Profile
Profile

Prof. Dr. Nicolas Perrin

Former Bonn Junior Fellow
Current position: Professor, Versailles Saint-Quentin-en-Yvelines University

E-mail: nicolas.perrin(at)uvsq.fr
Homepage: http://lmv.math.cnrs.fr/annuaire/nicolas-perrin/
Institute: Mathematical Institute
Research Area: Former Research Area F
Date of birth: 15.Apr 1974
Mathscinet-Number: 661087

Academic Career

2000

PhD, Versailles University, France

2001 - 2002

Postdoc, University of Cologne

2002 -

Maître de conférences, University of Paris VI, France

2007 -

Professor (W2, Bonn Junior Fellow), University of Bonn

2010

Habilitation, University of Paris VI, France

Research Profile

I work at the intersection of representation theory and algebraic geometry. I have been working on the singularities of Schubert varieties and the existence of small resolutions which allow to compute Kahzdan-Luzstig polynomials. I have also been working on enumerative geometry of homogeneous spaces and especially quantum cohomology. On this subject I computed the quantum cohomology of new varieties and discovered some unexpected symmetries.
I have also been interested in the geometry of low codimension subvarieties in homogeneous spaces (see [8] and [7]).

Research Projects and Activities

ANR (French National Research Agency) project: “New Symmetries in Gromov-Witten Theory”
Member

Contribution to Research Areas

Former Research Area F
In [1], we study the quantum cohomology of (co)minuscule homogeneous varieties under a unified perspective. We show that three points Gromov-Witten invariants can always be interpreted as classical intersection numbers on auxiliary varieties. Our main combinatorial tools are certain quivers, in terms of which we obtain a quantum Chevalley formula and a higher quantum Poincaré duality. In [2], we prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, once localized at the quantum parameter, has a nontrivial involution mapping Schubert classes to multiples of Schubert classes. This can be stated as a strange duality property for the Gromov-Witten invariants, which turn out to be very symmetric.
In [3], we prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, specialized at q=1, is semisimple. This implies that complex conjugation defines an algebra automorphism of the quantum cohomology ring localized at the quantum parameter. We check that this involution coincides with the strange duality defined in a previous paper. We deduce Vafa-Intriligator type formulas for the Gromov-Witten invariants.
For a G-variety X with an open orbit, we define its boundary <br>partial X as the complement of the open orbit. The action sheaf S_X is the subsheaf of the tangent sheaf made of vector fields tangent to <br>partial X. We prove in [4], for a large family of smooth spherical varieties, the vanishing of the cohomology groups H^i(X,S_X) for i>0, extending results of F. Bien and M. Brion.
In [5], we show that for any minuscule or cominuscule homogeneous space X, the Gromov-Witten varieties of degree d curves passing through three general points of X are rational or empty for any d. Applying techniques of A. Buch and L. Mihalcea to constructions of the authors together with L. Manivel, we deduce that the equivariant K-theoretic three points Gromov-Witten invariants are equal to classical equivariant K-theoretic invariants on auxiliary spaces.
In [6], we decompose the fibers of the Springer resolution for the odd nilcone of the Lie superalgebra <br>mathfrak{osp}(2n+1,2n) into locally closed subsets. We use this decomposition to prove that almost all fibers are connected. However, in contrast with the classical Springer fibers, we prove that the fibers can be disconnected and non equidimensional.
In [7], we prove Bertini type theorems for the inverse image, under a proper morphism, of any Schubert variety in a homogeneous space. Using generalizations of Deligne's trick, we deduce connectedness results for the inverse image of the diagonal in X^2 where X is any isotropic Grassmannian. We also deduce simple connectedness properties for subvarieties of X.

Selected Publications

[1] P. E. Chaput, L. Manivel, N. Perrin
Quantum cohomology of minuscule homogeneous spaces
Transform. Groups , 13: (1): 47--89
2008
DOI: 10.1007/s00031-008-9001-5
[2] Pierre-Emmanuel Chaput, Laurent Manivel, Nicolas Perrin
Quantum cohomology of minuscule homogeneous spaces. II. Hidden symmetries
Int. Math. Res. Not. IMRN (22): Art. ID rnm107, 29
2007
DOI: 10.1093/imrn/rnm107
[3] P. E. Chaput, L. Manivel, N. Perrin
Quantum cohomology of minuscule homogeneous spaces III. Semi-simplicity and consequences
Canad. J. Math.
, 62: (6): 1246--1263
2010
DOI: 10.4153/CJM-2010-050-9
[4] Boris Pasquier, Nicolas Perrin
Local rigidity of quasi-regular varieties
Math. Z. , 265: (3): 589--600
2010
DOI: 10.1007/s00209-009-0531-x
[8] Nicolas Perrin
Small codimension smooth subvarieties in even-dimensional homogeneous spaces with Picard group \Bbb Z
C. R. Math. Acad. Sci. Paris , 345: (3): 155--160
2007
DOI: 10.1016/j.crma.2007.06.012

Publication List

Selected Invited Lectures

2007

Schubert calculus and Schubert geometry, Banff, AB, Canada

2009

Combinatorial, Enumerative and Toric geometry, Mathematical Sciences Research Institute (MSRI), Berkeley, CA, USA

2009

Romanian-German symposium on moduli spaces in Geometry and Physics, Sibiu (Hermannstadt), Romania

2010

Algebraic Groups, Oberwolfach, Germany

2010

Classical Algebraic Geometry, Oberwolfach, Germany

2010

Complex Algebraic Geometry, Paris, France

2010

Complex Geometry, Vector bundles, Algebraic varieties with a group action, Algebraic Cycles, Hyderabad, India

Selected PhD students

Piotr Achinger (2015): “K(pi, 1) Spaces in Algebraic Geometry”,
now EPDI Postdoc, Banach Center, Warsaw, Poland
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