Mathematics Department

Colloquium

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.

2011-12 Schedule

Fall

4:40-5:30pm in Eliot 314 (unless marked otherwise). Directions to Reed.

Sept 1Meeting with majors. No talk this week.
Time: 4:10 PM
Sept 8Orientations, semiorders, arrangements, and parking functions
David Perkinson, Reed College
Time: 4:10 PM

This talk presents work done this summer with Sam Hopkins ('13) in enumerative combinatorics.  I will describe and relate four structures associated with a graph, three of which appear in the figure below.

 

structures on a graph

Sept 15Toward Modularity: the Simplest Non-Abelian Example
Jerry Shurman, Reed College
Time: 4:10 PM

The Modularity Theorem famously asserts that all rational elliptic curves arise from modular forms. This talk will not attempt to explain the theorem, much less to prove any of it, but only to touch on the fact that representations of Galois groups play a crucial role. We will sketch an example involving the simplest non-Abelian Galois group. The talk will not be easy to follow in close detail, but rather its purpose is to illustrate an astonishing connection among wide-ranging phenomena.

Sept 22High-Order Staggered Finite Difference Methods for Maxwell's Equations in Dispersive Media
Vrushali Bokil, Oregon State University
Time: 4:10 PM

We consider high order (in space) staggered finite difference schemes for Maxwell's equations coupled with a Debye or Lorentz polarization model. A novel expansion of the symbol of arbitrary (even) order finite difference approximations of the first order spatial derivative operator allows us to derive a concise formula for the numerical dispersion relation for all (even) order schemes applied to each model, including the limiting (infinite order) case. We further derive a closed-form analytical stability condition for these schemes as a function of the order of the method. Using representative numerical values for the physical parameters, we validate the stability criterion while quantifying numerical dissipation. Lastly, we demonstrate the effect that the spatial discretization order, and the corresponding stability constraint, has on the dispersion error.

Sept 29Conditional independence ideals and permanents
Irena Swanson, Reed College
Time: 4:10 PM

In the first part I present the structure of ideals that arise from some conditional independence models. These ideals live in polynomial rings over fields, and are generated by 2 x 2 determinants.

In the second part I present the differences between determinantal and permanental ideals, and especially as pertaining to the same conditional independence models.

The first part is joint work with Amelia Taylor, and the second part is joint work with Julia Porcino '11.

Oct 6Witt vectors and their generalizations
Lance Edward Miller, University of Utah
Time: 4:10 PM
In this talk we will introduce an interesting family of rings, called Witt vectors, which have applications throughout algebra and number theory. Their construction was generalized in the 1980s by Dress and Siebeneicher. These generalized Witt vectors share some properties with the classical Witt vectors, but we recently found there are also some surprising differences.
Oct 13Rees Algebra of Square-Free Monomial Ideals
Kuei-Nuan Lin, University of California at Riverside
Time: 4:10 PM

We start with the definition of varieties and singular points, then we explain what it means to blow up at a point and why people want to blow up at a point. Later we focus on the definition of a graph and how to use a graph to understand the Rees algebra in degree two square-free monomial ideals (the work of Villarreal). At the end, I present my joint work with Louiza Fouli on how to extend to degree 3 and bigger square-free monomial ideals.

Oct 20Fall break
Time: 4:10 PM
Oct 27Permanent Total Reality
Joe Buhler, CCR/Reed College (emeritus)
Time: 4:10 PM

A polynomial in one variable with real coefficients that has no non-real roots is said to be totally real. Julius Borcea and Petter Brändén explored a notion of "totally real" multivariate polynomials that has turned out to have an extraordinary array of applications. This talk will attempt to cover that idea and (a) its use by Leonid Gurvits to prove a vast generalization of the van der Waerden conjecture on permanents, (b) the proof by Haglund et al of the so-called Monotone Permanent Conjecture, and (c) a proof of a conjecture due to Ray Mayer.

