Pure Mathematics Colloquium

Tropical geometry is geometry over the tropical semiring, where multiplication is replaced by addition and addition is replaced by minimum. This "tropicalization" procedure turns algebraic varieties (solutions to polynomial equations) into polyhedral complexes, which are combinatorial objects. A surprisingly large amount of information about the variety is still present in tropical "combinatorial shadow". In this talk I will introduce tropical varieties, and indicate some of their applications, both inside and outside algebraic geometry.

We will introduce quiver varieties, which are geometric objects describing various matrix problems. We will discuss various results relating geometry and representation theory, in particular looking at some of the arithmetic of these varieties.

I will consider groups of M\"obius transformations generated by two elements and consider the geometry of the quotient space. This involves hyperbolic geometry in 3 dimensions and classical geometry in representing the generators as compositions of involutions.

‘This is a Mathematical War’ declared Sir George Greenhill, erstwhile Professor of Mathematics at the Royal Military Academy, in January 1915. Two years later his words were echoed at the front by a young British soldier who found himself fighting in a ‘war of guns and mathematics’. But were these accurate descriptions or isolated observations? In my talk, I shall consider the contributions of British mathematicians to the war effort as well as discussing the effect of the war on the mathematical community.

Young Brouwer was equally attracted to mathematics and philosophy. His definite choice for mathematics was made writing his PhD thesis, even though the thesis was strongly colored by philosophy. As an intermezzo Brouwer published a series of lectures "Life, Art, and Mysticism", which foreshadowed some of his foundational views. After the proclamation of a constructive program he turned to the second topic of his thesis - topology. Within two years he was the leading topologist, with strong ties to Göttingen. The isolation of the First World War caused Brouwer to return to the foundations. He laid down the principles of his mature intuitionism, which was developed to a viable mathematical discipline. From 1918 onward his publications kept the mathematical world in suspense. At the same time he started, parallel to Hardy's efforts, to campaign for a return to a fully international mathematical community. This, sadly, gave him a reputation of being pro-German. His life offered a fair number of actions of a social/political nature, much to the astonishment and irritation of the public. We will go into Brouwer's intuitionistic program and some of his extra-academic activities.