We will give a snap shot of the second edition of my book, "Mathematics for the Environment: Teaching Math As If Survival Matters." How is the Pandemic a warmup for other crises like global warming, collapse of industrial agriculture, universal pollution, the sixth extinction and more. A mathematical perspective implemented by elementary mathematics is useful in redesigning our civilization so that it (and we) shall survive.
Doing Math As If Survival Matters
Nov. 24, 2020 1pm (Zoom)
David Broodryk (University of Cape Town, South Africa)
X
Many categorical conditions can be translated into the language of universal algebra as Mal'cev conditions. For example, much of the theory of Mal'cev varieties can be reproduced in the purely categorical context of Mal'cev categories. Then, the famous result that a variety is congruence-permutable if and only if it has a Mal'cev term t(x,y,y)=x and t(x,x,y)=y can be seen as a syntactical characterisation of this categorical condition. In this talk, we present such a characterisation for the categorical condition of co-extensivity, where a category C is said to be co-extensive if for each pair of objects X,Y in C, the canonical functor ×:X/C×Y/C?(X×Y)/Cis an equivalence. As a motivating example, we begin by considering the property of commutative semirings that to present a commutative semiring S as a product is exactly to find two elements of S which sum to 1 and whose product is 0. This very naturally implies that the variety of commutative semirings is co-extensive, and it then becomes interesting to ask exactly which other varieties are co-extensive. In order to answer this question, we first characterise the weaker condition of left co-extensivity. We then show that any co-extensive variety must have what we call a 'diagonalising term'. Lastly, we complete the characterisation by finding a sufficient and necessary set of identities this term must satisfy in order for the variety to be co-extensive.
Characterisation of co-extensive varieties of universal algebras
Lusztig's -function on a Coxeter group is a function that takes a constant nonnegative integer value on each Kazhdan--Lusztig cell of . The function plays a key role in the study of the cells of and the cell representations of the Hecke algebra of , but it is also very difficult to compute. For an element in , we have if and only if is the identity and if and only if has a unique nonempty reduced word; moreover, the elements of -values 0 and 1 each form a single two-sided cell. We report on some extensions of these results about the set of elements with -value 2. In particular, we classify Coxeter groups where is finite, count , and describe the partition of into cells when is finite. If time allows, we will also discuss a connection between cell representations associated to and the reflection representation of in types . (Joint work with Richard Green.)