Barbara A. Shipman

The geometry of the full Kostant–Toda lattice of sl(4,C)

Abstract The full Kostant–Toda lattice is an integrable system whose geometry appears naturally in the setting of flag manifolds but is not easily apparent in the original phase space. Separatrices in the flows for two different families of integrals in the Toda lattice of sl(4, C ) appear in splittings of weight polytopes associated to partial flag manifolds of Sl(4, C ) . Each family of integrals gives rise to a C * -bundle of level set varieties with singular fibers. In a neighborhood of a separatrix, the monodromy is a single twist of the noncompact cycle around the cylinder, C * . Near another type of singularity, the cycle is twisted twice; this interchanges the source and sink of one of the Hamiltonian flows.

Determining Definitions for Comparing Cardinalities

Abstract Through a series of six guided classroom discoveries, students create, via targeted questions, a definition for deciding when two sets have the same cardinality. The program begins by developing basic facts about cardinalities of finite sets. Extending two of these facts to infinite sets yields two statements on comparing infinite cardinalities that contradict each other. The experiment “More circles or more squares?” resolves this dilemma in favor of the definition of “same cardinality” that Georg Cantor adopted over a century ago.

Simple yet Hidden Counterexamples in Undergraduate Real Analysis

Abstract We study situations in introductory analysis in which students affirmed false statements as true, despite simple counterexamples that they easily recognized afterwards. The study draws attention to how simple counterexamples can become hidden in plain sight, even in an active learning atmosphere where students proposed simple (as well as more complex) counterexamples daily in class. We provide perspective on these observations in the context of an informative body of related work in undergraduate mathematics studies.

ON THE FIXED-POINT SETS OF TORUS ACTIONS ON FLAG MANIFOLDS

This paper takes a detailed look at a subject that occurs in various contexts in mathematics, the fixed-point sets of torus actions on flag manifolds, and considers it from the (perhaps nontraditional) perspective of moment maps and length functions on Weyl groups. The approach comes from earlier work of the author where it is shown that certain singular flows in the Hamiltonian system known as the Toda lattice generate the action of a group A on a flag manifold, where A is a direct product of a non-maximal torus and unipotent group. As a first step in understanding the orbits of A in connection with the Toda lattice, this paper seeks to understand the fixed points of the non-maximal tori in this setting.

Monodromy near the singular level set in the sl(2, C) Toda lattice

Abstract Completing the flows of the sl(2, C ) Toda lattice and removing the two fixed points in each level set of the constant of motion produces a C∗ bundle with nontrivial monodromy. The monodromy causes a reversal in the character of the fixed points, one a source and the other a sink.

Limits of Infinite Processes for Liberal Arts Majors: Two Classic Examples

Abstract This paper presents guided classroom activities that showcase two classic problems in which a finite limit exists and where there is a certain charm to engage liberal arts majors. The two scenarios build solely on students existing knowledge of number systems and harness potential misconceptions about limits and infinity to guide their thinking. Through exploration of the monotonic convergence of a sequence to a finite limit and oscillatory convergence of a sequence to a finite limit, the two examples recast the essential mathematics in a way that allows liberal arts students to develop a mathematically correct appreciation for convergence to the limit.

A Comparative Study of Definitions on Limit and Continuity of Functions

Abstract Differences in definitions of limit and continuity of functions as treated in courses on calculus and in rigorous undergraduate analysis yield contradictory outcomes and unexpected language. There are results about limits in calculus that are false by the definitions of analysis, functions not continuous by one definition and continuous by another, and continuous functions with discontinuities. This paper uncovers these issues and resolves them in five guided classroom discoveries ideal for students in a first course in real analysis, an honors calculus course, or a course on transitions to higher-level mathematical thinking. The conclusion provides definitions consistent with analysis in a setting for the general student of calculus.

A UNIPOTENT GROUP ACTION ON A FLAG MANIFOLD AND "GAP SEQUENCES" OF PERMUTATIONS

There is a unipotent subgroup of Sl(n, C) whose action on the flag manifold of Sl(n, C) completes the flows of the complex Kostant–Toda lattice (a Hamiltonian system in Lax form) through initial conditions where all the eigenvalues coincide. The action preserves the Bruhat cells, which are in one-to-one correspondence with the elements of the permutation group Σn. A generic orbit in a given cell is homeomorphic to Cm, where m is determined by the gap sequence of the permutation, which lists the number inversions of each degree.

Fifty years of the finite nonperiodic Toda lattice: a geometric and topological viewpoint

In 1967, Japanese physicist Morikazu Toda published a pair of seminal papers in the Journal of the Physical Society of Japan that exhibited soliton solutions to a chain of particles with nonlinear interactions between nearest neighbors. In the fifty years that followed, Todas system of particles has been generalized in different directions, each with its own analytic, geometric, and topological characteristics. These are known collectively as the Toda lattice. This survey recounts and compares the various versions of the finite nonperiodic Toda lattice from the perspective of their geometry and topology. In particular, we highlight the polytope structure of the solution spaces as viewed through the moment map, and we explain the connection between the real indefinite Toda flows and the integral cohomology of real flag varieties.

Convergence and the Cauchy Property of Sequences in the Setting of Actual Infinity

Abstract Traditional definitions, language, and visualizations of convergence and the Cauchy property of sequences convey a sense of the sequence as a potentially infinite process rather than an actually infinite object. This has a deep-rooted influence on how we think about and teach concepts on sequences, particularly in undergraduate calculus and analysis. After characterizing this point of view, this paper reformulates the definitions of convergence and the Cauchy property in the setting of actual infinity. This yields a conceptually streamlined approach and simple proofs of classic results on sequences. The paper also presents pedagogical metaphors that guide students in defining limit and the Cauchy property from the actually infinite standpoint.