Amherst 2010

AGNES: Algebraic Geometry Northeastern Series

University of Massachusetts at Amherst,

Department of Mathematics and Statistics   

Saturday (Apr 10) and Sunday (Apr 11)

with a special lecture on Friday (Apr 9) 

Organizers: Paul Hacking and Jenia Tevelev (UMass); Jason Starr (Stony Brook).


AGNES at Amherst was supported by NSF grants DMS-0963853 and DMS-0734178,

Department of Mathematics and Statistics of the University of Massachusetts,

Research Leadership in Action grant of the University of Massachusetts, and 

Department of Mathematics and Statistics of Mt Holyoke college.

Open Problems 

Notes from the open problem session by Phillip Griffiths, Rahul Pandharipande, and Paul Hacking.

Abstracts of Lectures and Links to Videos 

(note: videos are high-quality and in some browsers will start playing only after a download is complete.)

Roman Bezrukavnikov (MIT)
Noncommutative resolutions from positive characteristic Link to Video

Quantizing an algebraic symplectic variety over a field of positive characteristic one gets interesting structures on the variety over a field of zero characteristic. The construction is inspired by representation theory and is expected to have applications in algebraic geometry.

Igor Dolgachev (Michigan) Configuration Spaces of Complex Spheres Link to Video

    A complex sphere is a nonsingular quadric in projective space intersecting the fixed hyperplane at infinity at a fixed quadric. A birational transformation from the space given by quadrics containing the fixed quadric identifies a sphere with a hyperplane section of a nonsingular quadric in the projective space of dimension one larger. The spherical geometry becomes the invariant theory of an orthogonal group. I will discuss the GIT-moduli spaces of configurations of N spheres, or equivalently, the moduli spaces of ordered sets of N points in a projective space with respect to the projective orthogonal group.

Phillip Griffiths (IAS) Mumford-Tate Groups and Mumford-Tate Domains Link to Video


Open problems

    Mumford-Tate groups M are the fundamental symmetry groups of Hodge theory, and Mumford-Tate domains are the natural parameter spaces for families of polarized Hodge structures whose generic member has M as its Mumford-Tate group.  There is vast theory (Shimura varieties) of Mumford-Tate domains in the classical case of weight one polarized Hodge structures, or equivalently complex Abelian varieties.  Much less is known in the non-classical case, and the purpose of this talk is to present some recent work. Specifically, the more fundamental notion of a Hodge group Mφ,  Hodge representation (M,ρ,φ), and a Hodge domain DM,φ will be introduced and their basic structure developed.  Applications include

(1)  An answer to the questions:  “What are all the possible Hodge groups?”.  For a given Hodge group Mφ, what are all possible realizations of Mφ as a Mumford-Tate group?

For example, in the classical case no exceptional Lie group is a Mumford-Tate group, whereas most exceptional groups, including the two real forms of G2, are interesting Mumford-Tate groups.

(2)  The result that the following are equivalent for a reductive, Q-algebraic group M:
  • M is a Mumford-Tate group
  • Discrete series representations occur in L2(M(R))
  • Cuspidal automorphic representation may be expected to occur in L2(M(Q)\M(A)/K)
(3)  A structure theorem, that loosely stated is the result that the natural spaces to which global variations of Hodge structure map are a quotient, Г\DM,φ , where the monodromy group Г is Q-Zariski dense in M; in particular, it has the same tensor invariants as a arithmetic group.

(4) An “algorithm” that, at least in low dimensional cases, enables one to classify all possible Mumford-Tate groups for polarized Hodge structures with given Hodge numbers.  This turns out to be a highly arithmetic and a very interesting story; e.g., for polarized Hodge structures of mirror quintic type.

Our basic point is that Hodge groups and Hodge domains are natural objects that encode:
  • Algebraic geometry
  • Hodge theory
  • Representation theory and arithmetic
Aside from the classical case of Shimura varieties, their integrated study is in a very early stage, and some preliminary results and some of the outstanding issues will be discussed.

Sean Keel (UT Austin)
Towards theta functions for polarized K3 surfaces 
Link to Video

I'll explain
joint work with Paul Hacking and Mark Gross on the following conjecture:

Conjecture (A. Tyurin) : Let (X,L) be a smooth polarized K3 surface over k((t)). Then the projective space P(H0(X,L)) has a canonical set of coordinate points, parameterized by the integer points in a canonical "lattice polygon without boundary" -- a piecewise integral affine structure on S2, provided X (thought of as the generic point of a 1-parameter family) has maximally unipotent monodromy (or equivalently, the family is a so called type III degeneration).

