2024 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 



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: Introduction to Derived Categories (By Dr. Franco Rota)


Lecture 1:  Main definitions and constructions of the derived category of an abelian category (in the spirit of my notes derived_category.pdf - Google Drive, and following Huybrechts' book as a reference).

Lecture 2: Develop the main examples: the categories of coherent sheaves on a variety, and of quiver representations.

Lecture 3: Examples and illustrations of the derived McKay correspondence for surfaces, with focus on perverse sheaves and t-structures.

For the last two lectures I would use Alistair Craw's notes tilting.dvi (utah.edu). 


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 



Organizing Committee

1.  David Ssevviiri  - Makerere University, Uganda (Chairperson,  LOC)

2. Tilahun Abebaw -  Addis Ababa University, Ethiopia

3. Alex S. Bamunoba  - Makerere University, Uganda

4. Alex B. Tumwesigye - Makerere University, Uganda

5. Adson Banda – University of Zambia, Zambia

6. Iara Goncalves – Eduardo Mondlane, Mozambique

7. Celestin Kurujyibwami – University of Rwanda, Rwanda

8. Jared Ongaro – University of Nairobi, Kenya

9. Layla Sorkatti – Khartoum University, Sudan





For previous workshops, see

2023:  https://sites.google.com/view/eaalg/2023-eaalg-workshop  

 2021: https://sites.google.com/view/eaalg/2021-eaalg-workshop