AGNES Workshop

Saturday 
Event 
Sunday 
Event 
8:30AM9:30AM  Registration and coffee  8:30AM9:00AM  Coffee 
9:30AM10:30AM  Arend Bayer, The local projective plane, a fractal curve and Γ_{1}(3)  9:00AM10:00AM  Ana–Maria Castravet, Rational curves of minimal degree on higher Fano manifolds 
10:30AM10:45AM  Break  10:00AM10:15AM  Break 
10:45AM11:15AM  Pre–lecture  10:15AM10:45AM  Pre–lecture 
11:15AM12:15PM  Jenia Tevelev, On the cone of effective divisors of M_{0,n}  10:45AM11:45AM  Herb Clemens, Exploring the Hodge problem 
12:15PM1:45PM  Lunch  11:45AM12NOON  Break 
1:45PM2:15PM  Pre–lecture  12NOON1:00PM  Amanda Knecht, Rationally connected varieties over ℚ_{p}^{nr} 
2:15PM3:15PM  Mike Roth, A localglobal principle for weak approximation of varieties over function fields  1PM  Workshop ends 
3:15PM3:45PM  Break 


3:45PM4:15PM  Pre–lecture 


4:15PM5:15PM  William Fulton, Character formulas 


7PM10PM  Dinner (John Harvard's) 



Arend Bayer, The local projective plane, a
fractal curve and Γ_{1}(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 autoequivalences, which includes a subgroup isomorphic to Γ_{1}(3). Finally, via mirror symmetry, it contains a universal cover of the moduli space of elliptic curves with Γ_{1}(3)–level structure.

AnaMaria 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?
 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.

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 padics. They `usually' have points, and we will define what `usually' means.
 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 DeligneMumford 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 (ℂ^{2})^{n} and q,tCatalan numbers (joint work with Kyungyong Lee).

Mike Roth, A localglobal 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 "localglobal" 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.

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 GrothendieckKnudsen moduli space M_{0,n} of stable rational curves. Roughly speaking, nonboundary extremal divisors correspond to total stable degenerations (or, equivalently, pairofpants decompositions) with some decorations of smooth curves of genus n3. Joint with AnaMaria Castravet.
Pre–lectures.
Preceding some of the lectures there will be an halfhour 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 GarciaRaboso, 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
 TsungYin 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 EssenDuisburg
 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
Jason Starr
4108 Math Tower
Department of Mathematics
Stony Brook University
Stony Brook, NY 117943651
Phone: 631 632 8270
FAX: 631 632 7631
Jason Starr