Nov 3An introduction to matroids
Steve Klee, University of California at Davis
Time: 4:10 PM

A matroid is a combinatorial object that generalizes the notions of independence that are studied in linear algebra (linear independence of a family of vectors) or graph theory (a cycle-free collection of edges in a graph). In this talk, I will introduce matroids and motivate them through these examples. We will study some interesting examples of matroids and see how they can be used to analyze a well-known mathematical game.

Nov 10Automorphisms of Surfaces
Aaron Wooton, University of Portland
Time: 4:10 PM

An automorphism f of a surface S is a bijective, continuous map from S to itself. An example of an automorphism is the trivial map i : S→ S defined as i(x)=x for all x on S. A non-trivial automorphism f is said to have order n if n is the smallest positive integer such that fn=i, the trivial map. Given an explicit surface S, it is easy to construct examples of automorphisms on S. For example, if S is a sphere, then rotation of S through an angle π about any axis through the center of S will be an automorphism of order 2. Unfortunately, though we may be able to find numerous automorphisms of a specific surface S, it is not clear how we could determine whether we have found all automorphisms of S.

In this talk, we shall introduce a new way of describing automorphisms through the use of algebra rather than geometry. Though perhaps not as aesthetically pleasing as describing automorphisms geometrically, we shall see that this algebraic description makes determination of all automorphisms of a given surface a (relatively) simple task. We shall finish by discussing how this idea can be generalized to find automorphism groups of a given surface S.

Nov 17The googol-th bit of the Erdős-Borwein constant
Richard Crandall, Director CAC, Reed College
Time: 4:10 PM

The Erdös-Borwein constant is the sum of reciprocals of Mersenne numbers:

E : =
n ≥ 1
1
2n−1
= 1.1001101101010000010111111... (binary).

E was proven irrational by Erdös in 1948; he discovered arbitrarily long but terminating strings of 0 bits in E. Later, Peter Borwein proved E irrational via Padé approximants. For this and other celebrated constants, it is possible to resolve "remote bits" without knowing the bits preceding. For example, using deeper properties of the classical divisor function, we now know the googol-th bit of E; that would be the bit in position 10100 to the right of the point.

Nov 24Thanksgiving break
Time: 4:10 PM
Dec 1Game of commuting matrices
Leila Khatami, Union College, Schenectady
Time: 4:10 PM

Suppose that we are given a square matrix B. We all know that there may be square matrices A of the same size such that A.B and B.A are not equal, in other words the multiplication of matrices is not "commutative". So it is quite natural to ask about properties of those matrices A that do "commute" with B, namely those A for which one has A.B=B.A. The goal of this talk is to present a refined version of this natural question, using one of the most important classical theorems in linear algebra, Jordan Decomposition Theorem. We are going to introduce a game which will help us understand the main idea of Jordan decomposition theorem as well as the question of commuting pairs of matrices.

Spring

4:40-5:30pm in Eliot 314 (unless marked otherwise). Directions to Reed.

Jan 26How to deliver goods using a vehicle on the plane
Aparna Das, Department of Computer Science, University of Arizona
Time: 4:10 PM

In the vehicle routing problem the objective is to find low cost delivery routes from warehouses to customers using  vehicles of limited capacity. The problem has real world applications to businesses with high transportation costs such as waste removal companies, city transportation planners, food and beverage distributors and package deliverers. The problem is also of theoretical interest as it is a generalization of the famous traveling salesman problem which is known to be NP-Hard. We focus on the Euclidean setting of the problem where all distances are defined by the Euclidean metric. We study the simple version where each customer requires delivery of a single identical item and present a quasi-polynomial time approximation scheme when the input points lie in low dimensional Euclidean space.

Feb 2Detecting subclusters in outliers
Dongseok Choi, Oregon Health and Science University
Time: 4:10 PM

Medical research is often interested in finding subgroups in an outlier group. For example, a certain medical condition can be more frequent in a small group that is different from the majority of population. One approach to find groups in a data set is using cluster analysis. Cluster analysis has been a popular tool in exploring potential group structure in complex data and has received greater attention in recent years due to data mining and high dimensional data such as microarrays. In this presentation, I will introduce split-and-recombine procedure and its application for a medical data set. In addition, analysis results of the same data using other clustering methods will be discussed.

