Turan's theorem is probably one of the most well-known theorem in graph theory. Let k be a small integer, say 5, and n be a large integer. Turan showed that one must delete at least a $1/(k-1)$-fraction of the edges of the complete graph $K_n$ in order to destroy all cliques of size k. For instance, one must delete at least half of the edges to destroy all triangles. It is natural to study the problem for graphs other than $K_n$. A graph G on n vertices is called k-Turan if one needs to delete at least a $1/(k-1)$- fraction of the edges of G in order to destroy all k-cliques in G. Which graphs are k-Turan ? For instance, is the Paley graph 5-Turan ? Despite the long and rich history of Turan's theory, which spans several decades, not much has been known about this question. Recently, Sudakov, Szabo, and Vu, based on earlier results of the last two researchers, found a general sufficient condition for a regular graph to be k-Turan. We discovered a surprising connection between the k-Turan property and the spectral of the graph: if the second eigenvalue of G is sufficiently small compared to the degree, then G is k-Turan. In particular, good expanders have the Turan properties. The proof is completely elementary.

I will start with an introduction to hyperbolic space and hyperbolictetrahedra. I will then show how to construct an arbitrary 3/4-idealhyperbolic tetrahedron out of 10 ideal tetrahedra. There will be lots ofpictures and not many equations in this talk!

Non-commutative conditional expectations and martingales arise inthe setting of von Neumann algebras, which are the naturalframework for non-commutative measure theory and integration.Analogues of classical martingale inequalities such asBurkholder-Gundy's square function inequalities and Doob'sinequality have recently been established for martingales innon-commutative $L_p$-spaces by Junge, Pisier and Xu. They alsoproved the analogue of the classical duality between $H^1$ and$BMO$ of martingales. We will discuss interpolation properties ofnon-commutative $BMO$ and show that it is a natural substitute for$L_infty,.$ As an application we establish boundedness ofnon-commutative martingale transforms.

If a group G acts on a tree X we may determine the precise structure of G byapplying machinery of Bass and Serre for reconstructing the group action. We use this to give structure theorems for certain subgroups of $SL_2 $ over a on-archimedean local field, and of groups associated to Kac-Moody Lie algebras over finite fields. Of special interest are lattice subgroups and their congruence subgroups.

We explore the geometric, algebraic, and computational implications of thepresence of continuous and discrete symmetries in semidefinite programs(SDPs). It is shown that symmetry exploitation allows a significantreduction in both matrix size and number of decision variables. To this end,we define a class of SDPs, that are invariant under the action of a finiteor continuous symmetry group. Using group averaging and linearrepresentation theory, it is shown that the feasible set can be restrictedto a specific invariant subspace, thus reducing the problem to a collectionof coupled semidefinite programs of smaller dimensions.We focus particularly on SDPs arising in the sum ofsquares/Positivstellensatz framework, where the group representation isinduced by an action on the space of monomials. It is shown that thecomplexity is significantly reduced, and the techniques are illustrated withnumerous examples. The results, reinterpreted from an invariant-theoreticviewpoint, provide a novel representation of nonnegative symmetricpolynomials. This alternative approach has as attractive features itscomputational efficiency and the natural connections with therepresentation-based approach developed earlier.Finally, the computational savings of the techniques are demonstrated insome large-scale problems. It is shown how the symmetry reduction techniquesenable the numerical solution of complicated instances, otherwisecomputationally infeasible to solve.The material in the talk is based on joint work with Karin Gatermann (ZIBBerlin).

We explore the group of 2x2 determinant 1 matrices with coefficients inthe ring of integers Z as well as an analog obtained by replacing Z by thering of polynomials F[t] where F is a finite field. Decomposition theoremsfor these groups will be obtained using the actions of the groups ontrees, which are connected graphs having no cycles, such as that of degree3 a part of which is pictured above.REFRESHMENTS WILL BE SERVED!