwitz correspondences. I will introduce Hurwitz correspondences and their connection to PCF rational maps, and discuss how the dynamical complexity of Hurwitz correspondence can be studied via combinatorial compactifications of M_{0,n}. |
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December 11: Mareike Dressler (University of California San Diego)
Title: A New Approach to Nonnegativity and Polynomial Optimization Abstract: Deciding nonnegativity of real polynomials is a key question in real algebraic geometry with crucial importance in polynomial optimization. It is well-known that in general this problem is NP-hard, therefore one is interested in finding sufficient conditions (certificates) for nonnegativity, which are easier to check. Since the 19th century, sums of squares (SOS) are a standard certificate for nonnegativity, which can be detected by using semidefinite programming (SDP). This SOS/SDP approach, however, has some issues, especially in practice if the problem has many variables or high degree. In this talk I will introduce sums of nonnegative circuit polynomials (SONC). SONC polynomials are certain sparse polynomials having a special structure in terms of their Newton polytopes and supports and serve as a nonnegativity certificate for real polynomials, which is independent of sums of squares. I will present some structural results of SONC polynomials and I will provide an overview about polynomial optimization via SONC polynomials. |
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.
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.