## Departmental PhD Thesis Exam – Lluis Vena

Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.

Friday, June 20, 2014
10:10 a.m.
BA6183, 40 St George St.

PhD Candidate
: Lluis Vena
Supervisor
: Balazs Szegedy
Thesis title
:  The removal property for linear configurations in compact abelian groups

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Abstract:

The combinatorial removal lemma states that, if a (hyper)graph $K$ has not many copies of the fixed (hyper)graph $H$, then $K$ can be made  free of copies of $H$ by removing a small set of (hyper)edges from $K$.

In this thesis we show an analogous of the combinatorial removal result for homomorphisms in finite abelian groups and for integer linear systems of compact abelian groups. The results states that,  given some subsets of the group $X_i$, if there are not many solutions to the system $Ax=0$, where the variables $x_i$ takes values in $X_i$, then there exist small subsets $X_i’$ inside $X_i$ such that there is no solution to the system $Ax=0$ with $x_i\in X_i \ X_i’$.

These results are shown by constructing an appropriate (hyper)graph that allows us to retrieve the information on the sets of elements in the groups to be removed from the set of edges.

These algebraic removal lemmas extend the first removal lemma for groups proved by Green in 2005. They also present a comprehensive approach to results involving finding non-trivial linear configurations in dense sets. Examples of these type of results are Szemer\’edi’s theorem that ensures finding arbitrarily long arithmetic progressions on the integers or its multidimensional version, first shown by Furstenberg and Katznelson in 1978 using ergodic theory and later on by Solymosi in 2004 using the hypergraph removal lemma.

A copy of the thesis can be found: http://www.math.toronto.edu/lvena/thesis_draft.pdf