Crecimiento en grupos y otras estructuras

Esta página contiene material para las escuelas de verano a la Universidad de Costa Rica (25 – 29 julio 2016) y a la Universidad National Autónoma de Nicaragua — Leon (1 – 5 agosto 2016) sobre crecimiento en grupos y otras estructuras.

Los niveles de los dos cursos seran un poco differentes, pero mucho de la material sera similar.

Lecturas

Las notas son aquí. Los subjetos son como sigue:

  • introducción al crecimiento;
  • acciones de grupos;
  • grafos de Cayley;
  • una conjetura de Babai.

Referencias sobre grupos

Esta material es más clasica, entonces hay muchas referencias posibles. Si no ha estado la teoría de grupos antes, recomiendo el libro de Fraleigh.

 

Referencias sobre crecimiento

La mayoría de estas referencias estan un poco avanzadas. Yo he incluido dos referencias generales (por Tao y por Tao–Vu) que contienen mucho material fondamental — malafortunadamente, el libro de Tao y Vu no es disponible en una forma gratuita en la web.

Mi primera recomendación es las lecturas de Helfgott, “Crecimiento y espansión en SL2″. Primeramente, son en español(!) pero también comenzan a un nivel bastante fácil y, rapidamente, presentan un resultado muy importante de Helfgott sí mismo, sobre crecimiento en el grupo SL(2,p).

Successful grant: on the Product Decomposition Conjecture

I have just found that my application for EPSRC-funding has been successful. I have been awarded funding via their “first grant” scheme. The grant will buy me out of 50% of my teaching and admin duties, thereby allowing me to focus on the problem of proving “The Product Decomposition Conjecture”.

The case for support document from my grant application gives details of this conjecture, its importance, and the strategies that I hope to employ to work on it.

Excitingly, the university has agreed to fund a PhD student as part of this research. I just drafted a short description of what the PhD would be about, and I’ll post this below. (Note that this description might be edited a little over the next few days. In any case, it should give an idea of what the project will be about.) If you are interested, please get in touch!

This programme of research is within the study of finite group theory (although some investigation of linear algebraic groups may also be involved). The aim is to prove, or partially prove, the Product Decomposition Conjecture which concerns “conjugate-growth” of subsets of a finite simple group: roughly speaking, given a finite nonabelian simple group G and a subset A in G of size at least 2, we would like to show that one can always write G as a product of “not many” conjugates of A.

This notion of conjugate-growth has interesting connections to many interesting areas of mathematics, including expander graphs, the product growth results of Helfgott et al, bases of permutation groups, word problems and more.

In the process of working on this conjecture, the student can expect to learn a great deal about the structure of finite simple groups (especially the simple classical groups) and, in particular, will study and make use of one of the most famous theorems in mathematics, the Classification of Finite Simple Groups.

Celebrating New Appointments

This two-day event will take place on 23rd and 30th June 2015 at the University of South Wales. The event is being held to celebrate the appointment of four new lecturers in the maths dept of USW. It will consist of four sessions of general talks, each hosted by a new lecturer. All are welcome!

There is no registration fee, but please register your attendance or obtain any further details by contacting Nick Gill. All events are held in rooms G310 and G311. Morning tea, lunch and afternoon tea are included and complementary. There are limited funds available for dinner — please let us know if you would like to join us.

New: A list of titles and abstracts for all talks is now available.

Tuesday 23rd June 2015

09:30 coffee
10:00 Session 1: Combinatorics and cryptography

  1. Siân K. Jones (USW)
  2. Derek Smith (Emeritus Professor)
  3. Hannah Davies (USW)
12:00 lunch
13:30 Session 2: Numerically modelling the atmosphere

  1. James Kent (USW)
  2. John Thuburn (University of Exeter)
  3. Ayoe Buus Hansen (Met Office)
15:30 coffee
18:00 dinner

Tuesday 30th June 2015

09:30 coffee
10:00 Session 3: Operational Research

  1. Penny Holborn (USW)
  2. Rhyd Lewis (Cardiff University)
  3. Izabela Komenda (Aneurin Bevan University Health Board)
12:00 lunch
13:30 Session 4: Group Theory

  1. Nick Gill (USW)
  2. Ian Short (Open University)
  3. John Britnell (Imperial)
15:30 coffee
18:00 dinner

The meeting is supported by an LMS Conference grant celebrating new appointments and the University of South Wales.

London Mathematical SocietyUniversity of South Wales