School of Mathematics

We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result for inclusion of vertex (or edge) subgroups in finite graphs of (strongly) relatively hyperbolic groups. This generalizes a result of Bowditch for punctured surfaces in 3 manifolds and a result of Mitra for trees of hyperbolic metric spaces.

We define metric bundles which provide a purely topological/coarsegeometric generalization of the notion of Trees of metric spaces a la Bestvina-Feighn in the case that the inclusions of the edge spaces into the vertex spaces are uniform (coarsely surjective) quasi-isometries. We prove the existence of q(uasi)-i(sometric) sections in this generality. Then we prove a combination theorem for metric bundles (including exact sequences of groups) that establishes sufficient conditions (particularly flaring) under which the metric bundles are hyperbolic. We also show that in typical situations, flaring is also a necessary condition.

We prove a combination theorem for trees of (strongly) relatively hyperbolic spaces and finite graphs of (strongly) relatively hyperbolic groups. This gives a geometric extension of Bestvina and Feighn's Combination Theorem for hyperbolic groups and answers a question of Swarup. We also prove a converse to the main Combination Theorem.

Cost functions in problems concerning the existence of Nash Equilibria are traditionally multilinear in the mixed strategies. The main aim of this paper is to relax the hypothesis of multilinearity. We use basic intersection theory, Poincar\'e Duality and the Dold-Thom Theorem to establish existence of Nash Equilibria under fairly general hypotheses. The Dold-Thom Theorem provides us with a homological version of a selection Theorem, which may be of independent interest.

For $i= 1,2$, let $G_i$ be cocompact groups of isometries of hyperbolic space $\Hyp^n$ of real dimension $n$, $n \geq 3$. Let $H_i \subset G_i$ be infinite index quasiconvex subgroups satisfying one of the following conditions: 1) limit set of $H_i$ is a codimension one topological sphere. 2) limit set of $H_i$ is an even dimensional topological sphere. 3) $H_i$

We prove a combination theorem for trees of (strongly) relatively hyperbolic spaces and finite graphs of (strongly) relatively hyperbolic groups. This gives a geometric extension of Bestvina and Feighn's Combination Theorem for hyperbolic groups and answers a question of Swarup. We also prove a converse to the main Combination Theorem.

Let I S I = n,
m(n ; kl ,k 2 ,k) respectively m'(n,k 1 ,k, k)
denote the cardinali.ty of the largest family of subsets A i C S
satisfying IAi I = k (respectively 1A í 1 S k) and 1Ai n Ai 1 = kl or
Z
2~
912* In this paper we prove
a) m(n,0,k2 ,k)
(n), mI(n,0,k 2 ,k) 5 (2) + ntl ; equality, iff k = 2 ;
b) m(n,0,k2' k) s n, if k 2 X k, with equality for an infinity of n .
For n a no (k) we show that :
n-k
n-k
a) m(nl ,kVR 2' k)
1 , m' (n,k l ,k 2 ,k) s
1 + (n-2 1 ) i- 1 ;
2
2
b) more exactly, m(n,k,l,R,2,k) s
infinity of n .
n-k l
ti-k2
k-k l
k-k2 with equality for an

It is shown that a pairwise balanced design on n points in which each block is of size at least n a/z c can be embedded in a projective plane of order n-t-i for some i~ c + 2 if n is sufficiently large. A m o n g other things this implies that if the projective plane conjecture is true, the conjecture of Erd6s and Larson will not be true.

A new method to study families of finite sets, in particular t-designs, by studying families of multisets (also called lists) and their relationships with families of sets, is developed. Notion of the tag for a subset defined earlier by one of the authors is extended to a submultiset. A new concept t-(v, k, λ) list design is defined and studied. Basic existence theory for designs is extended to a new set up of list designs. In particular tags are used to prove that signed t-list designs exist whenever necessary conditions are satisfied. The concepts of homomorphisms and block spreading are extended to this new set up. (

By the extremal number ex(v; {C 3 , C 4 , . . . , C n }) we denote the maximum number of edges in a graph of order v and girth at least g ≥ n + 1. The set of such graphs is denoted by EX(v; {C 3 , C 4 , . . . , C n }). In 1975, Erdős mentioned the problem of determining extremal numbers ex(v; {C 3 , C 4 }) in a graph of order v and girth at least five. In this paper, we consider a generalized version of the problem for any value of girth by using the hybrid simulated annealing and genetic algorithm (HSAGA). Using this algorithm, some new results for n ≥ 5 have been obtained. In particular, we generate some graphs of girth 6, 7 and 8 which in some cases have more edges than corresponding cages. Furthermore, future work will be described regarding the investigation of structural properties of such extremal graphs and the implementation of HSAGA using parallel computing. Crown