Mini-courses (Week 1 & 2)

Week 1: June, 13-17, 2022

The week 1 will focus on a main extended course (around 15h) on computational group theory, cohomology of groups and topological methods (including introduction the GAP and practical sessions), as well as several lectures on lattices and their computational aspects. There will be an introductory course with practical sessions on PARI/GP and a lecture on Illustrating the Mathematics of COGENT with an associated Art exhibit (e.g., 3D printed model from the geometry associated to arithmetic groups).

Monday 13th June:

Morning : welcome of the participants in room 6, ground floor of the Institut Fourier. Access to the institut Fourier

Afternoon: beginning of courses, in Chabauty lecture room, located on the ground floor of the Institut Fourier (entrance A). See the planning

Friday 17th June : end of courses no later than 13:30 (Paris time)

• Lattices, Perfects lattices, Voronoi reduction theory, modular forms, computations of isometries and automorphisms
  
• Renaud COULANGEON (Université de Bordeaux) and Gabriele NEBE (RWTH  Aachen)
  
• Abstract: The talks of Coulangeon will introduce the notion of perfect, eutactic and extreme lattices  and the Voronoi's algorithm to enumerate perfect lattices (both Eulcidean and Hermitian).  The talk of Nebe will build upon these notions, introduce Boris Venkov's notion of strongly perfect lattices  and show that these are perfect and eutactic and hence extreme. We will show how to construct such lattices using representation theory of finite groups and modular forms. 
  

• Introduction to PARI/GP for Explicit Number Theory and Lattices
  
• Bill ALLOMBERT (CNRS and Université de Bordeaux)
  
• Abstract: This lecture is a hands-on presentation of the PARI/GP computer algebra system focused on its applications to algebraic number theory and Euclidean lattices.
  

• Computational group theory, cohomology of groups and topological methods
  
• Bettina EICK (TU Braunschweig), Graham ELLIS (NUI Galway) and Alexander HULPKE (Colorado State University)
• Abstract: The lecture series will give an introduction to the computer algebra system GAP, focussing on calculations involving cohomology. We will describe the mathematics underlying the algorithms, and how to use them within GAP.  Alexander Hulpke's lectures will being with some general computational group theory (mainly focussing on permutation and matrix groups) and then move on to the cohomology used for the construction of perfect groups. Bettina Eick's lectures will again begin with some general computational group theory  (mainly focussing on polycyclic groups) and then move on to the computation of cohomology for polycyclic groups with a view towards the classification of Almost-Bieberbach-Groups and SmallGroups. Graham Ellis's lectures will begin with some general computational topology (mainly focussing on 3-manifolds) and then move on to computations in group cohomology with a view to calculations on congruence subgroups of SL(2,Z) and Bianchi groups.
  

• Computational Aspects of Euclidean Lattices
• Phong NGUYEN (Inria Paris & ENS PSL)
  
• Abstract: This is an introduction to the mysterious world of lattice algorithms, which have found many applications in computer science, notably in cryptography. We will explain how lattices are represented by computers. We will present the main hard computational problems on lattices: SVP, CVP and BDD, related to short and close vectors in lattices, and their average version known as SIS and LWE. We will present the celebrated LLL algorithm, Babai’s algorithm and discrete Gaussian sampling. If time allows it, we will also take a look at worst-case to average-case reductions.
  

• Illustrating the Mathematics of COGENT: A hyperbolic angle (including Art Exhibit at the Institut Fourier library on 15th and 16th june)
  
• Herbert GANGL (Durham University)
  
• Abstract: One of the COGENT topics concerns symmetric spaces and relations to arithmetic.  We obtain an example of this when we let a discrete group of isometries act on hyperbolic 3-space for which we can study a fundamental domain. It turns out that for the building blocks of such a fundamental domain one can often find hidden symmetries, giving rise  to aesthetically pleasing by-products like highly symmetric polytopes and variants thereof. In this lecture series we will discuss a little bit of the background story and illustrate it by way of a couple of those new kinds of polyhedra, in 3D-printed form. We briefly outline the underlying process of how they were generated and, most importantly, give you a few pointers and examples of how you can rather quickly design a 3D-printed object yourself--on the spot!--using the free software package "OpenSCAD" (similar to Python). We should find time to print some of your designs during the summer school: There is a Fab Lab onsite which we are planning to pay a visit a day later, hopefully allowing us to see it already in the process of realising some of your creations--ideally the latter should reflect some of the beautiful structures that you have encountered in your own studies.
  
• Part of the "practical work session" will use the devices of the fablab FabMSTIC location (thanks to Germain, the Fablab manager, and his team).
  
• Complement: "Jewellery from hyperbolic space", Article for Snapshots of Modern Mathematics, Oberwolfach
  
  

The week 2 will focus on two  main extended courses  on the cohomology of arithmetic groups (from low rank to higher ranks) from both theoretical and computational viewpoints. One course focusing on  geometric models, asymptotics and the computations, the other one on duality and stability. Both courses will emphasize on recent progress and open problems, in particular related to number theory.

Monday 20th June:

Morning : welcome of the participants, in room 6, ground floor of the Institut Fourier.

Afternoon: beginning of courses, in Chabauty lecture room, located on the ground floor of the Institut Fourier (entrance A).

Friday 24th June : end of courses no later than 13:30 (Paris time)

• Cohomology of arithmetic groups and number theory: geometric, asymptotic and computational aspects
• Philippe ELBAZ (U. Grenoble Alpes & Inria Bordeaux Sud-Ouest), Paul GUNNELLS (U. Massachusetts-Amherst), Aurel PAGE (Inria Bordeaux Sud-Ouest) and Haluk SENGUN (University of Sheffield)
  
• Abstract: In this lecture series, the first part will be dedicated to cohomology of arithmetic groups of lower ranks (e.g., Bianchi groups), their  associated geometric models (mainly from hyperbolic geometry) and
  connexion to number theory. The second part will deal with higher rank  groups, mainly using  Voronoi models and similar complexes (both  euclidean and hermitian). We will describe in details the geometric,  cohomological and topological tools, as well as Hecke actions on the  cohomology of modular groups and asymptotic results. Applications to number theory and K-theory  will also be detailled. We will also explain several computational  methods and algorithmic improvements in order to get explicit results.
  The last part of the lectures will be dedicated to open problems.
  

• High dimensional cohomology of SL_n(Z) and its principal congruence subgroups
  
• Peter PATZT (University of Oklahoma) and Jennifer WILSON (University of Michigan)
  
• Abstract: Group cohomology of arithmetic groups is ubiquitous in the study of arithmetic K-theory and algebraic number theory. Rationally, SL_n(Z) and its finite index subgroups don't have cohomology above dimension n choose 2. Using Borel-Serre duality, one has access to the high dimensions. Church, Farb, and Putman conjectured that the high dimensional cohomology of SL_n(Z) with trivial rational coefficients vanishes. In this lecture series, we will give an introduction to these notions, prove the aforementioned conjecture in codimensions 0 and 1. We will also study the top cohomology of principal congruence subgroups. In the final lecture, we summarize some further directions and open problems in the field.