• • Arend Bayer, The local projective plane, a fractal curve and Γ₁(3) .

    • I will report on joint work with E. Macri on the space of stability conditions for the derived category of the total space of the canonical bundle on the projective plane. It is a 3–dimensional manifold, with many chamber decompositions coming from the behaviour of moduli spaces of stable objects under change of stability conditions.

    • I will explain how this space is related to classical results by Drezet and Le Potier on stable vector bundles on the projective plane. Using the space helps to determine the group of auto-equivalences, which includes a subgroup isomorphic to Γ₁(3). Finally, via mirror symmetry, it contains a universal cover of the moduli space of elliptic curves with Γ₁(3)–level structure.

    • Hand-written notes by Jason Starr.

  • Ana-Maria Castravet, Rational curves of minimal degree on higher Fano manifolds.

    • This is joint work with Carolina Araujo. We will discuss a notion of higher Fano variety introduced by de Jong and Starr. We will especially study what one can say about the families of minimal degree rational curves that sweep out such a higher Fano variety. Related questions: Can one classify higher Fanos? Is there an inductive structure on the collection of all higher Fano varieties?

    • Hand-written notes by Jason Starr.

  • Herb Clemens, Exploring the Hodge problem.

    • I will give a brief explanation of the Hodge conjecture for complex projective manifolds of even dimension and the use of normal functions in studying it. I will then build on work of Voisin to generalize the notion of normal function in order to create a context for studying the Hodge problem for complex projective manifolds of dimension 2n by induction on n.

  • William Fulton, Character formulas.

    • In this expository talk, we will give a simple formula, with a simple proof, for the equivariant euler characteristic of an equivariant vector bundle on a complete, smooth variety with an action of a diagonalizable group. On homogeneous varieties this gives Weyl's character formula, and on toric varieties it gives Brion's formula for lattice points in polytopes. This is based on ideas of George Quart in the 1970's and recent conversations with Bill Graham.

    • Notes by Paul Hacking.

  • Amanda Knecht, Rationally connected varieties over ℚ_p^nr.

    • A result of Graber, Harris, and Starr shows that a rationally connected variety defined over the function field of a curve over the complex numbers always has a rational point. Similarly, a separably rationally connected variety over a finite field or the function field of a curve over any algebraically closed field has a rational point. We will discuss the case of rationally connected varieties over the maximally unramified extension of the p-adics. They `usually' have points, and we will define what `usually' means.

    • Hand-written notes by Jason Starr.

  • Li Li, Hilbert schemes of points.

    • I will talk on two problems that are related to Hilbert schemes of points. The first is to study the Hilbert schemes of points on a tame Deligne-Mumford stack. I shall talk on smoothness, connectedness and Betti numbers of such Hilbert schemes (joint work with Jason Starr). The second problem is about generators of the diagonal ideal of (ℂ²)ⁿ and q,t-Catalan numbers (joint work with Kyungyong Lee).

    • Li Li's slides

  • Mike Roth, A local-global principle for weak approximation of varieties over function fields

    • Brendan Hassett and Yuri Tschinkel introduced the study of the "weak approximation problem" for varieties over function fields, motivated by the analogous question over number fields. The question is whether local sections can be approximated to arbitrarily high order by global sections, and thus fits in to a well known (and fruitful) class of "local-global" type questions. The purpose of this talk is to study whether the weak approximation problem is itself purely local. This is joint work with Jason Starr.

    • Notes by Paul Hacking.

  • Jenia Tevelev, On the cone of effective divisors of M_0,n.

    • We give a conjectural description of the cone of effective divisors of the Grothendieck-Knudsen moduli space M_0,n of stable rational curves. Roughly speaking, non-boundary extremal divisors correspond to total stable degenerations (or, equivalently, pair-of-pants decompositions) with some decorations of smooth curves of genus n-3. Joint with Ana-Maria Castravet.

    • Hand-written notes by Jason Starr.

    • The transparency from the lecture.

Pre–lectures.

Preceding some of the lectures there will be an half-hour discussion to introduce the material discussed in the main lecture.

