It is proved that if $F$ is an infinite field with characteristic different from $2$, whose theory is supersimple, and $C$ is an elliptic or hyperelliptic curve over $F$ with generic ‘modulus’, then $C$ has a generic $F$-rational point. The notion of generity here is in the sense of the supersimple field $F$.

The paper contains a final identification theorem for the ‘generic’ $K^*$-groups of finite Morley rank.

Let $X$ be a subset in $[-1,1]^{n_0}\,{\subset}\,\Real^{n_0}$ defined by the formula \[ X=\{ {\bf x}_0\,{\mid}\,Q_1{\bf x}_1 Q_2{\bf x}_2 \cdots Q_{\nu}{\bf x}_{\nu} (({\bf x}_0,{\bf x}_1,\,{\ldots}\,,{\bf x}_{\nu}) \in X_{\nu})\}, \] where $Q_i \in \{ \exists, \forall \}$, $Q_i \neq Q_{i{+}1}$, ${\bf x}_i \in \Real^{n_i}$, and $X_{\nu}$ may be either an open or a closed set in $[-1,1]^{n_0+ \cdots +n_{\nu}}{\!}$, being the difference between a finite CW-complex and its subcomplex. An upper bound on each Betti number of $X$ is expressed via a sum of Betti numbers of some sets defined by quantifier-free formulae involving $X_{\nu}$.

In important particular cases of semialgebraic and semi-Pfaffian sets defined by quantifier-free formulae with polynomials and Pfaffian functions respectively, upper bounds on Betti numbers of $X_{\nu}$ are well known. The results allow to extend the bounds to sets defined with quantifiers, in particular to sub-Pfaffian sets.

The study of the pentagon equation leads to results on the structure and classification of finite quantum groups. It is proved that $L$ is a finite-dimensional Hopf algebra if and only if there exists an invertible matrix $R$, solution of the pentagon equation $R^{12}R^{13}R^{23}=R^{23}R^{12}$, such that $L\cong P(n, R)$; the Hopf algebra structure of $P(n, R)$ is explicitly described using generators and relations. Finally, it is proved that there exists a one-to-one correspondence between the set of types of $n$-dimensional Hopf algebras and the set of orbits of the action ${\rm GL}_n(k)\times (M_n(k)\otimes M_n(k)) \,{\longrightarrow}\, M_n(k)\otimes M_n(k), (u, R)\,{\longrightarrow}\, (u\otimes u)R(u\otimes u)^{-1}$, the representatives of which are invertible solutions of length $n$ for the pentagon equation.

Let $H$ be a finite dimensional non-semisimple Hopf algebra over an algebraically closed field $k$ of characteristic 0. If $H$ has no nontrivial skew-primitive elements, some bounds are found for the dimension of $H_1$, the second term in the coradical filtration of $H$. Using these results, it is shown that every Hopf algebra of dimension 14 is semisimple and thus isomorphic to a group algebra or the dual of a group algebra. Also a Hopf algebra of dimension $pq$ where $p$ and $q$ are odd primes with $p < q \leq 1 + 3p$ and $q \leq 13$ is semisimple and thus a group algebra or the dual of a group algebra. Some partial results in the classification problem for dimension 16 are also given.

A theorem of Kac on quiver representations states that the dimension vectors of indecomposable representations are precisely the positive roots of the associated symmetric Kac–Moody Lie algebra. Here this result is generalised to representations respecting an admissible quiver automorphism, and the positive roots of an associated symmetrisable Kac–Moody Lie algebra are obtained.

Also the relationship with species of valued quivers over finite fields is discussed. It is known that the number of isomorphism classes of indecomposable representations of a given dimension vector for a species is a polynomial in the size of the base field. It is shown that these polynomials are non-zero if and only if the dimension vector is a positive root of the corresponding symmetrisable Kac–Moody Lie algebra.

Let U be a domain, convex in $x$ and symmetric about the $y$-axis, which is contained in a centered and oriented rectangle $S$. It is proved that $H_t{(U^+)}/H_t{(U)}\,{\leq}\, H_t{(S^+)}/H_t{(S)}$ where $H_t$ stands for heat content, that is, the remaining heat in the domain at time $t$ if it initially has uniform temperature 1, with Dirichlet boundary conditions, where $A^+\,{=}\,A\,{\cap}\, \{(x,y)\,{:}\,x\,{>}\,0\}$. It is also shown that the analog of this inequality holds for some other Schrödinger operators.

Fundamental solutions to second-order parabolic and elliptic equations with lower-order terms are studied. New sufficient conditions on the coefficients of the lower-order terms are given that guarantee the validity of two-sided Gaussian bounds on the fundamental solution to the parabolic equation, locally and globally in time. As an application, the problem of existence and non-existence of positive solutions to a class of semi-linear equations is studied.

