We study a $q$-deformation for the crossed product of the symmetric group $S_n$ with the Clifford algebra on $n$ anticommuting generators. We obtain a $q$-version of the projective analogue for the classical Young symmetrizer introduced by the second author. Our main tool is an analogue of the Hecke algebra of complex-valued functions on the group $\mbox{GL}_n$ over a $p$-adic field relative to the Iwahori subgroup.

We characterise those concealed-canonical algebras which arise as endomorphism rings of tilting modules, all of whose indecomposable summands have strictly positive rank, as those artin algebras whose module categories have a separating exact subcategory (that is, a separating tubular family of standard tubes).

This paper develops further the technique of shift automorphisms which arises from the tubular structure.

It is related to the characterisation of hereditary noetherian categories with a tilting object as the categories of coherent sheaves on a weighted projective line.

A systematic study is made of the isomorphic properties of the Banach space ${\cal C}_0(\Upsilon)$ of continuous functions, vanishing at infinity, on a tree $\Upsilon$, equipped with its natural locally compact topology. Necessary and sufficient conditions, expressed in terms of the combinatorial structure of $\Upsilon$, are obtained for ${\cal C}_0(\Upsilon)$ to possess equivalent norms with various good properties of smoothness and strict convexity. These characterizations, together with the construction of appropriate trees, lead to counter-examples refuting a number of conjectures about renormings. It is shown that the existence of a Fr\'echet-smooth renorming is not inherited by quotients, that strict convexifiability is not a three-space property and that neither the Kadec property nor the MLUR property implies the existence of an equivalent norm which is locally uniformly rotund. An example is also given of a space with a smooth norm but no equivalent strictly convex norm. Finally, it is shown that ${\cal C}_0(\Upsilon)$ always admits a ${\cal C}^\infty$ `bump-function', even in cases where no good norms exist.

The main result of this paper is the resolvent similarity criterion which says that linear growth of the resolvent towards the spectrum is sufficient for a Hilbert space contraction with finite rank defect operators and spectrum not covering the unit disc to be similar to a normal operator. Similar results are proved for operators having a spectral set bounded by a Dini-smooth Jordan curve; in particular, a dissipative operator with finite rank imaginary part is similar to a normal operator if and only if its resolvent grows linearly towards the spectrum. Relevant results on the insufficiency of linear resolvent growth not accompanied by smallness of defect operators are presented. Also it is proved that there is no restriction on the spectrum, other than finiteness, which together with linear resolvent growth implies similarity to a normal operator. The construction of corresponding examples depends on a characterization of well-known Ahlfors curves as curves of linear length growth with respect to linear fractional transformations.

Let $k\ge 3$ be an integer. For $0<s<1$, let $D_s \subset {\Bbb R}^2$ be the set that is constructed iteratively as follows. Take a regular open $k$-gon with sides of unit length, attach regular open $k$-gons with sides of length $s$ to the middles of the edges, and so on. At each stage of the iteration the $k$-gons that are added are a factor $s$ smaller than the previous generation and are attached to the outer edges of the family grown so far. The set $D_s$ is defined to be the interior of the closure of the union of all the $k$-gons. It is easy to see that there must exist some $s_k>0$ such that no $k$-gons overlap if and only if $0<s\le s_k$. We derive an explicit formula for $s_k$.

The set $D_s$ is open, bounded, connected and has a fractal polygonal boundary. Let $E_{D_s}(t)$ denote the heat content of $D_s$ at time $t$ when $D_s$ initially has temperature $0$ and $\partial D_s$ is kept at temperature $1$. We derive the complete short-time expansion of $E_{D_s}(t)$ up to terms that are exponentially small in $1/t$. It turns out that there are three regimes, corresponding to $0<s<1/(k-1)$, $s=1/(k-1)$, and $1/(k-1)<s\le s_k$. For $s \neq 1/(k-1)$ the expansion has the form

$$ E_{D_s}(t)=p_s(\log t)t^{1-d_s/2}+A_st^{1/2}+Bt+O(e^{-r_s/t}), $$

where $p_s$ is a $\log (1/s^2)$-periodic function, $d_s=\log (k-1)/\log (1/s)$ is a similarity dimension, $A_s$ and $B$ are constants related to the edges and vertices, respectively, of $D_s$, and $r_s$ is an error exponent. For $s=1/(k-1)$, the $t^{1/2}$-term carries an additional $\log t$.

Suppose that $M$ is a hyperbolic 3-manifold which admits two Dehn fillings $M(r_1)$ and $M(r_2)$ such that $M(r_1)$ contains an essential annulus, and $M(r_2)$ contains an essential torus. It is known that $\Delta = \Delta (r_1, r_2) \leq 5.$ We will show that if $\Delta = 5$ then $M$ is the Whitehead sister link exterior, and if $\Delta = 4$ then $M$ is the exterior of either the Whitehead link or the 2-bridge link associated to the rational number $\frac{3}{10}$. There are infinitely many examples with $\Delta = 3$.

In this paper a completely integrable Hamiltonian system on $T^*M$ is constructed for every Riemannian symmetric space $M$. We show that the solutions of this system correspond to the solutions of Nahm's equations for suitably chosen maps. Nahm's equations were introduced by Nahm as a rewriting of Bogomolny equations for magnetic monopoles.

We represent our system as a degeneration of a certain case of Hitchin's algebraically integrable system. We prove the complete integrability of our system by means of this representation.

Some concrete examples of our Hamiltonian system on $T^*M$ are described. When $M= S^n$, we obtain the classical system of C. Neumann. If the configuration space $M$ of our system is the $n$-dimensional hyperbolic space, we get the Minkowskian analogue of the C. Neumann system. Other examples that we describe are a many-body C. Neumann system, a spherical pendulum, and a spherical pendulum with an additional magnetic force.