Feb 9Communication complexity and its applications
Paul Beame, Department of Computer Science, University of Washington, Seattle, Washington
Time: 4:10 PM

How much communication is required for cooperating agents to achieve a common aim in an ideal world where their messages are transmitted perfectly?

  • For example, if two cooperating agents, Alice and Bob, have inputs x and y, respectively, how many bits do they need to send to each other in order for them both to learn f(x,y) for some function f? How does this number depend on the choice of f?
  • How efficiently can a group of prisoners in a mirrorless room determine common properties of markings that have been scrawled on their foreheads by their captors?
  • How do the answers to these questions change if the agents are allowed to flip coins and can tolerate some probability of error?

These questions and more are the subject of communication complexity, a research topic that has had many applications in theoretical computer science. Communication complexity is analyzed uses ideas from combinatorics, algebra, and discrete probability. In this talk we give an introduction to communication complexity and its applications focusing on some of the example problems and puzzles that have guided its development.

Feb 16Cutting a square into triangles
Aaron Abrams, Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia
Time: 4:10 PM

Suppose you want to cut the unit square into triangles. What are the possibilities for the set of areas of the triangles? It turns out there are some restrictions: for instance, a theorem of P. Monsky says that if the number of triangles is odd, it is impossible for them all to have the same area. In general, what happens is that once you decide on the combinatorics of the triangulation, there will always be a polynomial relation that the areas are guaranteed to satisfy. This polynomial is different for different combinatorial triangulations, and it tends to be quite complicated (in particular the polynomial will have one variable for each triangle in the triangulation). The degree of the polynomial, however, is an integer that might be more tractable. In this talk we will discuss this polynomial and an algorithm for computing its degree.

Feb 23Markov chains and convex polytopes
James Berhnard, University of Puget Sound, Tacoma, Washington
Time: 4:10 PM

In this talk, we will show how the theory of (bounded) convex polytopes can be used to prove some of the fundamental theorems on the convergence of Markov chains. We will recast and investigate these standard theorems geometrically. One of the central themes of this talk will be the interrelatedness of mathematics, as we explore how these two seemingly unrelated areas of mathematics -- Markov chains and convex polytopes -- overlap.

We will develop what we need in both of these areas during the talk, so no prior familiarity with convex polytopes or Markov chains is necessary, only some familiarity with linear algebra.

Mar 1
Cancelled
Has the global risk of large earthquakes increased recently? (Joint work with Peter Shearer, UCSD)
Philip B. Stark, Department of Statistics, University of California at Berkeley
Time: 4:10 PM

No.

Mar 8Convex Sets and Cone Factorizations
Rekha Thomas, University of Washington, Seattle
Time: 4:10 PM

A central problem in optimization is to maximize or minimize a linear function over a convex set. For instance, linear programming concerns optimizing a linear function over a polyhedron. In such instances, it becomes crucial to have efficient representations of the underlying convex set. This brings us to the question of whether complicated convex sets can have simple representations. For example, can a complicated set be the projection of a simple set over which one can optimize efficiently? For polytopes, the answer to this projection question is controlled by nonnegative factorizations of a certain matrix associated with the polytope. In this talk I will explain this surprising link and the many open questions that surround it. I will also extend the result to all convex sets via cone factorizations of certain operators.

Mar 15Spring Break
Time: 4:10 PM
Mar 22Remarks on size, shape and boundary conditions effects on the zero-field superconducting transition temperature
Tiziana Giorgi, New Mexico State University
Time: 4:10 PM

Superconductors are materials that at low temperature exhibit some remarkable properties, such as zero electrical resistance. In this talk, we analyze a celebrated continuous model of superconductivity, the Ginzburg-Landau Energy functional, to study how the shape, size and external materials influence the temperatures at which these phenomena appear in the absence of an applied magnetic field.