The analogous result in the "log K3" case of a polarized toric surface is simple and familiar. In the pretalk I'll prove this by an argument which generalizes to the K3 case.

Radu Laza (Stony Brook) Moduli spaces birational to locally symmetric varieties of orthogonal or unitary type  Link to Video

A small number  of (geometrically important cases of) moduli spaces can be described as  arithmetic quotients of a bounded symmetric domains of type IV or I1,n (complex ball). In this talk, I will survey some consequences of this description, with a special focus on the birational geometry of the moduli space of genus 4 curves.

Alina Marian (UI Chicago) Strange duality on moduli of sheaves over a generic K3 surface Link to Video

Spaces of sections of determinant line bundles over moduli spaces of stable sheaves on a K3 surface are conjectured to obey a geometrically induced duality. I will show the statement is true on a generic K3 surface for a large class of topological types of the sheaves. The proof uses Fourier-Mukai techniques over elliptically fibered K3 surfaces. This is joint work with Dragos Oprea.

Rahul Pandharipande (Princeton) Algebraic cobordism for varieties, bundles, etc.  Link to Video

Open problems

I will talk about a theory of algebraic cobordism for varieties, bundles, etc. via normal crossings degenerations (with only double points). The results are joint work with M. Levine and with Y.-P. Lee.

Chenyang Xu (MIT) Boundedness of the Volumes and its Applications  Link to Video

(Joint work with Hacon and McKernan) I will talk about our recent investigation on the boundedness of the volumes of singular pairs (in the sense of birational geometry). As an application, we show that fixed a dimension n, any general type variety X of dimension n has the order of the automorphism group |Aut(X)|  less or equal to N · vol(KX), where N is a constant depending only on n.  This is a generalization of the classical Hurwitz's theorem for curves, and Xiao's theorem for surfaces. If time permits, I will also talk about the work-in-progress in this direction, which yields the boundedness of the moduli functor of slc canonically polarized varieties with a fixed numerical condition. This will be a generalization of Alexeev's theorem in the surfaces case.

Poster Presentations

Our poster session featured 17 posters by graduate students, postdocs, and undergraduate students. It was
organized by Milena Hering and Jessica Sidman.

Christine Berkesch (Purdue University) The rank of a hypergeometric system  Download Poster

An A-hypergeometric system is a system of PDEs determined by a toric ideal and certain homogeneity parameters. The dimension of its solution space, called its rank, is constant for generic parameters. I explain the combinatorial nature of this rank at non-generic parameters, producing a stratification of the parameter space with constant rank on each stratum.

Tristram Bogart (Queen's University)
The relative tropical lifting problem
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A common theme in the recently concluded semester on tropical geometry at MSRI was the tropical lifting problem: given a balanced rational polyhedral complex in real n-space, when is it the tropicalization of an algebraic variety? This is a difficult problem in general: even in the case of linear spaces it depends on realizability of matroids. A characterization in the case of tropical curves of genus zero and one was given by David Speyer. With Eric Katz, we develop a local obstruction to a related problem: lifting tropical curves inside tropical hypersurfaces in n-space.

Greg Burnham (Princeton undergrad) Graph Curves of High Degree and their Secant Varieties 
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We study graph curves of genus 2 and high degree (d at least 2g+3). We discuss ways of embedding these curves, and, using data obtained with the computer algebra system Macaulay 2, we obtain a conjectural description of features of their graded Betti diagrams. Finally, we discuss the secant varieties of these graph curves, and mention how to generalize our project to curves with genus greater than 2.

Qile Chen (Brown) Stable Maps Relative to Simple Normal Crossing Divisors 
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The idea of logarithmic stable maps was introduced by Bernd Siebert in 2001. But the program has been on hold for a while, since Mark Gross and Bernd Siebert were working on other projects in mirror symmetry. They have returned to it only recently. Their approach is to probe the moduli space using the standard log point. Next we describe our point of view of log stable maps. This covers the case where the target is a variety with a simple normal crossings divisor.

