2004 Application

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Institute for Advanced Study

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IAS/Park City Mathematics Institute
Undergraduate Faculty Program (UFP) 2004

For the faculty members whose main focus is teaching undergraduate students, PCMI offers the opportunity to renew excitement about mathematics, talk with peers about new teaching approaches, address some challenging research questions, and interact with the broader mathematical community.  Each year the theme of the UFP bridges the research and education themes of the Summer Institute.  This year's theme is:

Combinatorics in Concert: for Teaching, Research, Outreach and Recreation

Featuring two parts:

• a Baseline - a survey on "Geometric Combinatorics"

• Melody - using combinatorics to enhance professional activities

Workshop Leader:  Francis Edward Su, Harvey Mudd College

This year's UFP Program will use geometric combinatorics to weave together many aspects of a faculty member's professional life-- teaching, research, outreach, and recreation-- into a harmonious whole. 

The many beautiful yet accessible ideas in geometric combinatorics make this topic perfect for: (a) enriching a wide variety of undergraduate courses with examples from this field; (b) providing a source of research problems (for undergraduates or oneself); (c) generating topics for general lectures in the community or local high schools; and (d) sustaining recreation opportunities such as puzzle solving.

Geometric combinatorics refers to a growing body of mathematics concerned with counting properties of geometric objects described by a finite set of building blocks. Primary examples include polytopes (which are bounded polyhedra and the convex hulls of finite sets of points) and complexes built up from them. Other examples include arrangements and intersections of points, lines, and planes. There are many connections to linear algebra, discrete mathematics, analysis, and topology, and there are many exciting applications to economics, game theory, robotics, and biology.

Our concert will feature two concurrent parts, the "Baseline" and "Melody". Each will generally last one hour per day. The "Baseline" hour will be a course on "Geometric Combinatorics" (open both to undergraduate faculty as well as other PCMI participants) where we will have fun learning a selection of topics from this area, such as:

Combinatorial convexity: affine geometry and Radon's theorem 
Set intersection theorems: Helly and KKM theorems, and others
Many ways to cut a diamond: polyhedra and polytopes, duality
Thinking in high dimensions: Schlegel diagrams and other devices
The many faces of a polytope: Euler's theorem, upper bound theorem
Our LEGO's: triangulations and simplicial complexes
Counting points for volumes: Pick's theorem and Ehrhart polynomials.
Robot motion: configuration spaces in a simplicial complex
Combinatorial fixed point theorems & applications to fair division
Ham Sandwich type theorems, Kneser colorings of graphs
Tropical geometry and phylogenetic trees

The "Melody" hour will build on the Baseline material to address issues of particular interest to faculty in highly interactive and participatory ways. For example, various sessions (the "Verses") will focus on:

1. Developing ways that the Baseline material can enrich one's own undergraduate courses (such as discrete mathematics, linear algebra, geometry, and topology) or to teach a stand-alone course in the subject.

2. Discussing pointers for how to do research in this area, and how to find problems suitable for undergraduate research.

3. Working in groups to prepare and deliver an "outreach lecture" on one topic in this area. This will equip faculty members to give general lectures in their communities (as we are often asked to do).

4. Generating excitement for recreational mathematics, discussing how to run a successful problem-solving group at one's own college or university, preparing students for competitions such as the Putnam, and encouraging community among math students.

All faculty at PCMI, whether or not they are part of the UFP, are also welcome to participate in the Melody at any time.

Prerequisites:  College faculty with a strong interest in undergraduate education are encouraged to apply to PCMI's Undergraduate Faculty Program. At least two years of undergraduate mathematical teaching experience are recommended.  This program is generally not for graduate students or new PhD's. 

Please indicate in your application which aspects of the program most appeal to you and which you might have some experience with already, along with ideas about how you might share your experiences with other participants during PCMI and with your department when you return.

Not only does the UFP Program typically explore both the content and pedagogy of one undergraduate topic per year, we also help encourage and facilitate interaction among PCMI's constituent groups and programs. For example, some UFP participants like to attend the Graduate Summer School courses. A large number are attracted to the Undergraduate Program, both for the interesting mathematics in the courses and for the kind of research experiences for undergraduates that are often available. Some also work actively with the High School Teachers program, particularly concerning pedagogical, curricular, or articulation issues.