SINGREP: Linking Singularity Theory and Representation Theory with Homological Methods

A Marie Sklodowska-Curie Project at the University of Leeds (Eleonore Faber & Robert Marsh)

Project description

In algebraic geometry one tries to understand and explain geometric phenomena of zerosets of polynomial equations (algebraic varieties) with algebraic techniques. Singularities of algebraic varieties are, roughly speaking, points of indeterminacy, where most analytical methods collapse. Geometrically, this corresponds e.g. to cusps or crossing points. In a practical example, the arm of a robot can break if it passes through a singular point, which could result in a complete breakdown of the system. Such a situation should be avoided by theoretical considerations.
This project lies at the intersection of singularity theory, (non-commutative) algebraic geometry, commutative algebra, and representation theory. The main goal is to develop homological methods to understand singularities of algebraic varieties and use them to study representation theoretic concepts such as cluster categories. The project will provide a bridge between these seemingly distant areas that can be exploited in both directions.

Some visualizations of discriminants of reflection groups (and other surface singularities) are now on Instagram . Check out my profile!

I gave a talk on frieze patterns in Middlesbrough, in the Tesside University Maths Club. For some more info see here .

Published articles/preprints

  1. R.-O. Buchweitz, E. Faber, and C. Ingalls: A McKay correspondence for reflection groups, accepted for publication in Duke Math. J. 2019, see arXiv:1709.04218[math.AG] .
  2. E. Faber, G. Muller, and K.E. Smith: Non-commutative resolutions of toric varieties, Adv. Math. 351 (2019), 236--274. See arXiv:1805.00492[math.AC] .
  3. E. Faber: Trace ideals, normalization chains, and endomorphism rings, accepted for publication in PAMQ 2019, see arXiv:1901.04766[math.AC].
  4. K. Baur, E.Faber, S. Gratz, K. Serhiyenko, and G. Todorov: Friezes satisfying higher SL$_k$-determinants, to appear in Algebra Number Theory arXiv:1810.10562[math.RA]
  5. R.-O. Buchweitz, E. Faber, C. Ingalls, and M. Lewis: McKay quivers and Lusztig algebras of some finite groups arXiv:2009.06674[math.RT]
  6. J. August, M.-W. Cheung,E. Faber, S. Gratz, and S. Schroll: Grassmannian category of infinite rank arXiv:2007.14224[math.RT]
  7. R.-O. Buchweitz, E. Faber, and C. Ingalls: The magic square of reflections and rotations, PDF , submitted.


Photo credit: P.-G. Plamondon

I gave a series of four lectures on the McKay correspondence and noncommutative desingularizations at the School ISCRA (Isfahan School on Representations of Algebras) in Isfahan (Iran) in April 2019 and at the Advanced Course Crossing Cohomological bridges in frame of the Follow up of IRTATCA in Barcelona (Spain) in June 2019.

More talks:

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