Inspired by the work of B. Kim on logarithmic stable maps, our starting point is the notion of minimal log structures. However, instead of using expensions as in B. Kim's situation, we put minimal log structures on the source curves, but fix the target with the canonical log structure associated to the simple normal crossing divisors. Our log stable maps will be the usual stable maps equipped with the minimal log structures. The stack parametrizing log stable maps under our definition is shown to be a proper Deligne-Mumford stack. This is in part joint work with Dan Abramovich.

Maria Angelica Cueto (University of California, Berkeley) Tropical secant graphs of monomial curves
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We construct and study four graphs parameterized by n coprime numbers i1, ..., in. We start with an abstract graph and a particular embedding as a weighted graph in Rn+1, which we call the tropical secant surface graph. We identify the latter with the tropicalization of a surface in Cn+1 parameterized by binomials. Using this graph, we construct a third graph which, after an appropriate embedding as a graph in a sphere, coincides with the tropicalization of the first secant variety of the projective monomial curve (1 : ti1: ... : tin). The combinatorics involved in computing the degree of these classical secant varieties is by no means trivial and was developed earlier by K. Ranestad. By strengthening the theory of Geometric Tropicalization developed by Hacking, Keel and Tevelev, we give algorithms to effectively compute the degree (as well as its multidegree) and the Chow polytope of the first secant variety of any monomial curve in Pn.

Dragos Deliu (University of Pennsylvania)  Homological Projective Duality for G(3,6)  
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Homological projective duality(HPD) was introduced in 2005 by A. Kuznetsov. The construction gives a powerful method of producing semiorthogonal decompositions of the derived category of coherent sheaves on algebraic varieties. More precisely it allows to describe the derived categories of all complete linear sections of an algebraic variety. For this, one has to introduce a special kind of a semiorthogonal decomposition, which Kuznetsov calls a Lefschetz decomposition because of the similarity with taking cohomology of hyperplane sections. Then one can define the homological projective dual variety. Among the interesting examples where HPD is well understood are the projective space in the double Veronese embedding and G(2,n) (for n lower or equal to 7) with the Plucker embedding. In this poster I will explain the general context of HPD and then describe some results regarding G(3,6).

Olivia Dumitrescu (Colorado State University)
Progress in Nagata's Conjecture for ten points   
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The virtual dimension of a homogeneous linear system of plane curves of degree d passing through n points with multiplicity m is e(Ld(mn))=max{-1; d(d+3)/2-nm(m+1)/2}. One of the related conjectures regarding interpolation problems is related to Nagata's conjecture in the projective plane. Nagata conjectured that if the leading term of the virtual dimension is negative for large values of m and d, then the linear system is empty. The result is known if n<10 or when the number of points is a perfect square, so the case when n=10 appears to be a boundary case for this conjecture. By performing a series of two throws (blowing up -1 curves twice and contracting them) we construct a degeneration of a family of planes to a union of nine surfaces in the central fiber and we calculate all possible limits of the linear system
Ld(m10) on the central fiber of the family. Such a limit line bundle is a line bundle on each surface, agreeing on all of the double curves of the degeneration. We prove that if d/m<117/37 then Ld(m10) is empty. The rational number 117/37 is a close approximation of 101/2 and is the best ratio known for ten points.

Tamar Friedmann (MIT) Orbifold Singularities, Lie Algebras of the Third Kind (LATKes), and Pure Yang-Mills with Matter   

Since the advent of dualities in string theory, it has been well-known that codimension 4 orbifold singularities that appear in extra-dimensional spaces, such as Calabi-Yau or G2 spaces, may be interpreted as ADE gauge theories. As to orbifold singularities of higher codimension, there has not been an analog of this interpretation. In this poster, I show how the search for such an analog led me from the singularities to the creation of "Commutator-Intersection Relations" and Lie Algebras of the Third Kind ("LATKes"). I will introduce an example of a LATKe that arises from the singularity C3/Z3, and prove it to be simple and unique. The root space of this LATKe is 1-dimensional and its Dynkin diagram consists of one point. I will explain that the uniqueness of the LATKe serves as a vacuum selection mechanism. I will also show how the LATKe leads to a new kind of gauge theory in which the matter field arises naturally and which is tantalizingly close to the Standard Model of particle physics.

