Abstract:

Schubert problems, in their simplest form, ask for the number of linear spaces incident to fixed general linear spaces. A typical example asks for the number of lines in space intersecting four general lines. Schubert problems are among the most classical in algebraic geometry and have been studied intensely since the nineteenth century. They have surprising and beautiful connections to other branches of mathematics like representation theory and combinatorics.

In these lectures, I will describe geometric approaches to several problems in the theory of homogeneous varieties and Schubert calculus. I will first discuss the problem of studying the cohomology of Type A Grassmannians and flag varieties using degenerations. Degenerations allow one to obtain positive algorithms for computing the cohomology of Grassmannians and greatly clarify Schubert geometry. I will outline several applications of degeneration techniques to computing the quantum cohomology of Grassmannians and the monodromy groups of Schubert problems. I

will then examine the singularities of Schubert varieties, Richardson varieties and certain generalizations. I will show that these varieties are normal, Cohen-Macaulay with rational singularities. Then I will discuss the rigidity problem for Schubert classes. Finally, I will discuss corresponding problems in the geometry of Type B, C, D Grassmannians and flag varieties. I will solve the restriction problem and explore its

relation with the rigidity problem.

These lectures will be relatively elementary and should be accessible to anyone with a basic background in algebraic geometry or topology.

Some references in case people need them (They are all available on line, either through the arxiv or my webpage)

S. Billey and I. Coskun, Singularities of generalized Richardson varieties, Comm. Alg. 40:4 2012, 1466-1495.

M. Brion, Lectures on the geometry of flag varieties. Topics in cohomological studies of algebraic varieties. Trends Math. Birkhauser, Basel,

2005, 33--85

R. Bryant, Rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric spaces, Ann. Math. Studies, AM 153 (2005), Princeton University Press.

I. Coskun, A Littlewood-Richardson rule for two-step flag varieties, Invent. Math. 176 (2009), 325--395

I. Coskun, Rigid and non-smoothable Schubert classes, J. Differential Geom. 87 no.3 (2011), 493-514.

I. Coskun, Restriction varieties and geometric branching rules, Adv. Math. 228 no. 4 (2011), 2441-2502.

I. Coskun, Symplectic restriction varieties and geometric branching rules, A celebration of algebraic geometry, Clay Math Proc 18 (2013), 205-239.

I. Coskun and R. Vakil, Geometric positivity in the cohomology of homogeneous spaces and generalized Schubert calculus, Algebraic Geometry Seattle 2005, 77--124.

R. Vakil, A geometric Littlewood-Richardson rule, Ann. Math, 164 (2006), 371--421.

R. Vakil, Schubert induction, Ann. Math, 164 (2006), 489--512.

Schubert problems, in their simplest form, ask for the number of linear spaces incident to fixed general linear spaces. A typical example asks for the number of lines in space intersecting four general lines. Schubert problems are among the most classical in algebraic geometry and have been studied intensely since the nineteenth century. They have surprising and beautiful connections to other branches of mathematics like representation theory and combinatorics.

In these lectures, I will describe geometric approaches to several problems in the theory of homogeneous varieties and Schubert calculus. I will first discuss the problem of studying the cohomology of Type A Grassmannians and flag varieties using degenerations. Degenerations allow one to obtain positive algorithms for computing the cohomology of Grassmannians and greatly clarify Schubert geometry. I will outline several applications of degeneration techniques to computing the quantum cohomology of Grassmannians and the monodromy groups of Schubert problems. I

will then examine the singularities of Schubert varieties, Richardson varieties and certain generalizations. I will show that these varieties are normal, Cohen-Macaulay with rational singularities. Then I will discuss the rigidity problem for Schubert classes. Finally, I will discuss corresponding problems in the geometry of Type B, C, D Grassmannians and flag varieties. I will solve the restriction problem and explore its

relation with the rigidity problem.

These lectures will be relatively elementary and should be accessible to anyone with a basic background in algebraic geometry or topology.

Some references in case people need them (They are all available on line, either through the arxiv or my webpage)

S. Billey and I. Coskun, Singularities of generalized Richardson varieties, Comm. Alg. 40:4 2012, 1466-1495.

M. Brion, Lectures on the geometry of flag varieties. Topics in cohomological studies of algebraic varieties. Trends Math. Birkhauser, Basel,

2005, 33--85

R. Bryant, Rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric spaces, Ann. Math. Studies, AM 153 (2005), Princeton University Press.

I. Coskun, A Littlewood-Richardson rule for two-step flag varieties, Invent. Math. 176 (2009), 325--395

I. Coskun, Rigid and non-smoothable Schubert classes, J. Differential Geom. 87 no.3 (2011), 493-514.

I. Coskun, Restriction varieties and geometric branching rules, Adv. Math. 228 no. 4 (2011), 2441-2502.

I. Coskun, Symplectic restriction varieties and geometric branching rules, A celebration of algebraic geometry, Clay Math Proc 18 (2013), 205-239.

I. Coskun and R. Vakil, Geometric positivity in the cohomology of homogeneous spaces and generalized Schubert calculus, Algebraic Geometry Seattle 2005, 77--124.

R. Vakil, A geometric Littlewood-Richardson rule, Ann. Math, 164 (2006), 371--421.

R. Vakil, Schubert induction, Ann. Math, 164 (2006), 489--512.