TOPOLOGY OF SCHUBERT VARIETIES AND THEIR DEGENERATIONS

Instructor:

The geometry of Flag manifolds -- a class which generalises complex projective spaces and Grassmannians -- has remarkable connections to the combinatorics of symmetric groups and the representations of the unitary group. Generalizations exist for other (orthogonal, symplectic and exceptional) Lie groups. One handle on these manifolds is their decomposition into Schubert cells; for the Grassmannian, these reflect the combinatorics of the intersection of a linear subspace with various coordinate subspaces. The closures of these Schubert cells are the Schubert varieties, whose singularities include a wealth of information about representations.

These varieties can be degenerated to a class of more singular varieties that carry more symmetry, called toric varieties. Everything about such varieties can be understood combinatorially, using linear properties of convex polytopes. The first constructions, due to Gonciulea-Lakshmibai for the unitary group, involved polynomial algebra techniques, but subsequent generalisation by Caldero exploited deep features of representation theory.

This project will provide an introduction to Schubert varieties and toric geometry and investigate how the topology of a Schubert variety changes under toric degeneration. We will use combinatorial approaches to topology of singularities ((co)homology calculations) and/or investigate Caldero’s degeneration in particular low rank cases. Explicit calculations get difficult rather quickly, and there will be opportunities for `conjecture testing’ using computer software.

Lectures on the material will be delivered throughout the program, although students will have opportunity to begin working on their research projects relatively quickly and present what they have learned/discovered at weekly colloquia. The aim is to foster a comfortable, open working environment for all participants.

Prerequisites: A strong background in linear algebra and a course in abstract algebra; further courses in basic topology, commutative algebra, differential geometry would be helpful but not required - motivated students may complete background reading prior to the program’s commencement.

Dummit & Foote ‘

Bjorner-Brenti ‘

Chipalkatti ‘

W. Fulton ‘

D. Cox ‘

P. Caldero ‘

Alexeev-Brion ‘

Gonciulea-Lakshmibai ‘

J. Tymozcko ‘

Nohara-Ueda ‘

Instructor:

__George Melvin__The geometry of Flag manifolds -- a class which generalises complex projective spaces and Grassmannians -- has remarkable connections to the combinatorics of symmetric groups and the representations of the unitary group. Generalizations exist for other (orthogonal, symplectic and exceptional) Lie groups. One handle on these manifolds is their decomposition into Schubert cells; for the Grassmannian, these reflect the combinatorics of the intersection of a linear subspace with various coordinate subspaces. The closures of these Schubert cells are the Schubert varieties, whose singularities include a wealth of information about representations.

These varieties can be degenerated to a class of more singular varieties that carry more symmetry, called toric varieties. Everything about such varieties can be understood combinatorially, using linear properties of convex polytopes. The first constructions, due to Gonciulea-Lakshmibai for the unitary group, involved polynomial algebra techniques, but subsequent generalisation by Caldero exploited deep features of representation theory.

This project will provide an introduction to Schubert varieties and toric geometry and investigate how the topology of a Schubert variety changes under toric degeneration. We will use combinatorial approaches to topology of singularities ((co)homology calculations) and/or investigate Caldero’s degeneration in particular low rank cases. Explicit calculations get difficult rather quickly, and there will be opportunities for `conjecture testing’ using computer software.

Lectures on the material will be delivered throughout the program, although students will have opportunity to begin working on their research projects relatively quickly and present what they have learned/discovered at weekly colloquia. The aim is to foster a comfortable, open working environment for all participants.

Prerequisites: A strong background in linear algebra and a course in abstract algebra; further courses in basic topology, commutative algebra, differential geometry would be helpful but not required - motivated students may complete background reading prior to the program’s commencement.

__References & Complements__:Dummit & Foote ‘

*Abstract Algebra*’ Ch. 1-4, 7-9Bjorner-Brenti ‘

*Combinatorics of Coxeter Groups*’Chipalkatti ‘

*Notes on Grassmannians and Schubert Varieties*’ http://server.maths.umanitoba.ca/~jaydeep/Site/Papers_files/GrSc.pdfW. Fulton ‘

*Young Tableaux: with Applications to Representation Theory...'*Ch. IIID. Cox ‘

*Lectures on Toric Varieties*’ http://www.cs.amherst.edu/~dac/lectures/coxcimpa.pdfP. Caldero ‘

*Toric Degenerations of Schubert Varieties*’ http://arxiv.org/abs/math/0012165Alexeev-Brion ‘

*Toric Degenerations of Spherical Varieties*’ http://arxiv.org/abs/math/0403379Gonciulea-Lakshmibai ‘

*Degenerations of flag and Schubert varieties to toric varieties*’ Transformation Groups, Vol. 1, Issue 3 , pp 215-248J. Tymozcko ‘

*Introduction to Equivariant Cohomology...*’ http://arxiv.org/abs/math/0503369Nohara-Ueda ‘

*Toric degenerations of integrable systems on Grassmannians...*’ http://arxiv.org/abs/1111.4809v1CURVATURE SHORTENING FLOWS IN THE PLANE

Instructor: Patrick Wilson

A geometric flow is a way of deforming a geometric object from one shape into another, possibly nicer, shape. More precisely, a geometric flow is a partial differential equation that describes how a geometric object deforms or evolves in time. The most famous example of such a flow is the Ricci flow, used in Perelman's proof of the Poincare Conjecture. This research program will focus on the curvature shortening flow (CSF), which describes how a curve evolves in the plane. Grayson's Theorem states that under the CSF, any embedding of a circle in the plane will shrink to a point in finite time, while becoming more circular. After understanding this result, we will look at extensions such as computer algorithms for solving this PDE, thereby giving visual examples of the flow, and evolutions of networks.

Instructor: Patrick Wilson

A geometric flow is a way of deforming a geometric object from one shape into another, possibly nicer, shape. More precisely, a geometric flow is a partial differential equation that describes how a geometric object deforms or evolves in time. The most famous example of such a flow is the Ricci flow, used in Perelman's proof of the Poincare Conjecture. This research program will focus on the curvature shortening flow (CSF), which describes how a curve evolves in the plane. Grayson's Theorem states that under the CSF, any embedding of a circle in the plane will shrink to a point in finite time, while becoming more circular. After understanding this result, we will look at extensions such as computer algorithms for solving this PDE, thereby giving visual examples of the flow, and evolutions of networks.

__Prerequisites__: Students should be familiar with undergraduate real analysis. Some background in differential geometry and/or partial differential equations would be helpful. In addition, some programming knowledge may be useful.__Background reading__: Differential geometry of curves, as in Chapters 1-3 of "Elementary Differential Geometry" by Andrew Pressley