3rd EAALG Workshop

We are pleased to announce the second EAALG workshop on Some topics in Algebra and Geometry which is scheduled for November 25 - 29, 2024 at Makerere University, Uganda. Just like in the previous EAALG workshops, the target audience comprises of academic staff and graduate students from the Eastern Africa region and beyond who are interested in Algebra and Geometry.

Speakers

Dr. Giulia Gugiatti (University of Edinburgh, United Kingdom)
Dr. Daniel Kaplan (UMass Boston, United States)
Dr. Franco Rota (University of Glasgow, United Kingdom)

Topic 1: Deformation of Algebras (By Dr. Daniel Kaplan)

Lecture 1: Define algebras by generators and relations where the relation has a variable t. Investigate the behavior after setting t=0 and t=1. Do the examples: (a) C[x]/(x^2-t), (b) a polynomial algebra deforming to a universal enveloping algebra C<x, y>/( xy -yx - tx) and to a Weyl algebra (of differential operators) C<x, y>/( xy-yx -1), (c) possibly the exterior algebra deforming to a Clifford algebra. Highlight what properties stay the same and which ones change as t goes from 0 to 1.

Lecture 2: Compute that a perturbation of an associative multiplication mu_0 to mu = mu_0 + t mu_1 is still associative if mu_1 is a Hochschild 2-cocycle. In the commutative case, this is a Poisson bracket. Show that two such multiplications are equivalent if they differ by a derivation (a Hochschild 1-cocycle). (Possibly discuss the issue of extending from 1st order to second order where the obstruction class is a Hochschild 3-cocycle.) Revisit the examples in Lecture 1, writing down each deformation again, now identifying the Poisson bracket that gives the deformation.

Lecture 3: Again revisit the examples in Lecture 1, this time view each algebra as filtered (the relations are not homogenous so these algebras are no longer graded like the polynomial algebra). The associated graded algebra is commutative so in that sense these algebras are "almost commutative." Explain how this kind of deformation (filtered) is related to the previous kind (infinitesimal or formal).

Topic 2: McKay Correspondence (By Dr. Franco Rota)

Lecture 1: the classification of finite subgroups of SL2. How to construct their McKay quiver (using rep theory) and how to construct the associated singularity and its resolution. The example of D4. Statement of classical McKay correspondence.

Lecture 2: Categories associated with the two sides of McKay: quiver representations and coherent sheaves.

Construction of the derived category of an abelian category. Axioms of triangulated categories and derived functors.

Lecture 3: The derived McKay principle. Moduli spaces of clusters, with at least one explicit example. Fourier-Mukai functors, and the statement of Bridgeland-King-Reid.

Lecture 4: Open ended: discussion of more examples, ideas from the proof, and generalizations (Craw-Ishii conjecture, Gl2 correspondence)

Topic 3: Introduction to Mirror Symmetry (By Dr. Giulia Gugiatti)

Lecture 1: Define lattice polytopes, and discuss the existence of a duality between certain lattice polytopes. Show a few examples.

Lecture 2: Sketch how the duality reflects a duality in geometry known as Mirror Symmetry. Discuss a few examples (in dimension 2, perhaps higher).

Lecture 3: Outline the categorical version of Mirror Symmetry. Show how this works for the 2-dimensional projective space and its mirror (discussed in 2).

This would be based on material from Cox-Katz's book “Mirror Symmetry and Algebraic Geometry”, and the papers: arXiv:1212.1722, arXiv:math/0506166. Lectures 2 and 3 make contact with Topics 1 and 2.

Scientific Committee