Mar 29How many subgroups of index 2
Jean Bernard Nganou, University of Oregon
Time: 4:10 PM

Lagrange's Theorem is one of the simplest, yet crucial results in the study of finite groups. It states that the order of any subgroup is a divisor of the order of the group. The converse of the result is known to be false: not every divisor of the order of a group is the order of a subgroup. The smallest counterexample to this converse is the alternating group A4 on four letters. In fact, A4 has no subgroups of order 6, or equivalently, no subgroups of index 2. In most of the proofs of this fact, the subgroup of squares of A4 plays a crucial role. We investigate the role of this subgroup in the existence of subgroups of index 2 in a general group. Starting with any finite group G, we consider the factor group G/G2, which is elementary Abelian of exponent 2. Therefore, G/G2 can be viewed as a vector space over Z2. We can then use elementary linear algebra to compute the number of subspaces of any dimension in G/G2, which correspond to the number of subgroups of appropriate index. This yields a formula for the number of subgroups of index 2 in any arbitrary finite group. We apply the result to the dihedral groups and other well known groups. Finally, we obtain a new class of counterexamples to the converse of Lagrange's Theorem.

Apr 5Modular forms of half-integral weight and representation theory
Brooks Roberts '87, University of Idaho
Time: 4:10 PM

Classical modular forms are analytic functions on the complex upper half plane that respect many symmetries, and their Fourier coefficients are involved in number theoretic problems. After a slow start, the contemporary theory of modular forms now occupies a central position in mathematics. In most of this talk I will provide a conceptual introduction to classical modular forms  and their connection to representation theory. In the remainder of the talk I will discuss my current work with Ralf Schmidt of the University of Oklahoma concerning modular forms of half-integral weight.

Apr 12The Kohn algorithm, an instance of symbiosis between analysis and algebra
Andreea Nicoara, University of Pennsylvania, Philadelphia
Time: 4:10 PM

The dbar equation is by far the most important partial differential equation in the field of several complex variables. When posed on a domain in C^n, the corresponding boundary value problem is called the dbar-Neumann problem. This problem, which turned out not to be elliptic, was solved in the 1960s by Joseph J. Kohn in the strongly pseudoconvex case and by Lars Hormander in the pseudoconvex case. The next line of inquiry was trying to gauge when the solution to the dbar-Neumann problem gained in differentiability with respect to the data, a property called subellipticity. In the late 1970s, Joseph J. Kohn proposed a new approach involving subelliptic multipliers. Much to his and everybody else's surprise, these subelliptic multipliers have amazing algebraic properties. At each point, they form an ideal, so globally they give a sheaf of ideals closed under certain algebraic operations that come very naturally from the set-up of the dbar-Neumann problem. Joseph J. Kohn restated these properties in a way that gives an algorithm for testing subellipticity, nowadays called the Kohn algorithm. He found a complete characterization of subellipticity for domains with real-analytic boundary, where the local rings are well-behaved, whereas the corresponding characterization for the even more important case of smooth domains is still open. I will attempt to give enough background about the different pieces of this puzzle so that the remarkable relationship between analysis and algebra in this instance can be fully appreciated.

Apr 19Symplectic geometry: the geometry of area
Strom Borman '07, University of Chicago
Time: 4:10 PM

Symplectic geometry is the geometry based on the measurement of area and it is the mathematical setting for both classical mechanics and string theory. In this talk I will introduce some basic notions in symplectic geometry and explain the idea of Gromov's non-squeezing theorem, which says that one cannot symplecticly squeeze a large 4-dimensional ball into a thin cylinder.

Apr 26Connections between Invariant Theory and Number theory
Mara Neusel, Texas Tech University, Lubbock, Texas
Time: 4:10 PM

In 1916 Noether showed that the ring of polynomial invariants (over the complex numbers) of a finite group is generated by polynomials of degree at most equal to group order. In 1989 B. Schmid showed in her PhD thesis that we need polynomials of degree equal to the group order exactly if the group is cyclic. Her proof sheds light on a connection between invariant theory and number theory: for an abelian group the maximal degree of a generator (in a minimal generating set of the ring of invariants) is equal to the Davenport constant. In recent work with A. Geroldinger (Graz) we study what happens to the two constants in the non-abelian case.

Contact Us

Phone: 503-777-7710
Fax: 503-788-6691

Join the Math-at-Reed Email List

Follow @ReedCollegeMath on Twitter