Noah Giansiracusa (Brown) GIT Compactifications of M0,n from Conics 
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Joint with Matthew Simpson. Inspired by recent work of the second author, we study GIT quotients parametrizing n-pointed conics that generalize the GIT quotients (P1)n//SL(2). Our main result is that the moduli space of stable rational pointed curves M0,n admits a morphism to each such GIT quotient, analogous to the well-known result of Kapranov for the simpler
(P1)n quotients. Moreover, for most linearizations these morphisms factor through Hassett's moduli space of weighted pointed rational curves, where the weight data comes from the GIT linearization data. Additionally, we discuss the hypersimplex that parametrizes the linearizations.

Jose Gonzalez (University of Michigan) Toric vector bundles and Okounkov bodies   
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The global Okounkov body of a projective variety is a closed convex cone that encodes asymptotic information about every big line bundle on the variety. In the case of a rank two toric vector bundle E on a smooth projective toric variety, we use its Klyachko filtrations to give an explicit description of the global Okounkov body of P(E). In particular, we show that this is a rational polyhedral cone, and that P(E) has a finitely generated Cox ring.

Wenchuan Hu (IAS) On Additive Invariants of Actions of Additive and Multiplicative Groups   

Bialynicki-Birula has described a simple method to study the fixed point varieties of actions of additive and multiplicative  group varieties. He obtained a fixed point formula of Euler characteristic. We generalize his method and apply it to compute  additive invariants (e.g. Euler characteristic, Hodge polynomial, counting points, etc.) of varieties admitting such actions.  In particular, we show that the virtual Hodge (p,0) and (0,q)-numbers of the Chow varieties and affine group varieties are zero for all p,q positive.

Mehmet Umut Isik (University of Pennsylvania) Equivalence of the derived category of a variety with a singularity category

In the last decade, building on the previous work of Eisenbud and Buchweitz, Orlov defined the singularity category of a variety to be the quotient of its derived category by the subcategory of perfect complexes and established many properties of it. This invariant is trivial for smooth varieties and contains information about the singularities of a variety. The equivariant version of this invariant is the category of B-type objects for the mirror of a Fano variety in the homological mirror symmetry programme.

It is known that in some cases, the derived category of one variety is equivalent to the equivariant singularity category of another; this is true in particular between a Calabi-Yau hypersurface and its affine cone. In this poster, I explain the proof of such an equivalence for all local complete intersection varieties -- an equivalence between the derived category of any local complete intersection variety and the equivariant singularity category of an associated singular variety.

Mikhail Mazin (University of Toronto) Geometric Theory of Parshin Residues 
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In the early 70's Parshin introduced his notion of the multidimensional residues of meromorphic top-forms on algebraic varieties. Parshin's theory is a generalization of the classical one-dimensional residue theory. The main difference between the Parshin's definition and the one-dimensional case is that in higher dimensions one computes the residue not at a point but at a complete flag of irreducible subvarieties X=Xn
⊃...X0, dim(Xk)=k. Parshin, Beilinson, and Lomadze also proved the Reciprocity Law for residues: if one fixes all elements of the flag, except for Xk, where 0<k<n, and consider all possible choices of Xk, then only finitely many of these choices give non-zero residues, and the sum of these residues is zero. Parshin's constructions are completely algebraic. In fact, they work in very general settings, not only over complex numbers. However, in the complex case one would expect a more geometric variant of the theory.

In my thesis I study Parshin residues from the geometric point of view. In particular, the residue is expressed in terms of the integral over a smooth cycle. Parshin-Lomadze Reciprocity Law for residues in the complex case is proved via a homological relation on these cycles. The thesis consists of two parts. In the first part the theory of Leray coboundary operators for stratified spaces is developed. These operators are used to construct the cycle and prove the homological relation. In the second part resolution of singularities techniques are applied to study the local geometry near a complete flag of subvarieties.

Yijun Shao (University of Arizona) A Compactification of the Space of Algebraic Maps from P1 to a Grassmannian  
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This is a report on my thesis work, which provides an explicit construction of a smooth compactification of the space of algebraic maps from
P1 to a Grassmannian such that the boundary is a normal crossing divisor. We start with the Quot-scheme compactification, which is an irreducible smooth projective variety, and then we perform a sequence of blowing-ups along certain subschemes of the boundary. We first blow up along the "worst" locus, and then blow up along the proper transform of the "second worst" locus, and so on. While this construction is simple to describe, it requires more work to show that the end result is the desired compactification.

