This conference is devoted to the broad area of research interests of Professor S. Ramanan (Chennai Mathematical Institute, India) in celebration of his 70th birthday.

This is a joint activity of the Spanish Semester on Moduli Spaces (January-June 2008) and the international research group Vector Bundles on Algebraic Curves (VBAC).

The conference will take place at Residencia La Cristalera in Miraflores de la Sierra (Madrid, Spain). Participants will be accommodated in this hall of residence and at Hotel Miraflores. Talks will start on the morning of Monday 16th of June and will end by lunch time on Friday 20th of June.

Programme

You can click on the lecture titles below to view the corresponding abstracts. You can also view or download some lectures slides and click here to download a pdf file with the programme, the schedule and the abstracts.
  • V. Balaji (Chennai Mathematical Institute, India) [lecture slides]
    Principal bundles in positive characteristics
    The aim of the talk is to give the proof of existence of stable principal bundles over surfaces over fields of positive characteristics.
  • A. Beauville (Université de Nice, France) [lecture slides]
    The theta map for principal bundles on curves
    The moduli space of G-bundles on a curve (for G a simple algebraic group) carries a natural line bundle, the determinant bundle, which generates its Picard group; the map to projective space defined by this line bundle is called the theta map. I will discuss some recent results on this map for classical groups.
  • D. Ben-Zvi (Univesity of Texas at Austin, USA) [lecture slides]
    Geometric Function Theory
    I will describe some basic features of a geometric version of function theory and harmonic analysis, inspired by the Fourier-Mukai transform and the geometric Langlands program. In this theory spaces of functions on manifolds are replaced by derived categories of sheaves on spaces such as the moduli of vector bundles on a curve. I will describe my recent work with John Francis and David Nadler (building on ideas of Toen, Lurie and others) establishing geometric analogues of some standard tools, such as the representation of operators by integral kernels, the description of modules over convolution algebras and the affineness of affine varieties.
  • U. Bhosle (TIFR, Mumbai, India) [lecture slides]
    Picard groups of moduli spaces of torsionfree sheaves on curves
    The lecture presents old and new results on the Picard groups of the moduli spaces of vector bundles of (any rank) on irreducible curves. In case of low ranks we also study the Picard groups of some subschemes associated to the moduli spaces of torsionfree sheaves on a nodal curve.
  • R. Donagi (University of Pennsylvania, USA)
    On the Geometric Langlands Conjecture
    We will state, explain and outline a proof of the classical limit of the geometric Langlands conjecture. Concretely, we show that the Hitchin integrable system for a complex reductive group G is dual to the Hitchin system for the Langlands dual group LG. We will discuss the relation of this result to the full "quantum" conjecture.
  • N. Hitchin (University of Oxford, UK) [lecture slides]
    Vector bundles and the icosahedron
    We observe that a point in an irreducible representation space V of SO(3) of dimension 2n+1 defines a plane curve of degree n together with a rank two semi-stable bundle over it. When n=3, we use Atiyah's classification of vector bundles over an elliptic curve to describe distinguished subvarieties of P(V), which are closely related to the icosahedral group.
  • K. Hulek (Leibniz Universität Hannover, Germany) [lecture slides]
    Moduli of polarized symplectic manifolds
    In many ways irreducible symplectic manifolds behave similar to K3-surfaces, although it is known that the global Torelli theorem fails in general. Nevertheless, it is possible to relate moduli spaces of polarized irreducible symplectic manifolds to quotients of type IV domains by an arithmetic group. We analyse the situation and prove that certain moduli spaces of polarized irreducible symplectic manifolds are of general type. This is joint work with V. Gritsenko and G.K. Sankaran.
  • J.-M. Hwang (KIAS, Seoul, Korea) [lecture slides]
    Base manifolds for fibrations of projective irreducible symplectic manifolds
    Given a projective irreducible symplectic manifold M of dimension 2n, a projective manifold X and a surjective holomorphic map f: M→X with connected fibers of positive dimension, we prove that X is biholomorphic to the projective space of dimension n. The proof is obtained by exploiting two geometric structures at general points of X: the affine structure arising from the action variables of the Lagrangian fibration f and the structure defined by the variety of minimal rational tangents on the Fano manifold X.
  • J. N. Iyer (IMSc, Chennai, India) [lecture slides]
    Chern-Simons classes of canonical extensions of flat bundles
    In this talk, we will discuss the various constructions of the Chern-Simons class of flat bundles. Some of these constructions can be extended to the Deligne's canonical extension of flat bundles. We discuss some questions and partial results on these classes. This is a joint work with C. Simpson.
