Department of

Mathematics


Seminar Calendar
for events the day of Wednesday, April 4, 2018.

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Wednesday, April 4, 2018

4:00 pm in 243 Altgeld Hall,Wednesday, April 4, 2018

Irving Reiner lectures: Lectures on Quantum Schubert Calculus III

Leonardo C. Mihalcea (Virginia Tech )

Abstract: The quantum cohomology ring of a complex projective manifold X is a deformation of the ordinary cohomology ring of X. It was defined by Kontsevich in the mid 1990’s in relation to physics and enumerative geometry. Its structure constants - the Gromov-Witten invariants - encode numbers such as how many conics pass through 3 general points in the Grassmann manifold of 2-planes in the 4-space. The quantum cohomology ring is best understood when X has many symmetries, or good combinatorial properties, and these lectures will focus to the case when X is a Grassmann manifold or a flag manifold. The subject is quite rich, and intensively studied, with connections to algebraic combinatorics, algebraic and symplectic geometry, representation theory, and integrable systems. My goal is to introduce the audience to some of the basic ideas and techniques in the subject, such as how to calculate effectively in the quantum cohomology rings, and what are the geometric ideas behind the calculations, all illustrated by examples. The lectures are intended for graduate students, in particular I am not assuming prior knowledge of quantum cohomology. I plan to include the following topics. The ‘quantum = classical’ phenomenon of Buch, Kresch and Tamvakis: how a ‘quantum’ calculation can be performed in the ‘classical’ cohomology of an auxiliary space - this leads to formulas based on Knutson and Tao’s puzzles; the technique of curve neighborhoods and the quantum Chevalley formula: what are these, and how they help to get recursive formulas for the equivariant Gromov-Witten invariants; the quantum Schubert Calculus of Grassmannians: a presentation for the quantum ring and polynomial representatives for Schubert classes; quantum K-theory: what is it, what we know, and why is everything so much harder in this case. If time permits, I may briefly mention the connection between quantum cohomology and Toda lattice (B. Kim’s theorem), and the ‘quantum=affine’ phenomenon (D. Peterson’s conjecture, proved by T. Lam and M. Shimozono).

4:00 pm in 245 Altgeld Hall,Wednesday, April 4, 2018

Measurable Differentiable Structures

Chris Gartland (Illinois Math)

Abstract: We'll discuss Lipschitz differentiable structures on a metric measure spaces. The structures consist of a Borel cover of the space, together with Lipschitz maps from the elements of the cover to $\mathbb{R}^n$, such that any real-valued Lipschitz function is differentiable almost everywhere with respect to the maps. Lipschitz differentiable structures were defined in a paper by Cheeger in '99, where he also proved the fundamental theorem that any doubling metric measure space satisfying a Poincare inequality admits such a structure. A corollary of this theorem is a criterion for the non-biLipschitz embedability of these metric spaces in $\mathbb{R}^n$ which generalizes the known results for Carnot groups and Laakso space. These structures allow for interesting constructions on the space such as the $L^\infty$ cotangent bundle and $L^\infty$ cohomology.