# Mathematics Department

## Colloquium

#### Upcoming Seminar

October 30, 4:10 PM in Physics 123
Computing Siegel Modular Forms by the Pullback-Genus Method
Jerry Shurman, Mathematics Department, Reed College

Siegel modular forms are important.  Also, they are enormous and unwieldy, prohibitive to work with by hand and computationally daunting. A formula due to Garrett shows that all Siegel modular forms of a given degree are encoded by one particular modular form of twice the degree, the Eisenstein series.  Work of Siegel and Katsurada shows that the Eisenstein series is more computationally tractable than a general Siegel modular form, providing some counterbalance to its having twice the degree.  Poor and Yuen have devised and implemented an algorithm that uses these ideas to compute Siegel modular forms, finding new examples of degree 3 and identifying them. This talk will attempt to explain some of these matters.

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 $$\mathbb{R}^2$$ Abstract: A polynomial $$p(x,y)$$ on a region $$S$$ in the plane is called a packing polynomial if the restriction of $$p(x,y)$$ to $$S\cap \mathbb{Z}^2$$ yields a bijection to $$\mathbb{N}$$. In this talk, we classify all quadratic packing polynomials on rational sectors of $$\mathbb{R}^2$$. Joshua Gancher: Weierstrass Points on Graphs Abstract: TBA Justin Katz: A primary decomposition in computer vision ﻿Abstract: The pinhole model for a projective camera is given by a projective plane $$P$$ and a focal point $$p$$, embedded in an ambient 3-dimensional projective space. The image of a point $$x$$ in the projective space under the pinhole camera is given by the unique point of intersection between the line connecting $$x$$ and $$p$$ and the image plane $$P$$. Given a collection of distinct pinhole cameras, the image of a point in any one image plane constrains the possible corresponding points in the other image planes. Analyzing a camera system by considering pairs of cameras determines a bilinear constraint polynomial on the image coordinates, and considering triples of cameras determines a related trilinear polynomial constraint, known as the trifocal tensor. This talk will consider the ideal generated by selecting a particular subset of bilinear constraints, and demonstrate that the whole ideal generated by the trilinear constraints arises as a primary component. Such a primary decomposition has the potential to speed up image matching algorithms. Higher-Dimensional Pick's Theorems and Strange Ways of CountingBen Fischer '09, Department of Mathematics & Statistics, Boston UniversityPick's Theorem relates the area of a two-dimensional polyhedron to the number of integer points in each face. The theorem is patently false in three or more dimensions, but in this talk I will describe (infinitely many) ways of constructing higher-dimensional analogues. In order to do so, it will be helpful to generalize the notions of "volume" and "number of integer points", in order to give us something meaningful even when the polyhedron is unbounded. This is joint work with Jamie Pommersheim that evolved out of my senior thesis at Reed in 2009. No prior familiarity with polyhedra will be assumed - only a basic knowledge of linear algebra and calculus. Fall break Computing Siegel Modular Forms by the Pullback-Genus MethodJerry Shurman, Mathematics Department, Reed CollegeSiegel modular forms are important.  Also, they are enormous and unwieldy, prohibitive to work with by hand and computationally daunting. A formula due to Garrett shows that all Siegel modular forms of a given degree are encoded by one particular modular form of twice the degree, the Eisenstein series.  Work of Siegel and Katsurada shows that the Eisenstein series is more computationally tractable than a general Siegel modular form, providing some counterbalance to its having twice the degree.  Poor and Yuen have devised and implemented an algorithm that uses these ideas to compute Siegel modular forms, finding new examples of degree 3 and identifying them. This talk will attempt to explain some of these matters. 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 TBAMohamed Omar, Department of Mathematics, Harvey Mudd College Spring Break TBA TBAAna Rita Pires, Department of Mathematics, Fordham University TBA TBA TBA TBA