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Einstein Institute of Mathematics
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Prof. Dan Freed (University of Texas at Austin)
Sunday, Nov. 20th, 14:00 Ross 70A|
Title: Twisted matrix factorizations and loop groups
Abstract: The data of a compact Lie group G and a degree 4 cohomology class on its classifying space leads to invariants in low-dimensional topology as well as important representations of the infinite dimensional group of loops in G. The former are the famous quantum invariants of Reshetikhin-Turaev-Witten; the latter are the positive energy representations. The important algebraic data is encoded in a braided tensor category, constructed in special cases (G finite, G simply connected) by various methods. Previous joint work with Mike Hopkins and Constantin Teleman gave a unified description of the Grothendieck ring of that category in terms of topological K-theory. After reviewing these ideas I will describe joint work with Teleman which gives a geometric construction of the category, though not of its tensor structure.
Monday, Nov. 21st, 16:00 Ross 70A|
Title: A differential geometer's BG
Abstract: The classifying space of a group has different incarnations in topology and algebraic geometry, for example. In joint work with Mike Hopkins we introduce a differential geometric manifestation whose de Rham complex consists precisely of the Chern-Weil forms. This differential geometric BG also provides a natural home for equivariant de Rham theory. The definitions and proofs use abstract homotopy theory. We illustrate the effectiveness of this BG in implementing the equivariance → families maneuver in differential geometric contexts.
Thursday, Nov. 24th, Colloquium, 14:00
Einstein Institute of Mathematics, Lecture Hall 2|
Title: Bordism and topological phases of matter
A gathering in memory of Prof. Alexander Zabrodsky will be held after the lecture, in the faculty lounge.
Abstract: Topological ideas have at various times played an important role in condensed matter physics. This year's Nobel Prize recognized the origins of a particular application of great current interest: the classification of phases of a quantum mechanical system. Mathematically, we would like describe them as path components of a moduli space, but that is not rigorously defined as of now. In joint work with Mike Hopkins we apply stable homotopy theory (Adams spectral sequence) to compute the group of topological phases of "invertible" systems. We posit a continuum field theory and then use the Axiom System for field theory initiated by Segal and Atiyah, and various refinements, to prove a theorem which underlies the computations.
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