# Mathematics Department

## Colloquium

#### Upcoming Seminar

October 2, 4:10 PM in Physics 123
Triangulations of a square and Monsky polynomials
Aaron Abrams, Mathematics Department, Washington and Lee University

This will be a sequel to Jamie's talk from 9/25, but his talk is not a prerequisite. The main theorem of his talk is that it is impossible to cut a square into an odd number of triangles of equal area. In my talk we will explore, more generally, what restrictions there are on the areas of triangles in a triangulation of a square. It turns out that for each combinatorial type of triangulation, there's a polynomial relation that must be satisfied by the areas. I will describe some of the things we know about this polynomial.

Most Thursday afternoons during the academic year, the Reed College Department of Mathematics hosts a math talk. The talks are directed to our mathematics majors but are usually accessible on a variety of levels. Refreshments are served before the talks. For more information, please email davidp at reed.edu.

### 2014-15 Schedule

#### Fall

4:10-5:00 in Physics 123 (unless marked otherwise). Directions to Reed.

Sept 4 Meeting with majors. No talk this week. The Combinatorics of CAT(0) Cubical Complexes and Robotic Motion PlanningFederico Ardila, Department of Mathematics, San Francisco State UniversityA cubical complex is CAT(0) if it has global non-positive curvature; informally, "all its triangles are thin". These complexes play an important role in pure mathematics (group theory) and in applications (phylogenetics, robot motion planning, etc.). In particular, as Abrams and Ghrist observed, when one studies the possible states of a discrete robot, one often finds that they naturally form a CAT(0) cube complex. Gromov gave a remarkable topological/combinatorial characterization of CAT(0) cube complexes. We give an alternative, purely combinatorial description of them, allowing a number of applications. In particular, for many robots, we can use these tools to find the fastest way to move from one position to another one. The talk will describe joint work with Tia Baker, Megan Owen, Seth Sullivant, and Rika Yatchak. It will require no previous knowledge of the subject, and be accessible to undergraduate students. Leadership Summit Dissecting a square into trianglesJamie Pommersheim, Mathematics Department, Reed CollegeYou can do this in many ways. You can even arrange for all the triangles have exactly the same area. However, an old theorem (1970) of Paul Monsky asserts that it is impossible to cut a square into an odd number of triangles of equal area.  The proof of this theorem is a delightful blend of geometry, combinatorics, and number theory. This aim of this talk is to prove Monsky’s Theorem and understand some of the geometrical, combinatorial, and algebraic ingredients that go into its proof. The talk may also serve as motivation for Aaron Abrams’s talk next week (October 2), in which Aaron will probably discuss a current research project which grew out of a desire to better understand Monsky’s Theorem. Triangulations of a square and Monsky polynomialsAaron Abrams, Mathematics Department, Washington and Lee UniversityThis will be a sequel to Jamie's talk from 9/25, but his talk is not a prerequisite. The main theorem of his talk is that it is impossible to cut a square into an odd number of triangles of equal area. In my talk we will explore, more generally, what restrictions there are on the areas of triangles in a triangulation of a square. It turns out that for each combinatorial type of triangulation, there's a polynomial relation that must be satisfied by the areas. I will describe some of the things we know about this polynomial. Student PresentationsMaddie Brandt, Joshua Gancher, Justin KatzMaddie Brandt: Packing Polynomials on Sectors of $\mathbb{R}^2$ Joshua Gancher: Weierstrass Points on Graphs Justin Katz: A primary decomposition in computer vision Ben Fischer, Department of Mathematics & Statistics, Boston University Fall break TBA TBA TBA TBA Thanksgiving. No talk this week. Anna Marie Bohmann, Mathematics Department, Northwestern University

#### Spring

4:10-5:00 in Physics 123 (unless marked otherwise). Directions to Reed.

Jan 29 TBA TBA TBA TBA TBA TBA TBA Spring Break TBA TBA TBA TBA TBA TBA