  • A. King ( University of Bath, UK) [lecture slides]
    Dimers, quivers and Calabi-Yau algebras
    Dimer models have recently found a new use in theoretical physics to describe branes on toric 3-fold Calabi-Yau (i.e. Gorenstein) singularities. I will explain how this can be understood within the framework of `non-commutative crepant resolutions' of such singularities.
  • S. Kumar (University of North Carolina at Chapel Hill, USA) [lecture slides]
    Equivariant K-theory of flag varieties
    This is a report on my joint work with W. Graham. We prove some general results on the T-equivariant K-theory of flag varieties G/B, where G is a semisimple complex algebraic group and B is a Borel subgroup. We also make a conjecture on some positivity phenomenon in the T-equivariant K-theory of G/B for the product of two basis elements written in terms of the basis of the T-equivariant K-theory of G/B introduced by Kostant-Kumar. This conjecture is parallel (but different) from the conjecture of Griffeth-Ram for the product of structure sheaf of schubert varieties. We give an explicit expression for the product in the T-equivariant K-theory of projective spaces. In particular, we establish our conjecture in this case.
  • I. Mundet i Riera (Universitat de Barcelona, Spain) [lecture slides]
    Hilbert-Mumford criterion for Kaehler manifolds and boundaries of symmetric spaces
    The Hilbert-Mumford criterion in GIT characterizes (semi)stability in terms of maximal weights. An analogue of it for Kaehler manifolds (endowed with a Hamiltonian action by isometries of a compact Lie group K) characterizes which orbits of the complexification G of K meet the zero locus of the moment map. We will explain that in this situation the analogue of the maximal weights can be seen as a function on the boundary at infinity of the symmetric space G/K. This point of view (which was adopted by Kapovich, Leeb and Millson in a particular case) leads to a very simple characterization of (semi, poli) stable points and allows to recover and extend results of Bruasse and A. Teleman which are Kaehler analogues of results of Ramanan and Ramanathan.
  • M. S. Narasimhan (TIFR, Bangalore, India) [lecture slides]
    S. Ramanan and his work
    In this lecture I propose give a survey of the mathematical contributions of Ramanan and their impact.
  • D. Oprea (Stanford University, USA) [lecture slides]
    Theta dualities on moduli spaces of sheaves
    The moduli spaces of sheaves carry natural theta divisors, which are higher-rank analogues of the classical theta divisors on Jacobians. I will explain the general setting of Beauville/Le Potier's strange duality conjecture which relates sections of theta line bundles on two moduli spaces of topologically complementary sheaves. In particular, I will discuss numerical evidence in the case of sheaves over curves, K3 and abelian surfaces. This is joint work with Alina Marian.
  • C. Pauly (Université de Montpellier 2, France)
    Opers and Frobenius-destabilized vector bundles over curves
    The Frobenius map induces by pull-back a rational self-map of the moduli space of rank-r semi-stable vector bundles over a smooth projective curve defined over a field of characteristic p>0. The indeterminacy locus I of this self-map corresponds to Frobenius-destabilized vector bundles. The notion of oper, introduced by Beilinson and Drinfeld, turns out to be very useful in the study of I. Our main result is a lower bound on the dimension of I. This is joint work with Kirti Joshi.
  • M. S. Raghunathan (TIFR, Mumbai, India) [lecture slides]
    Remarks on the Atiyah-Singer Index Theorem
    In this talk we take a look at the first proof provided by Atiyah and Singer for the theorem. We will show that the ``bordism invariance'' proved by them can be deduced from qualitative information on the heat kernel. We also offer some further comments on the proof that lead to technical simplifications in the original proof.
  • T. R. Ramadas ( ICTP, Trieste, Italy) [lecture slides]
    An approach to the Hodge conjecture for certain abelian fourfolds
  • C. S. Seshadri (Chennai Mathematical Institute, India) [lecture slides]
    Some remarks on Parabolic Structures
    A motivation for introducing parabolic structures on vector bundles (say on curves) was to interpret vector bundles on ramified galois coverings with the action of the galois group. When we consider semisimple groups as structure groups, the usual definition of parabolic structures has to be modified.
  • C. Simpson (Université de Nice, France) [lecture slides]
    Iterated destabilizing modifications of vector bundles with connection
    Given a vector bundle with integrable connection (E,∇) on a curve, if E is not itself semistable as a vector bundle then we can iterate a construction involving modification by the destabilizing subobject to obtain a Hodge-like filtration F p of E which satisfies Griffiths transversality. The associated graded Higgs bundle is the limit of (E,t ∇) under the de Rham to Dolbeault degeneration. We get a stratification of the moduli space of connections, such that one of the minimal strata is the space of opers.

