Singularities in Analytic PDE

In this project we will study solutions to differential equations in several complex variables with analytic coefficients. In particular, we will be interested the types of singularities such solutions can have, if any. A basic question, for example, is if you are given a domain U and a differential equation, is there a solution which is analytic in U but has singularities on the boundary of U, and thus is not analytic in a larger domain? This question depends on the interaction between the equation and the geometry of U, and is not yet completely answered. We will learn some of the partial results that have been obtained thus far, and hopefully come up with some of our own.

Written by:  Cory Barnes, Michael Knudson, and Tommy McCauley

Statistical Analysis for the Minnesota Pollution Control Agency

We will analyze data on water quality in Minnesota waterways. I do not have the exact details yet, but we will work with scientists at the MPCA.

Written by:  Landon Chan, Daniel Levy, Charlie Liu, and Amartya Sinha

A Seminar in the Combinatorics or Symmetric Functions

A symmetric function is a polynomial in n variables which is unchanged by every permutation of those variables.  For example, if our variables are x, y, and z, then x3y + y3z + z3x is not a symmetric function, but x3y + y3z + z3x + x3z + y3x + z3y is a symmetric function.  The set of symmetric functions in which every term has degree k is a vector space, with several useful bases. Remarkably, transition matrices between these bases, a natural dot product on this space, products of various basis elements, and the bases themselves, can all be beautifully described using familiar combinatorial objects, including partitions and permutations. This project is a two-term seminar (for 12 students) devoted to this deep and amazing connection between combinatorics and symmetric functions. In the first half of the seminar we will work together to learn the basic combinatorics of these symmetric functions.  In the second half of the seminar we will divide into groups of three, and each group will read recent research papers on a special topic involving the combinatorics of symmetric functions.  Some of these topics will be almost completely combinatorial, but for those who have taken Abstract Algebra II, another possible topic is the representation theory of the symmetric group, and how this theory is related to symmetric functions. Each group will write an expository paper on their topic and give a public talk on their work.  The seminar will also include some training in LaTeX, which is the industry standard mathematical typesetting software. To register for the seminar you must have taken a course in combinatorics which includes some work with partitions and generating functions; two such courses are the Budapest program’s Combinatorics I and our Math 333 (Combinatorial Theory).  This is a winter/spring project, and Math 333 will be offered in the fall, so if you don’t yet have the required background, you can still choose this project, as long as you also plan to take Math 333 in the fall.

The Combinatorics of Symmetric Functions:  (3 + 1)-free Posets and the Poset Chain Conjecture written by Mary Bushman, Alex Evangelides, Nathan King, and Sam Tucker

Plane Partitions – written by Amy Becker, Lilly Betke-Brunswick, and Anna Zink

An Open Problem in the Combinatorics of Macdonald Polynomials – written by Gabe Davis, Aaron Mauer, and Julie Michelman

Bid-to-play games

We’re going to play tic-tac-toe, but with a twist.  In addition to a pile of Xs and Os we each have ten dollars.  To decide who moves first we each will secretly write down the number of dollars (whole numbers only) that we think the first move is worth.  The person who bids the most wins the right to move first, but pays the amount bid to the other player.   Then each player again, secretly, records a bid for the right to move second and the higher bidder earns the right to move and pays the amount bid to the non-mover.  The goal is to win the tic-tac-toe game.  Our project will be to find the best strategy in this game (or prove that no such thing exists). Of course, there is nothing special about ten dollars and we will seek a general theory that explains the game for all initial dollar amounts.  (Oh, in case you want to play this game, you need a mechanism to break ties between bids.  The easiest thing to do is to have the analogue of a possession arrow in basketball: one player is arbitrarily given the tie-breaking advantage at the beginning of the game.  In the event of tied bids that player moves, pays the amount bid to the other player, and cedes the advantage for the next tie.)

Written by:  Boye Akinwade, Andrew Bergman, Lizzie Cross, and Veronica Tan

Directed reading in algebraic geometry: You can help pick the topic!

This is an opportunity for one or two (preferably two) people to delve deeply into material beyond the standard undergraduate curriculum, and to get substantial (weekly) experience in presenting that material at the blackboard.

In algebraic geometry, techniques from abstract algebra are used to describe and investigate “varieties”: sets that are defined by polynomial equations (in several variables).  The interplay between algebraic and geometric points of view makes this a very rich topic, that has expanded enormously in the last hundred years or so; while it is traditionally considered “pure” mathematics, the more computational parts have applications in such areas as robotics and computer-aided design.  The minimal prerequisite is abstract algebra I.

The first major goal of the reading would almost certainly be Hilbert’s “Nullstellensatz” (theorem of the zeros), which establishes a correspondence between varieties and certain ideals in polynomial rings.  After that there is a variety (pun intended) of possibilities, depending on the background, interests, and tolerance for abstraction of the participants.

Written by:  Elizabeth Collins-Wildman and James Munson

Directed reading in analytic number theory

This is an opportunity for one or two (preferably two) people to delve deeply into material beyond the standard undergraduate curriculum, and to get substantial (weekly) experience in presenting that material at the blackboard.

Analytic number theory uses functions of a complex variable to investigate and solve problems in number theory.  Prerequisites are functions of a complex variable and at least one of number theory and abstract algebra I.

Recent comps topics in this area have included the prime number theorem and Dirichlet’s theorem (on primes in an arithmetic progression).  The most likely topic for 2010-11 would be elliptic functions and modular forms (elliptic functions are functions of a complex variable which have “two” periods, unlike trigonometric functions, which have only one; the definition of a modular form does not fit here), but there are other possibilities.

Written by:  Daniel Ehrenberg

Spontaneous Synchronicity

Emergence is the study of the natural phenomenon of how complex structures, behaviors, and patterns arise out of interactions among relatively simple parts. Examples include the complex and adaptive behavior of ants and other communal insects, the immune system, and the collection of neurons that comprise the brain. In our project, we will study the spontaneous synchrony that sometimes arises in a collection of weakly coupled oscillators.  In particular we will examine what kinds of conditions that do and do not give rise to this synchronicity.  We will consider different types of simple oscillators and different ways the individual oscillators in a large population can be connected.

The tools we will bring to bear on this problem include chaotic dynamics, simulations, and elementary graph theory.  A pre-requisite is Math 251 which will be offered in the fall.  At least one member of the team will need to know some elementary computer programming.  Some facility with Mathematica will also be helpful.

Written by:  Andrew Mering, Hang Nguyen, Chris Wilen, and Celine Yeh

Chemical and Biological Knot Theory

Though a well-established mathematical area of research, knot theory had its origins in chemistry.  In the 1880s, it was believed that atoms were tiny knots in space and so began the search for a list of all possible knots.  Mathematicians were hooked and have been searching for a classification of knots ever since, applying techniques from many branches of mathematics to this difficult problem.

In this comps project, we will apply these mathematical techniques to modern day chemistry and biology problems (with suitable abstractions, of course!).  Particularly, we consider the knottedness of chemical molecules and circular DNA, and the effect of chemical and enzyme actions on it.  We will begin with a basic overview of knot theory and recent journal articles, and time permitting, work on research problems in this area.

No biology or chemistry is required for this project.   The direction of the project may change depending on the group, but Abstract Algebra I, Topology, and/or Probability would be recommended, though not required.

Written by:  Jeremy Grevet, Qi Li, and Daisy Sun