Gromov-Witten Theory, geometry and representations

• Applicant: Professor Dr. Benjamin Klopsch, since 9/2015
• Subject Area: Mathematics
• Term: from 2013 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 240029351

• Final Report Year: 2018

Final Report Abstract

The project is concerned with fundamental research at the crossroad of Algebraic Geometry, Representation Theory and Algebraic Combinatorics. Algebraic Geometry studies geometric objects that can be described by polynomial equations, i.e. equations only involving the simplest mathematical operations + and ×. A very classical aspect of Algebraic Geometry is studied in so-called Enumerative Geometry which consists in counting the number of solutions in particular geometric situations (e.g., in ordinary 3-dimensional space, there is a unique line passing through 2 points). In Modern Enumerative Geometry, the above numbers are used to define a new product giving rise to the Quantum Cohomology Ring. These Quantum Cohomology Rings are still mysterious even for simple geometric objects. The objective of the founded project was to study such rings for geometric objects that posses many symmetries; they are called homogeneous spaces and spherical varieties (geometric objects built on the model of a sphere). The rich supply of symmetries (by its group action) makes the study simpler; sometimes it translates the geometric problem into an explicit problem which can be solved with the help of combinatorics and/or the aid of a computer implementation. The Principal Investigator and his collaborators made important advances on the above subject: they created an understanding of the structure of quantum cohomology for new geometric objects and developed new geometric techniques to attack harder problems in the field.

Publications

• (2019) On quantum cohomology of Grassmannians of isotropic lines, unfoldings of An -singularities, and Lefschetz exceptional collections. Ann. inst. Fourier (Annales de l’institut Fourier) 69 (3) 955–991
  Cruz Morales, John Alexander; Mellit, Anton; Perrin, Nicolas; Smirnov, Maxim
  
• Algebraic rational cells, equivariant intersection theory, and Poincaré duality. Math. Zeit., 282 (2016), no. 1-2, 79–97
  R. Gonzales
  
• Fano varieties in Mori fibre spaces, IMRN (2016)
  G. Codogni, A. Fanelli, R. Svaldi and L. Tasin
  
• Rational connectedness implies finiteness of quantum K-theory. Asian Journal of Mathematics, 20 (2016), no. 1, 117–122
  A. Buch, P.-E. Chaput, L. Mihalcea, and N. Perrin
  
• A Chevalley formula for the equivariant quantum K-theory of cominuscule varieties. Algebraic Geometry, 5 (2018), no. 5, 568–595
  A. Buch, P.-E. Chaput, L. Mihalcea, and N. Perrin
  
• Projected Gromov-Witten varieties in cominuscule spaces. Proceedings of the AMS, 146 (2018), no. 9, 3647–3660
  A. Buch, P.-E. Chaput, L. Mihalcea, and N. Perrin