Vivek Shende (Princeton University) The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link   

The intersection of a complex plane curve with a small three-sphere surrounding one of its singularities is a non-trivial link. The refined punctual Hilbert schemes of the singularity parameterize subschemes supported at the singular point of fixed length and whose defining ideals have a fixed number of generators. We conjecture that the generating function of Euler characteristics of refined punctual Hilbert schemes is the HOMFLY polynomial of the link. The conjecture is verified for irreducible singularities yk = xn, whose links are the k,n torus knots, and for the singularity y4=x7-x6+4x5y+2x3y2, whose link is the 2,13 cable of the trefoil.

Zhiyu Tian (Stony Brook)
Two Questions About Rationally Connected Varieties 
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A variety is called rationally connected if two general points can be connected by a rational curve. They are the higher dimensional analogues of rational surfaces and have many interesting geometric and arithmetic properties. I will discuss two questions about them. The first one is the weak approxiamtion conjecture, which roughly says that a smooth projective rationally connected variety defined over the function field of a curve should have lots of rational points. The second question asks if rationally connectedness can be seen in Gromov-Witten theory. It is also related to the so called symplectic birational geometry program.

Jie Wang (Ohio State University) On the maximal rank conjecture for line bundles of extremal degree 
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For a general curve C embedded into projective space by a general linear series of degree d dimension r, the maximal rank conjecture gives a prediction about the number of independent hypersurfaces of each degree a general curve C lies on. In this poster,  we propose a new method, using deformation theory, to study the maximal rank conjecture. For line bundles of extremal degree (i.e line bundles with Cliff(L)=Cliff(C), which can be thought as the first case to test the maximal rank conjecture, we prove that this conjecture holds using our new method.

Yi Zhu (Stony Brook) Fano Hypersurfaces in Positive Characteristic 
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Rationally connected varieties generalize rational varieties, which are much easier to deal with. Campana and Kollar-Miyaoka-Mori proved that Fano varieties are rationally connected when the characteristic is zero. In this poster, we use deformation theory to show that general Fano hypersurfaces are separably rationally connected in all characteristic.

