Lowell Abrams

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Associate Professor
Department of Mathematics
The George Washington University

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Contact Information

Lowell Abrams
c/o Department of Mathematics
The George Washington University
Monroe Hall, Room 240

2115 G Street NW

Washington, DC 20052

office: Monroe 271
phone: (202) 994-8119
email: labrams@gwu.edu

 

Office Hours

Fall 2013: I’m on sabbatical – please be in touch by email.

 

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Teaching Activities

 

Fall 2013


Sabbatical!

 

 

 

From Previous Courses:

 

I was very excited in Fall 2012 to be teaching a special “Extreme” Calculus II course which was open only to first-year students. Here’s the flier.

 

In Fall 2008 I taught a Dean’s Seminar for freshmen entitled “A topologist’s view of digital images.” I introduced the students to the digital topology and topological graph theory necessary to understand the ideas in my papers with Donniell Fishkind.

 

 

In Fall 2006 and Spring 2007, I taught a special course (in the GWU School of Professional Studies) called “Higher Algebra” for middle-school teachers from various DC schools. Here is a blurb about that course:

 

When thinking algebraically, and even more generally, people routinely make use of a variety of properties of their object of thought, and shift between a variety of perspectives, without even noticing. This ease can be positive, allowing one to operate efficiently, but it can also lead to muddled thinking and poor communication. The purpose of this course is to accustom the students to thinking explicitly and abstractly about the properties and perspectives that they already use, and about the interelationships among those, with the goal of developing a sophistication, clarity of thought, and depth of understanding which they can pass on to their students.

 

In Spring 2005 and Spring 2006, I taught a “Writing in the Disciplines” course entitled “Math as a Language.” Here is a short description:

 

This course will provide students with the specialized language skills they need to work effectively with ideas of mathematics and communicate them to a variety of audience. Writing with multiple review/revision cycles will play a prominent role; the course will involve creating original arguments and producing written records of arguments presented orally, as well as presenting oral explanations of written arguments. The focus will be on concepts, vocabulary, and syntactic constructions which are ubiquitous in mathematics.

 

In Spring 2005 I also taught a graduate course in Topological Graph Theory. We used the text by Gross and Tucker, but I presented the material from the vantage point of algebraic topology. Essentially, this course was part of my work to develop a broader version of the perspective introduced in my work with Daniel Slilaty.

 

In Fall 2005 I taught a dean’s seminar for freshmen entitled "Games: an introduction to mathematical reasoning." It was a slightly altered version of the course I taught in Fall 2004 and Fall 2003. Here is a one-paragraph "blurb" about it:

 

The patterns and methods of mathematical reasoning have a wide range of applicability. In this course, we will apply mathematical reasoning to the analysis of a variety of games by playing them, reflecting on them more abstractly, and writing about them. In this context, we will study such fundamental notions as axiomatic system, specialized notation, symbolic manipulation, proof, rigor, heuristic, refinement of ideas, and effective communication of technical ideas.

 

In Fall 2005 I spent time on Nimbers, which you can learn about in Winning Ways for Your Mathematical Plays by Berlekamp, Conway and Guy.

 

One of the things we discussed in the Games course in Fall 2003 was how strategies for and the general "nature" of a game changes when the rules are modified. (This is essentially a metaphor for the question of modifying a set of axioms.) As one type of example, here are various boards on which one can play dots-and-boxes:

Regular (Square)

Triangular

Pentagonal

Hexagonal

Please note that I obtained the Hexagon board from here.

 

We also spent some time learning about the game of Go. This is a great example of an axiomatic system with which students can become comfortable, thus allowing them to prove theorems while avoiding issues of “math anxiety.” A really nice program to play on a 9x9 board against a computer can be obtained here.

 

In Summer of 2004 I taught a “math for liberal arts course” which  introduced mathematical ideas relevant to the study of growth of various kinds of individual organisms as well as of populations, and relevant to the study of crystals and other physical phenomena from the natural world. A running theme throughout the course was the issue of how mathematical models for observable phenomena are developed. Here is a document with some sample population sequences for the logistic-growth model and the logistic-growth-with-harvesting model.

 

In the Spring of 2003 I taught a “math for liberal arts” course in which we discussed the combinatorial-topological properties of polyhedra. Here are templates for building the five regular polyhedra (Platonic solids). Print them out, cut out the shapes, fold along the dotted lines, and attach the tabs. Suggestions: (1) First attach sides that are adjacent in the template. (2) Use paper that is as stiff as possible

Tetrahedron

Cube

Octahedron

Dodecahedron

Icosahedron

 

 

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Research

I have finally come to the conclusion that I am a low-dimensional combinatorial topologist (think: cellular complexes and topological graph theory) who happens to also spend some of his research time on combinatorial algebra. (Not to be confused with algebraic combinatorics!)

 

 Here are some general things I am thinking about now:

  • An application of loop and quasigroup theory to a particular family of recursively generated arrays. (With Dena Morton)
  • Embedding graphs in surfaces. (With Dan Slilaty)

 

If you are interested in any of these things, send me email at labrams@gwu.edu!

 

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Publications and Preprints

 

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Some Fun Stuff

A picture of my second-oldest playing volleyball.

 

Here’s a picture of my wife and youngest son. Our family took a trip to the National Mall, here’s a picture from the day.  Our youngest is no longer a baby, but sometimes…

 

I’m still fond of these pictures: Here is a picture of number 4 with his Mommy, within an hour of birth. Here’s the baby again, doing one of the two things which, at the time, he did best. There’s a lot of guys in the family, here they are!

 

Here is a picture of me on the phone with Donniell Fishkind, working on the proof of the main theorem in our first “spherical homeomorphism” paper. I am the big guy; the little one is part of my support team (number 3).

 

Here is a picture of my three older boys and my three nephews, from before the birth of our youngest. The three in the lower left (i.e. the two oldest in the picture and the youngest) are my nephews, the others are my sons!

 

Here is a picture of me with some visitors from a high school in Charleston, SC. If you want to know what we talked about, look at the blackboard.

 

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Links

(list always under construction)

o   Math

o   MathSciNet

o   JSTOR

o   Encyclopedia of Mathematics

o   GWU

o   Main GW Page

o   Gelman Library

o   Banner

o   Supply Chain Forms

o   Blackboard

o   GWeb

o   P-card

o   Reference

o   Web programming tutorials (e.g. HTML)

o   Computational Topology Project (based at Stanford)

o   MedLine Plus (dictionary, encyclopedia, etc.)

o   Crystallographic Topology 

o   Imaging Site, Cognition and Brain Sciences Unit, UK Medical Research Council

o   Whole Brain Atlas (by Keith A. Johnson, M.D.  and J. Alex Becker, Ph.D.)

o   Miscellaneous

o   Washington D.C. Metro

o   Radio on the web

o   Hebrew texts of tanakh, talmud, midrash, rambam

o   Talmud on line (with daf shiurim)

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Page last updated 10/01/13.