Diophantine Problems and Homogeneous Dynamics

We will discuss some classical questions that have their origins in the work of Gauss from 1863 [16, pp. 269–291], along with some more recent developments due primarily to Duke, Rudnick and Sarnak [12], Eskin and McMullen [15]. Rather than aiming for maximal generality, we will try to expose the striking connection between certain equidistribution problems in dynamics and some asymptotic counting problems. In the second part we will discuss the connection between the theory of Diophantine approximation and homogeneous dynamics. In these notes we will make use of several concepts from ergodic theory and dynamical systems. For most of these a suitable source is [13], and for some of the material on the space of lattices particularly a suitable source is [14]

Fitting ideals for finitely presented algebraic dynamical systems

We consider a class of algebraic dynamical systems introduced by Kitchens and Schmidt. Under a weak finiteness condition - the Descending Chain Condition - the dual modules have finite resentations. Using methods from commutative algebra we show how the dynamical properties of the system may be deduced from the Fitting ideals of a finite free resolution of the finitely presented module. The entropy and expansiveness are shown to depend only on the initial Fitting ideal (and certain multiplicity data) which gives an easy computation: in particular, no syzygy modules need to be computed. For "square" presentations (in which the number of generators is equal to the number of relations) all the dynamics is visible in the initial Fitting ideal and certain multiplicity data, and we show how the dynamical properties and periodic point behaviour may be deduced from the determinant of the matrix of relations

Isomorphism rigidity in entropy rank two

We study the rigidity properties of a class of algebraic Z^3-actions with entropy rank two. For this class, conditions are found which force an invariant measure to be the Haar measure on an affine subset. This is applied to show isomorphism rigidity for such actions, and to provide examples of non-isomorphic Z^3-actions with all their Z^2-sub-actions isomorphic. The proofs use lexicographic half-space entropies and total ergodicity along critical directions

Asymptotic geometry of non-mixing sequences

The exact order of mixing for zero-dimensional algebraic dynamical systems is not entirely understood. Here we use valuations in function fields to exhibit an asymptotic shape in non-mixing sequences for algebraic Z^2-actions. This gives a relationship between the order of mixing and the convex hull of the defining polynomial. Using this result, we show that an algebraic dynamical system for which any shape of cardinality three is mixing is mixing of order three, and for any k greater than or equal to 1 exhibit examples that are k-fold mixing but not (k+1)-fold mixing

Entropy geometry and disjointness for zero-dimensional algebraic actions

We show that many algebraic actions of higher-rank abelian groups on zero-dimensional compact abelian groups are mutually disjoint. The proofs exploit differences in the entropy geometry arising from subdynamics and a form of Abramov-Rokhlin formula for half-space entropies

A Journey Through the Realm of Numbers: From Quadratic Equations to Quadratic Reciprocity

This book takes the reader on a journey from familiar high school mathematics to undergraduate algebra and number theory. The journey starts with the basic idea that new number systems arise from solving different equations, leading to (abstract) algebra. Along this journey, the reader will be exposed to important ideas of mathematics, and will learn a little about how mathematics is really done. Starting at an elementary level, the book gradually eases the reader into the complexities of higher mathematics; in particular, the formal structure of mathematical writing (definitions, theorems and proofs) is introduced in simple terms. The book covers a range of topics, from the very foundations (numbers, set theory) to basic abstract algebra (groups, rings, fields), driven throughout by the need to understand concrete equations and problems, such as determining which number are sums of squares. Some topics usually reserved for a more advanced audience, such as Eisenstein integers or quadratic reciprocity, are lucidly presented in an accessible way. The book also introduces the reader to open source software for computations, to enhance understanding of the material and nurture basic programming skills. For the more adventurous, a number of Outlooks included in the text offer a glimpse of possible mathematical excursions. This book supports readers in transition from high school to university mathematics, and will also benefit university students keen to explore the beginnings of algebraic number theory. It can be read either on its own or as a supporting text for first courses in algebra or number theory, and can also be used for a topics course on Diophantine equations