  • Abramovich, Dan (Brown University)
  • Aiello, Domenico (University of Massachusetts, Amherst)
  • Alper, Jarod (Columbia)
  • Bakker, Ben (Princeton)
  • Banerjee, Soumya (Yale)
  • Bayer, Arend (University of Connecticut)
  • Belcastro, Sarah-Marie (Mount Holyoke College)
  • Berkesch, Christine (Purdue University)
  • Bezrukavnikov, Roman (MIT)
  • Bhatt, Bhargav (Princeton)
  • Bogart, Tristram (Queen's University)
  • Braden, Tom (University of Massachusetts, Amherst)
  • Burnham, Greg (MIT)
  • Castravet, Ana-Maria (University of Arizona)
  • Cattani, Eduardo (University of Massachusetts, Amherst)
  • Chen, Qile (Brown University)
  • Cooper, Yaim (Princeton)
  • Cox, David (Amherst College)
  • Cueto, Maria Angelica (University of California, Berkeley)
  • Dalakov, Peter (University of Massachusetts, Amherst)
  • Deliu, Dragos (University of Pennsylvania)
  • de Cataldo, Mark (Stony Brook University)
  • DeLand, Matt (Stony Brook University)
  • Deng, Wei (Washington University in St. Louis)
  • Deopurkar, Anand (Harvard)
  • Di Brino, Gennaro (Yale)
  • Diemer, Colin (University of Pennsylvania)
  • Dolgachev, Igor (University of Michigan)
  • Dumitrescu, Olivia (Colorado State University)
  • Dwivedi, Shashank (MIT)
  • Fedorchuk, Maksym (Columbia University)
  • Fortuna, Giorgia (MIT)
  • Foster, Tyler (Yale)
  • Frailey, Conor (Yale)
  • Friedlander, Holley (University of Massachusetts, Amherst)
  • Friedmann, Tamar (MIT)
  • Garcia-Raboso, Alberto (University of Pennsylvania)
  • Giansiracusa, Noah (Brown University)
  • Gillam, William Danny (Brown University)
  • Gonzalez, Jose (University of Michigan)
  • Grieve, Nathan (Queen's University)
  • Griffiths, Phillip (IAS)
  • Grushevsky, Samuel (Stony Brook University)
  • Guerra, Stefano (Duke University)
  • Hacking, Paul (University of Massachusetts, Amherst)
  • Hao, Cheng (Stony Brook University)
  • Hasson, Hilaf (University of Pennsylvania)
  • Hatley, Jeffrey (University of Massachusetts, Amherst)
  • Hering, Milena (University of Connecticut)
  • Ho, Wei (Harvard)
  • Howald, Jason (SUNY Potsdam)
  • Hu, Wenchuan (IAS)
  • Humphreys, Jim (University of Massachusetts, Amherst)
  • Hyun, Yoonsuk (MIT)
  • Isik, Mehmet Umut (University of Pennsylvania)
  • Kamenova, Ljudmila (Stony Brook University)
  • Kazanova, Anna (University of Massachusetts, Amherst)
  • Keel, Sean (University of Texas, Austin)
  • Kumar, Abhinav (MIT)
  • Laza, Radu (Stony Brook University)
  • Lee, Jaepil (Stony Brook University)
  • Li, Long (Stony Brook University)
  • Li, Zhiyuan (Rice University)
  • Maddock, Zachary (Columbia University)
  • Marian, Alina (University of Illinois, Chicago)
  • Markman, Eyal (University of Massachusetts, Amherst)
  • Maulik, Davesh (MIT)
  • Mayanskiy, Evgeny (Pennsylvania State University)
  • Mazin, Mikhail (University of Toronto)
  • McKernan, James (MIT)
  • Mehrotra, Sukhendu (University of Wisconsin, Madison)
  • Molcho, Samouil (Brown University)
  • Mirkovic, Ivan (University of Massachusetts, Amherst)
  • Muthiah, Dinakar (Brown University)
  • Negut, Andrei (Harvard)
  • Norman, Peter (University of Massachusetts, Amherst)
  • Oblomkov, Alexei (University of Massachusetts, Amherst)
  • Oloo, Stephen (University of Massachusetts, Amherst)
  • Pandharipande, Rahul (Princeton)
  • Patel, Anand (Harvard)
  • Piercey, Victor I. (University of Arizona)
  • Pixton, Aaron (Princeton)
  • Preygel, Anatoly (MIT)
  • Qi, You (Columbia University)
  • Saccà, Giulia (Princeton)
  • Saltman, David (IDA-Princeton)
  • Schiffler, Ralf (University of Connecticut)
  • Sekhon, Gagan (University of Connecticut)
  • Sengupta, Tathagata (MIT)
  • Setayesh, Iman (Princeton)
  • Shao, Yijun (University of Arizona)
  • Shen, Mingmin (Columbia University)
  • Shende, Vivek (Princeton)
  • Shumway, Julie (University of Massachusetts, Amherst)
  • Sidman, Jessica (Mount Holyoke College)
  • Singh, Bhairav (MIT)
  • Starr, Jason (Stony Brook University)
  • Teixidor i Bigas, Montserrat (Tufts)
  • Tevelev, Jenia (University of Massachusetts, Amherst)
  • Tian, Zhiyu (Stony Brook University)
  • Tsymbaliuk, Oleksandr (MIT)
  • Urzua, Giancarlo (University of Massachusetts, Amherst)
  • Venkatram, Kartik (MIT)
  • Wang, Botong (Purdue University)
  • Wang, Jie (Ohio State University)
  • Wen, Jun (Stony Brook University)
  • Xu, Chenyang (MIT)
  • Xu, Fei (Rice University)
  • Yang, Yanhong (Columbia)
  • Yang, Yaping (Northeastern University)
  • Yao, Yuan (University of Texas, Austin)
  • Yuen, Cornelia (SUNY Potsdam)
  • Zhang, Letao (Rice University)
  • Zhang, Shizhuo (Kansas State University)
  • Zhang, Yongsheng (Stony Brook University)
  • Zhang, Zheng (Stony Brook University)
  • Zhao, Gufang (Northeastern University)
  • Zhu, Zhixian (University of Michigan)
  • Zhu, Yi (Stony Brook University)

Our beautiful poster (courtesy of Milena Hering)
This document contains information about AGNES reimbursement.