Workshop participants

• • Dan Abramovich, Brown University

  • Jarod Alper, Columbia University

  • Somnath Basu, Stony Brook University

  • Arend Bayer, University of Connecticut

  • Soumya Benerjee, Yale University

  • Ilke Canakci, University of Connecticut

  • Eduardo Cattani, University of Massachusetts Amherst

  • Qile Chen, Brown University

  • Herb Clemens, The Ohio State University

  • Keith Conrad, University of Connecticut

  • Peter Dalakov, University of Massachusetts Amherst

  • Matt DeLand, Stony Brook University

  • Alicia Dickenstein, Universidad de Buenos Aires and MSRI

  • Gennaro Di Brino, Yale University

  • Colin Diemer, University of Pennsylvania

  • Abigail Ebin, Yale University

  • Robert Findley, Stony Brook University

  • Maksym Fedorchuk, Columbia University

  • Conor Frailey, Yale University

  • William Fulton, University of Michigan Ann Arbor

  • Alberto Garcia-Raboso, University of Pennsylvania

  • Noah Giansiricusa, Brown University

  • Danny Gillam, Brown University

  • Samuel Grushevsky, Stony Brook University

  • Weixin Guo, Stony Brook University

  • Jan Gutt, Stony Brook University

  • Paul Hacking, University of Massachusetts Amherst

  • Hilaf Hasson, University of Pennsylvania

  • Milena Hering, University of Connecticut

  • Wenchuan Hu, IAS

  • YoonSuk Hyun, Massachusetts Institute of Technology

  • Mikhail Kapranov, Yale University

  • Anna Kazanova, University of Massachusetts Amherst

  • Leila Khatami, Northeastern University

  • Alexander Kirillov, Jr., Stony Brook University

  • János Kollár, Princeton University

  • Radu Laza, Stony Brook University

  • Brian Lehman, Massachusetts Institute of Technology

  • Long Li, Stony Brook University

  • Li Li, University of Illinois at Urbana–Champaign

  • Shuijing Li, Rice University

  • Zhiyuan Li, Rice University

  • Tsung-Yin Lin, Stony Brook University

  • Zachary Maddock, Columbia University

  • Eyal Markman, University of Massachusetts Amherst

  • Evgeny Mayanskiy, Penn State

  • Sukhendu Mehrotra, University of Wisconsin Madison

  • Ian Morrison, Fordham University

  • Dinakar Muthiah, Brown University

  • Alexei Oblomkov, University of Massachusetts Amherst

  • Svyatoslav Pimenov, Yale University

  • Alexandra Popa, Stony Brook University

  • You Qi, Columbia University

  • Alice Rizzardo, Columbia University

  • Joe Ross, Universität Essen-Duisburg

  • Mike Roth, Queen's University

  • Ryan Schwarz, University of Connecticut

  • Gagan Sekhon, University of Connecticut

  • Yijun Shao, University of Arizona

  • Mingmin Shen, Columbia University

  • Vivek Shende, Princeton University

  • David Speyer, Clay Mathematics Institute and Massachusetts Institute of Technology

  • Jason Starr, Stony Brook University

  • Dennis Sullivan, Stony Brook University and CUNY

  • Jingzhou Sun, Johns Hopkins University

  • Jenia Tevelev, University of Massachusetts Amherst

  • Zhiyu Tian, Stony Brook University

  • Gueorgui Todorov, Princeton University

  • Giancarlo Urzua, University of Massachusetts Amherst

  • Kartik Venkatram, Massachusetts Institute of Technology

  • Botong Wang, Purdue University

  • Jun Wen, Stony Brook University

  • Loy Weng, Stony Brook University

  • Jie Xia, Columbia University

  • Fei Xu, Rice University

  • Chenyang Xu, Massachusetts Institute of Technology

  • Pan Xuanyu, Columbia University

  • Yanhong Yang, Columbia University

  • Yaping Yang, Northeastern University

  • Matt Young, Stony Brook University

  • Letao Zhang, Rice University

  • Yongsheng Zhang, Stony Brook University

  • Zheng Zhang, Stony Brook University

  • Gufang Zhao, Northeastern University

  • Yi Zhu, Stony Brook University

  • Zhixian Zhu, University of Michigan Ann Arbor

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Jason Starr

4-108 Math Tower

Department of Mathematics

Stony Brook University

Stony Brook, NY 11794-3651

Phone: 631 632 8270

FAX: 631 632 7631

Jason Starr