This talk will introduce a method for learning to visualize 4‑dimensional space, give participants a chance to work on some 4D visualization exercises in small groups, and then present a few solutions using interactive 4D graphics software. The exercises range from elementary to advanced, so everyone from first-year undergraduates to experienced geometers should find something they like.
A formal power series h(x) = sum a_n x^n is called algebraic if it satisfies a polynomial equation with coefficients in K[x]. Typical examples are rational series like (1+x)^(-1) or roots of polynomials like sqrt(1+x). In characteristic 0, it is an open problem to characterize the coefficient sequence (a_n) of an algebraic series, whereas in positive characteristic, the theory of automata provides a complete answer (which is, though, not entirely satisfactory).
Polyominoes are edge-connected sets of cells on the square lattice. The study of polyominoes originated in statistical physics and is now a popular field in combinatorial geometry. A major goal in this area is to determine the limit growth rate of polyominoes, also known as ``Klarner's constant'' and usually denoted by lambda. Until recently, the best
known lower and upper bounds on lambda were 3.98 and 4.65, resp.
Over the last 15 years, it has been noted that many combinatorial structures, such as real and complex hyperplane arrangements, interval greedoids, matroids, oriented matroids, and others have the structure of a left regular band. The representation theory of the associated band has had a major influence on understanding these objects along with related structures such as finite Coxeter groups and various Markov processes.
Proof search constitutes an interesting form of computation, for example, at the heart of logic programming languages or theorem provers. In reductive proof search (proof search based on reduction of conclusions of inference rules to their premises), proofs are naturally generalised by solutions, comprising all (possibly infinite) structures generated by locally correct, bottom-up application of inference rules.
I will give a survey of some old and some new results about Hopf algebras, starting by discussing some very classical results about algebraic groups (work of Lazard from 1955) and Lie algebras (classical theorem of Cartier and Kostant), and progressing to current research. I will explain all needed background during the talk.
Stephen Watt, University of Western Ontario, Canada
Friday, 11 July, 2014 - 14:30
Mathematical handwriting provides a number of challenges beyond what is required for the recognition of handwritten natural languages. For example, it is usual to use symbols from a range of different alphabets and there are many similar-looking symbols. Mathematical notation is two-dimensional and size and placement information is important. Additionally, there is no fixed vocabulary of mathematical words'' to disambiguate symbol sequences. On the other hand, there are some simplifications. For example, symbols do tend to be well-segmented.
In a classical approach to dynamical systems, one frequently uses certain geometric structures (Markov partitions) which allow, under a codification of the system, to deduce certain statistical properties of the system, such as the existence of invariant measures with stochastic-like behaviour,...
We prove a general fixed point theorem for inverse transducers, which implies that the fixed point subgroup of an endomorphism of a finitely generated virtually free group is finitely generated. Furthermore, if the endomorphism is uniformly continuous for the hyperbolic metric, we can prove that...