W*-rigidity paradigms for embeddings of II1 factors
…finite index. This is a preorder relation and we prove that it is as complicated as it can be: under the appropriate separability assumptions, we concretely realize any partially ordered…
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The Qualifying Exam Syllabus
…and cohomology Relative homology CW complexes and the homology of CW complexes. Mayer-Vietoris Universal coefficient theorem Kunneth formula Poincare duality Lefschetz fixed point formula Hopf index theorem Cech cohomology and…
CMSA New Technologies: Statistical mechanics of neural networks: from the geometry of high dimensional error landscapes to beating power law neural scaling
…able to use the Kac-Rice formula and the replica method to calculate the number, location, energy levels, and Hessian eigenspectra of all critical points of any index. Second we will…
The unbounded denominators conjecture
The unbounded denominators conjecture, first raised by Atkin and Swinnerton-Dyer, asserts that a modular form for a finite index subgroup of SL_2(Z) whose Fourier coefficients have bounded denominators must be…
Super-rigidity and Castelnuovo’s bound
…it. Nevertheless, super-rigidity has important consequences. I will present one of these consequences, namely, a genus bound for index zero pseudo-holomorphic curves. This is joint work with Aleksander Doan and,…