A $5$-local characterization of the Lyons sporadic simple group is presented.

A computer construction of the Monster is used to prove that PSL(2, 59) is a subgroup of the Monster.

The classification theorem of compact surfaces states that every topological orientable compact surface is homeomorphic to a sphere or to a ‘torus’ of genus $g$, with $g\,=\,1,2,\dots$. It is proved in the paper that every hyperbolic Riemann surface except for $\bold D\setminus\{0\}$ can be decomposed into basic pieces of only a few different types: Y-pieces, funnels and half-disks. As a corollary of this result, the generalization of the classification theorem to non-compact surfaces is obtained.

A family of planar discrete electrostatic systems on the unit circle with finitely atomic external fields is considered. The geometry of particles in the external field yielding a given minimum energy configuration is studied. As an application, the wandering vectors of the shift operator in the Dirichlet spaces associated with finitely atomic measures are also studied. In particular, the zero locus of a wandering vector is discussed.

The $(C,F)$-construction from a previous paper of the first author is applied to produce a number of funny rank one infinite measure preserving actions of discrete countable Abelian groups $G$ with ‘unusual’ multiple recurrence properties. In particular, the following are constructed for each $p\in\Bbb N\cup\{\infty\}$:

1. (i) a $p$-recurrent action $T=(T_g)_{g\in G}$ such that (if $p\ne\infty$) no one transformation $T_g$ is $(p+1)$-recurrent for every element $g$ of infinite order;

2. (ii) an action $T=(T_g)_{g\in G}$ such that for every finite sequence $g_1,\dots,g_r\in G$ without torsion the transformation $T_{g_1}\times\cdots\times T_{g_r}$ is ergodic, $p$-recurrent but (if $p\ne\infty$) not $(p+1)$-recurrent;

3. (iii) a $p$-polynomially recurrent $(C,F)$-transformation which (if $p\ne\infty$) is not $(p+1)$-recurrent.

As a generalization of the dynamics of a single polynomial, Hinkkanen and Martin studied the dynamics of polynomial semigroups of finite type. Among the issues addressed were possible generalizations of the filled-in Julia set of a single polynomial and the basin of attraction for infinity. In the paper, these concepts are further refined, and the connections between the dynamics of a single function and the dynamics of semigroups are strengthened.

A correspondence between the equivariant degree introduced by Ize, Massabó, and Vignoli and an unstable version of the equivariant fixed point index defined by Prieto and Ulrich is shown. With the help of conormal maps and properties of the unstable index, a sum decomposition formula is proved for the index and consequently also for the degree. As an application, equivariant homotopy groups are decomposed as direct sums of smaller groups of fixed orbit types, and a geometric interpretation of each summand is given in terms of conormal maps.

Given a smooth family ${\hbox{\ac L}}$ of real or complex variable taking values within the class of Fredholm operators of index zero in a Banach space, there are some available definitions in the literature of the concept of algebraic multiplicity of the family ${\hbox{\ac L}}$ at a point $x_0$ of the parameter at which the operator ${\hbox{\ac L}}(x_0)$ becomes non-invertible. The purpose of the paper is to show that the algebraic multiplicity is uniquely determined by a few of its properties, independently of its construction. The main technical tools to obtain this uniqueness result are a Lyapunov–Schmidt reduction, the local Smith form and a new factorization result for general families at non-algebraic eigenvalues obtained in the paper.

The paper presents a construction of fibered links $(K, \Sigma)$ out of chord diagrams $\sL$. Let $\Gamma$ be the incidence graph of $\sL$. Under certain conditions on $\sL$ the symmetrized Seifert matrix of $(K, \Sigma)$ equals the bilinear form of the simply-laced Coxeter system $(W, S)$ associated to $\Gamma$; and the monodromy of $(K, \Sigma)$ equals minus the Coxeter element of $(W, S)$. Lehmer's problem is solved for the monodromy of these Coxeter links.

Two functions on finitely additive probability spaces that behave well under products are introduced: discrepancy, which measures how close one space comes to extending another, and bi-discrepancy, which is a pseudo-metric on the collection of all spaces on a given set, and a metric on the collection of complete spaces. These are then applied to show that the Loeb space of the internal product of two internal finitely additive probability spaces depends only on the Loeb spaces of the two original internal spaces. Thus the notion of a Loeb product of two Loeb spaces is well defined. The Loeb operation induces an isometry from the nonstandard hull of the space of internal probability spaces on a given set to the space of Loeb spaces on that set, with the metric of bi-discrepancy.