Series title: From representations of quivers and posets to applied real commutative algebra

This series of lectures explores recent interactions between algebra, topology, combinatorics, and geometry, with applications to statistics and biology. All three lectures are in colloquium style, aimed at wide audiences with limited mathematical background. Since the final lecture serves a dual purpose as CRM-ISM Colloquium, it is independent of the first two lectures, although the first two provide context and connections that enrich the third.

Monday, September 17, 2018 4:00 pm, UQAM, Pavillon President-Kennedy, room PK-5115

Lecture 1. Bar codes for quiver representations

At the turn of the millenium, algebraic geometers developing combinatorics to describe cohomology classes associated to representations of quivers happened upon the same "bar codes" that the nascent applied topology community was independently inventing for the purpose of summarizing topological information extracted from data via persistent homology. This is a historical overview of multiple views on bar codes, drawing connections to topology, representation theory, algebraic geometry, combinatorics, and commutative algebra.

Tuesday, September 18, 2018 4:00 pm, UQAM, Pavillon President-Kennedy, room PK-5115

Lecture 2. Encoding modules with posets

Bar codes arise from exploration of data using one multiscale parameter. When the context allows or demands more than one parameter, partially ordered sets step in to encode topological summaries in finite fashion. The theory governing the tame behavior of topological summaries of data depends on the algebra of graded modules over rings of polynomials where the exponents are allowed to be real numbers instead of integers. The finiteness condition that gives rise to effective data structures for statistical analyses replaces the noetherian hypothesis, which routinely fails in this context. Out of this tameness surprisingly falls much of ordinary commutative algebra, including primary decomposition and syzygy theorems.

Colloque des sciences mathématiques du Québec (CSMQ)

Friday, September 21, 2018 4:00 pm, UQAM, Pavillon President-Kennedy, room PK-5115

Lecture 3. Algebraic structures for topological summaries of data

The combinatorial algebra in the previous lecture has wide potential applications but was designed to serve a specific question in evolutionary biology. Here data and statistics take center stage. The main dataset comprises images of fruit fly wing veins, which amount to embedded planar graphs with varying combinatorics. Additional motivation comes from statistics more generally, the goal being to summarize unknown probability distributions from samples. The theoretical developments of modules over posets and polynomial rings with real exponents are seen to grow naturally from there, culminating in effective data structures for module presentations and a syzygy theorem.