Schedule

 
 Monday 16th June
 Tuesday 17th June
 Wednesday 18th June
 Thursday 19th June
 Friday 20th June
10:00- 11:00
 M. S. Narasimhan
 R. Donagi
 A. Beauville C. Simpson
 9:30-10:30
 N. Hitchin
11:00- 11:30
 Coffee
 Coffee
 Coffee Coffee
 10:30- 11:00
 Coffee
11:30-12:30
 U. Bhosle S. Kumar C. Pauly
 J. N. Iyer
 11:00-12:00
 A. King
13:00-15:00
 Lunch
 Lunch
 Lunch Lunch 12:00-13:00
 C. S. Seshadri
15:00-16:00
 K. Hulek
 M. S. Raghunathan D. Oprea
 D. Ben-Zvi
 13:00-15:00
 Lunch
16:00-16:30
 Coffee
 Coffee Coffee
 Coffee
   END OF 
  CONFERENCE
16:30-17:30
 J.-M. Hwang I. Mundet
 V. Balaji
 T. R. Ramadas
 
   SOCIAL DINNER 
   (starts at 20:30)  		 

The social dinner will take place on Wednesday 18th June, 20:30, at Residencia La Cristalera.

Venue and accommodation

The meeting will take place at a Residence and Conference Hall called La Cristalera. This is about 2 Km from Miraflores de la Sierra, a small village located 50 Km North of Madrid.
    Residencia La Cristalera
    Address: Carretera a Rascafría, Km 10. 28792, Miraflores de la Sierra, Spain
    Phone: +34 918443989 / +34 918443920
    Fax: +34 918444464
    Email:

    Map
    Road map
Participants will be accommodated in this hall of residence (where the meeting will take place) and at Hotel Miraflores. This hotel is in Miraflores de la Sierra.
    Hotel Miraflores
    Address: Calle del Rio, S/N. Miraflores de la Sierra, Spain.
    Phone: +34 918449090
    Fax: +34 918449091
    Email:

    Map
Breakfast, lunch and dinner will be provided at La Cristalera for all the participants:
  • Breakfast: 8:00-9:30 (Monday-Friday).
  • Lunch starts at 13:00 (Monday-Friday).
  • Dinner starts at 20:30 (Monday-Thursday) and at 21:00 (Sunday the 15th).
Since the conference venue is 2 Km away from the hotel, tansportation between the hotel and the conference venue will be provided:
  • From the Hotel to La Cristalera: every 15 minutes, from 8:00 to 9:00 (Monday-Friday) and from 20:00 to 21:00 (Sunday the 15th).
  • From La Cristalera to the Hotel: every 15 minutes, from 22:00 to 23:00 (Monday-Thursday), from 22:30 to 23:30 (Sunday the 15th) and from 14:30 to 15:30 (Friday the 20th).
If you are staying at Residencia La Cristalera, note that on Sunday 15th June, you will not be allowed to get in your room until 16:00. You can wait in the cafeteria or in the gardens of La Cristalera, but the restaurant and the cafeteria will not be working until the evening.

Travel information

Click here to obtain information on how to get to Miraflores de la Sierra and to the venue of the conference.

Registration fee

A fee of €100, to cover coffee and other refreshments, is payable in cash on arrival at the conference on Sunday the 15th (credit cards and cheques cannot be accepted).

Click here for information on registration to participate in this conference.

List of participants

Click here for to see the current list of participants.

Conference photographs

Click here to view a group photograph.

More Conference photographs are available at the following websites: site #1, site #2 and site #3.

Computer facilities

There are no computers in the venue of the conference, but you may want to bring your laptop, since there is Wi-Fi connectivity available throughout in Residencia La Cristalera. Hotel Miraflores also has Wi-Fi connectivity and ADSL. You should ask for these services at the reception desks of the Residence and/or the Hotel.

Application for NSF financial support

Click here to apply for NSF financial support to participate in this conference. Note that NSF financial support is restricted to participants with current USA affiliation.

Organizing committee

Sponsors

     CSIC   Ingenio Mathematica      MEC    NSF

Poster

Click here to download a conference poster in A3 format (263K pdf file).

Contact

For further information on this conference, please contact Oscar García-Prada .