Organizers: Paul Hacking and Jenia Tevelev

Confirmed Speakers

About AGNES

AGNES is a new series of weekend workshops in algebraic geometry. One of our goals is to introduce graduate students to a broad spectrum of current research in algebraic geometry. AGNES is held biannually at the participating universities in the Northeast.

Scientific Committee:

Dan Abramovich (Brown), Joe Harris (Harvard), Aise Johan de Jong (Columbia), Mikhail Kapranov (Yale), János Kollár (Princeton), James McKernan (MIT)

Organizing committee:

Arend Bayer (UConn), Sam Grushevsky (Stony Brook), Paul Hacking (UMass Amherst), Milena Hering (UConn), Jason Starr (Stony Brook), Jenia Tevelev (UMass Amherst).

Registration

Registration for the workshop is now closed. Please contact the organizers directly for further assistance.

Tentative Schedule

AGNES schedule

Abstracts

Igor Dolgachev (Michigan) Configuration Spaces of Complex Spheres

    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

    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 M_φ,  Hodge representation (M,ρ,φ), and a Hodge domain D_M,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 M_φ, what are all possible realizations of M_φ 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\D_M,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.

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

(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(K_X), 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.

Videotaping

All lectures will be videotaped and streaming videos will be made available at this webpage.

Pre-lectures

Preceding some of the lectures there will be an half-hour informal discussion to introduce the material discussed in the main lecture. The primary audience for pre-talks are graduate students and non-experts.

Poster Session

We invite graduate students and postdocs to participate in our poster session, organized by Jessica Sidman and Milena Hering. Posters will be available for viewing and discussion during breaks as well as during a wine and cheese reception (generously funded by Mount Holyoke College).  We believe the posters will help to start many great mathematical conversations over the course of the weekend. We will make titles and abstracts of posters available to participants at the start of the meeting.  We will also provide poster board and stands.  We encourage poster presenters to think about readability and presentation; your goal is not to present complete proofs of theorems but rather to pique enough interest in your results to motivate people to read the details in your paper later.  It may be helpful to organize the poster as three panels: (1) background and motivation, (2) the statement of the main result together with an illustrative example, and (3) a brief sketch of key proof ingredients.   We will send more logistical information to participants in advance of the workshop.

Professional Development Event

During the lunch break on Saturday, graduate students are invited to an informal talk Mathematicians can contribute by David Saltman (IDA-Princeton). David will talk about employment opportunities outside of academia for recent PhDs in mathematics. All UMass graduate students in mathematics are welcome as well.

Hotel

Participants who have been offered accommodation will be staying at the Campus Center Hotel.

We will send you an e-mail confirming your reservation. The hotel is located within very short walking distance to both the Lederle Graduate Research Tower and the Integrated Science Building where the lectures will be held.

Parking

Parking is available at the Parking Garage. If you stay in the hotel you get a parking pass for one car per room.  If more than one person needs to park the hotel sells them at $5 per night which is better than $20 that the garage sells them for. Parking is free on UMass campus between 5:00 p.m. and 7:00 a.m., Monday - Friday, and on weekends in any non-24 hour lot, in any space that is not marked as a special reserved space.

Parking Garage

Traveling to Amherst

If you are driving, you can use the map above for driving directions to UMass. If you prefer public transit, Amherst can be reached from pretty much anywhere in North East by a Peter Pan bus. Local PVTA buses serve the Pioneer Valley area (Northampton, etc). We will try to cover reasonable local transportation costs. However, our funds are very limited, so we would like to encourage participants to use their own grants for transportation and to carpool if possible. In case you are flying, the closest airport is BDL. Please let us know your itinerary and we will try to arrange your pick-up at the airport (or you can contact the Valley Transporter shuttle). We will cover airfare for speakers only.

Participants

• 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)
• D'Ambroise, Jennie (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)
• 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)
• Sekhon, Gagan (University of Connecticut)
• 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)
• 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)

Acknowledgements

AGNES at Amherst is 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.