It is well-known and easy to prove that the variety of groups abstractly captures algebras of permutations under composition and inverse, that the variety of inverse semigroups capture algebras of partial injective functions under composition and inverse, and that the variety of semigroups abstractly capture the algebras of any of total functions, partial functions or binary relations under the operation of composition. In contrast to this, a landmark result of Hirsch and Hodkinson showing the undecidability of determining when a finite algebra is isomorphic to an algebra of binary relations under Tarski’s signature: the usual set theoretic Boolean operations, composition, converse and identity. This is a very rich signature, and it has subsequently been discovered that undecidability of representability begins in weaker signatures. This talk will survey some of the very extensive literature in this area, and an overview of the approaches to undecidability, possibly touching on some new results for one of the weakest known algebraic signature to experience undecidability of representability as binary relations.
Undecidability of representability as binary relations
Borel subgroup orbits of classical symmetric spaces are parametrized by families of signed involutions called clans, which provide a combinatorial model for studying questions related to Schubert calculus on symmetric spaces. For symmetric spaces of Hermitian type, clans are grouped into “sects” corresponding to Schubert cells of an associated Grassmannian variety, yielding a cell decomposition of the symmetric space and facilitating a combinatorial description of the closure (Bruhat) order on the orbits. This decomposition reveals coincidences of clans in the largest sect with other well-studied posets of matrix Schubert varieties. We further describe explicit bijections between clans for Hermitian type symmetric spaces and several other combinatorial families of objects, including certain rook placements, set partitions, and weighted Delannoy paths. (Based on joint work with Mahir Can and Ozlem Ugurlu.)
The Bruhat order on symmetric spaces
Tue, Oct. 19 4pm (Zoom)
Jess Ellis Hagman (Colorado State University)
In this talk I will share my current research that addresses student success in STEM from a critical perspective. A critical perspective on student success means locating student challenges in STEM not within the students' themselves but within the systems at play, and that these systems are failing students not because they are broken but because they are functioning perfectly well based on their design. I will share my research teams' current work focused on student experiences in college precalculus and calculus from a critical perspective, discussing both recent qualitative research and quantitative research studies. I care about these courses personally because I think the content is fun, but more importantly because of the role they play in STEM students' college experiences. This talk is intended for a broad audience to gain a new perspective on student success in STEM and learn a bit about research coming from this perspective.
Speaker bio: Jess Ellis Hagman is an Associate Professor in the Department of Mathematics at CSU. She completed her PhD in Mathematics Education from the joint program between San Diego State University and the University of California, San Diego. Her area of research is undergraduate mathematics education. Her work is focused on dramatically increasing the number and diversity of people who succeed in undergraduate mathematics-especially introductory mathematics courses that often function as a roadblock for STEM intending students. Her current research includes studying characteristics of successful precalculus and calculus programs, focusing on investigating ways departments can create diverse, equitable, and inclusive introductory mathematics programs.
Talking Critically about Student Success in STEM
Wed, Oct. 20 4pm (Math 350)
Robin Deeley (CU-Boulder)
Informally, a dynamic system is any physical system that evolves with time (e.g., a pendulum, a planet orbiting the sun, the weather, etc). From a more mathematically precise perspective, one can consider a function mapping a space to itself. For example, f(x)=x^2 defined on the set of real numbers. Using this formulation, time is represented by iterating the function. In the example f(x)=x^2, if the initial value is 2, then after one unit of time, the value is f(2)=4, after two units of time, the value is f(f(2))=f(4)=16 and so on.
We will discuss a number of explicit examples of dynamical systems and the precise mathematical formulation of chaos. We will also learn the answer to the question: why is my dad's favorite number 6174?
Dynamical systems, chaos, and my dad's favorite number
Thu, Oct. 21 2pm (MATH 350)
Robin Deeley (CU Boulder)
Scarparo has constructed counterexamples to Matui's HK-conjecture. These counterexamples are essentially principal but not principal. For group actions the difference between essentially principal and principal is related to the difference between the action being topologically free and free. I will discuss a counterexample to the HK-conjecture that is principal. Like Scarparo's original counterexample, this counterexample is the transformation groupoid associated to a particular odometer. However, the relevant group is the fundamental group of a flat manifold (and hence is torsion-free) and the associated odometer action is free.
A counterexample to the HK-conjecture that is principal
Machine learning methods aimed at learning the underlying geometry of data predominantly assume that the underlying geometry is a smooth embedded manifold. This assumption fails for many physical models where the underlying geometry contains singularities. In this talk, we discuss learning the underlying geometry of data within the scope of algebraic geometry and we use this to detect singularities in data.