Algebraic

Persistence modules are the algebraic objects at the heart of various
theories of persistence. I will discuss some recent work with Bubenik and
Scott on generalised persistence modules (viewed categorically), as well
as some other studies with Chazal, Oudot and Glisse on the structure of
1-parameter persistence modules (the classical case). The idea is to move
beyond the usual restriction to modules with finitely many critical
points to allow much broader families of examples.

The tree structure is currently the accepted paradigm to represent evolutionary relationships between organisms, species or other taxa. However, horizontal, or reticulate, genomic exchanges are pervasive in nature and confound characterization of phylogenetic trees. Drawing from algebraic topology, we present a unique evolutionary framework that comprehensively captures both clonal and reticulate evolution. We show that whereas clonal evolution can be summarized as a tree, reticulate evolution exhibits nontrivial topology of dimension greater than zero.

Abstract: An area of interest in statistical mechanics is the study of
statistical distributions of stochastic currents generated in graphs.
It turns out that this problem amounts to the study of periodic families
of Markov processes that evolve according to the master equation.
Physicists have observed that, for almost every generated current,
quantization occurs in the adiabatic and low temperature limits.

My main goal in this talk will be to explain how this story can be understood using

Overview of the numerical algebraic geometry group agenda