Faculty of Mathematics
Ekaterina Melikhova
Grassmannians and Littlewood-Richardson rule (continued)
The second part of my talk will be mostly devoted to the combinatorial description of the Littlewood–Richardson rule via Young tableaux. More details can be found in the abstract of the previous talk.
Ekaterina Melikhova
Grassmannians and Littlewood-Richardson rule
The complex Grassmanian G_{m,n} is the space of linear subspaces of dimension m in the complex (m+n)-dimensional vector space. Last time for each partition contained in the m by n rectangle, Anastasiya defined the corresponding Schubert cell as a certain subset of G_{m,n}. This time we discuss that Schubert cells form a cellular decomposition of the Grassmanian and that the Poincar\’e duals of the fundamental classes of their closures (Schubert classes) form a basis of the integral cohomology of G_{m,n}. The main objective of the talk is to describe the cup-product in this ring. To this end we describe an epimorphism from the ring of symmetric polynomials in m variables with integer coefficients to the integral cohomology ring of G_{m,n}. This epimorphism sends the Schur polynomial corresponding to a partition \lambda to the Schubert class corresponding to \lambda, as long as \lambda contained in the m by n rectangle and otherwise to zero. The Littlewood–Richardson rule enables one to represent the product of two Schur polynomials as a linear combination of Schur polynomials and a similar rule works for Schubert classes due to the said epimorphism. We give a combinatorial description of this rule via Young tableaux.
Anastasia Bazykina
Schubert calculus and Hilbert’s 15th problem
Enumerative geometry is a branch of mathematics concerned with counting the number of geometric objects with given properties. In the 19th century, German mathematician Hermann Schubert developed the method for solving such problems, which is called Schubert calculus. Its rigorous foundation is a topic of Hilbert’s 15th problem. Via Schubert calculus one can solve the problem of finding a number of lines intersecting four given lines in a three-dimensional complex projective space, which is called Schubert problem. I will demonstrate the method used by Schubert to solve it. Then I am going to present two other ways to solve this problem: via hyperboloid and Grassmannian.
Arina Arkhipova (HSE, UNIGE)
Mixed Fiber Polytopes
Let f_1=…=f_n=0 be a system of polynomial equations with the same Newton polytope N. The theorem proven by A. Kushnirenko allows us to compute the number of roots in (C\0)^n for the system f_1=…=f_n=0 provided that the coefficients of the latter are in general position. Namely, such a system has Vol(N) roots. The Bernstein–Kushnirenko theorem is a generalization of this result. It expresses the number of roots for a generic system with the Newton polytopes N_1,…,N_n that are not necessarily all the same. In this case, the answer is the so-called mixed volume of the polytopes N_1,…,N_n. The talk concerns the following generalization of the Bernstein-Kushnirenko theorem. Suppose X={f_1=…=f_{k}=0} is a complete intersection in (C\0)^n and p:(C\0)^n–>(C\0)^m is a projection. A.Esterov and A.Khovanskii proved that if the image p(X) in (C\0)^m is a hypersurface {g=0}, then the Newton polytope of g can be described in terms of the Newton polytopes of the polynomials defining X. Namely, if their Newton polytopes are all equal to N, then the answer is the so-called fiber polytope of N w.r.t. the projection p. If their Newton polytopes N_1,..,N_k are not all the same, then the answer is the mixed fiber polytope of N_1,..,N_k w.r.t. to the projection p. We will outline the proof of this result as well as existence, construction and some important properties of the (mixed) fiber bodies.
No prerequisites required: I will gently introduce all necessary notions during the talk and give many examples. The talk will consist of two parts with a short break in the middle.