Title:  Introduction to trace ideals and centers of endomorphism rings

Abstract:  In many branches of mathematics, the full set of "functions" between two objects exhibits remarkable structure; it often forms a group and in some special cases it forms a ring. In this talk, we will discuss this phenomenon in Commutative Algebra. In particular, we will talk about the endomorphism ring formed by the homomorphisms from a module to itself by first looking at commuting square matrices with real entries. The well-known trace map on matrices can be generalized to a map on any module over a commutative ring. The image of such a map is a trace ideal. I'll introduce the trace ideal and explain its role in the question "What properties of a module does its endomorphism ring detect?"

Title:  Galois groups in Enumerative Geometry

Abstract: In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem.  Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems.  He posited that a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem.

I will describe this background and discuss some work of many to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry.

Title:  The mathematics of secrets

Abstract:  Information permeates our lives. We send texts, shop online, pay bills, type emails, and store pictures and medical data in “the cloud”. Hidden from view is the mathematics needed to make information transfer efficient and secure. This talk will give an overview of the history and the evolution of coding (turning information into numbers) and encryption (securing the information), from ancient times to the modern era, and we will look at how mathematics is the key to it all.

Title:  Making connections across algebra, analysis, and geometry

Abstract:  In the fall of 1988 Paul Halmos hired me, then a 19-year-old undergraduate, to do a bit of research.  He was writing the paper "Has Progress in Mathematics Slowed Down?" (Monthly, 1990) in which he identifies 22 mathematical concepts, explosions, and developments that had occurred during the first 75 years of the MAA's existence.  My specific task was to find and report on every article in the Monthly where these 22 subjects had appeared.  It was an eye-opening experience for me: almost nothing that I had learned as an undergraduate was among the 22 subjects.  Why had I not been exposed to more of these subjects, at least in passing?   I think there are many good reasons, including the fact that there is quite a bit of pre-1915 mathematics that is beautiful, valid, and useful in essential ways.  However, I think that some of the fault also lies with how many (most?) math instructors silo their teaching: calculus is about calculus, algebra is about algebra, analysis is about analysis...and so it is unnatural for teachers to discuss mathematical ideas that fall outside the subject matter of the class.  I will discuss an example where I try to break this approach to teaching in my introductory analysis courses.

Title:  Every Paper Tells a Story: Reflections on the Monthly

Abstract: Over its 130-year history, the American Mathematical Monthly has not only featured a wide array of mathematics on its pages but also a host of other insights related to the discipline. From Nobel prize winning ideas to careers inspired by a local drugstore to mathematical menus created by students, this talk showcases the riches of the Monthly and what we can learn about the profession.  This talk also includes tips for publishing in the journal today.

Brief bio: Della Dumbaugh is the Robert Edwin Gains Professor of Mathematics at the University of Richmond and Editor of the American Mathematical Monthly. She feels at home in a mathematics classroom where her teaching has been celebrated by the University of Richmond, the State Council of Higher Education of Virginia, and the Mathematical Association of America. Along with her friend and collaborator, Deanna Haunsperger, she recently published Count Me In: Community and Belonging in Mathematics. She enjoys writing letters the old fashioned way, exercising, and spending time with her family.