Gradate Analysis Seminar
18.199 -- Spring 2024

• Lectures in 2-146, Tuesday and Thursdays 11:00-12:30.
• Notes updated 24 April, 2024. SpinC
• AnLoSp old notes that I am revising
• Loop-Coll Notes from a colloquium
• Bott-Virasoro Notes for Lecture 8 Feb

Lecture topics:

• 6 February: Brief overview
• 8 February: Compression onto Hardy space
• 13 February: Bogus snow day!
• 15 February: Omar on manifolds. Parse the statement that the tangent bundle of the loop space of a manifold is the loop space of the tangent bundle. As time permits, more on the Bott-Virasoro group.
• 22 February (20 February is an MIT Monday): Alain on Riemann metric and exponential map. The direct application here is that we can find coordinate patches on the loop space this way.
• 27 February: Dain on Lie groups Notes
• 29 February: Isaac on Clifford algebra
• 5 March: Kevin on principal bundles
• 7 March: Spin and loop-orientation
• 12 March: Isaac on spin structures
• 14 March: Alain on Cech cohomology Notes
• 19 March: Dain on holonomy and circle bundles
• 21 March: Kevin on K-theory
• 2 April: Isaac on spin Dirac Notes
• 4 April: Loop-spin structures - central extension of LSpin
• 9 April: Eclipse vacation day for me! Mike on SpinC canceled Notes
• 11 April: The central extension of LSpin is fusive
• 16 April: Dain on the index of the spin Dirac operator
• 18 April: Dain continues
• 23 April: Mike on SpinC
• 25 April: Loop-spin structures - overview

Possible topics for the last four or five lectures. Some of these are short and most of the material can be gleaned from the paper of Spera and Wurzbacher plus my notes on the torus.
• Loop space of the torus
• String measure for the torus
• Positive energy representations of loop-spin. (See Toledano Laredo).
• Fock space
• Dirac-Ramond on the torus

References
• Spera-Wurzbacher review
• Spera-Wurzbacher paper
• Toledano Laredo thesis
• Witten index