# 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.

### 2010-11 Schedule

#### Fall

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

Sept 2Meeting with majors. No talk this week.
Time: 4:10 PM
Sept 9Time-frequency analysis for sound modeling: blinded by Fourier Transform
Michael J. O'Donnell, Department of Computer Science, University of Chicago
Time: 4:10 PM

Lots of manipulations of sounds would benefit from the right analysis of each sound into sinusoidal components with varying amplitudes and frequencies.  Windowed Fourier Transforms (WFT) generate time-frequency distributions ρ(f,t) with lovely mathematical properties, and lots of very effective uses.

Clearly, we should diddle the details of WFT until we get distributions where ρ(f,t) represents as precisely as possible what's happening at frequency f and time t

Or not.  I argue that the notational association of WFT data ρ(f,t) with frequency f and time t may blind us to a better use of the information.  Rather than specifying what happens at f and t, ρ(f,t) tells us something about the behavior near frequency f at a time near t.  Techniques from polynomial interpolation, used with an attitude that I got from estimation theory, offer an attractive alternative to distributions.

There is a good baccalaureate thesis in here.

Sept 16
Cancelled
TBA
Drew Endy, Department of Bioengineering, Stanford University
Time: 4:10 PM
Location: TBA
Sept 23 Obstacle Numbers of Graphs
Josh Laison, Department of Mathematics, Willamette University
Time: 4:10 PM

Arrange some vertices in the plane, and some polygon "obstacles", and construct a graph by drawing a straight line edge between vertices whenever it doesn't intersect an obstacle.  Which graphs can be constructed in this way?  Given a graph, how many obstacles do you need?  What if the obstacles are convex polygons?  In this talk we'll answer these questions using some surprising connections to a few different subfields of graph theory.  We'll also pose a number of questions which are still open.  This work was done with undergraduate students in the Willamette Valley REU program.

Sept 30Variational calculus for breakfast
Richard E. Crandall, Director, Center for Advanced Computation, Reed College
Time: 4:10 PM
The variational calculus is one of those special mathematical tools that often works out well on just a breakfast-table napkin.  The lecture will tour accessible variational problems ranging from zoological to quantum-theoretical.  Indeed, many such problems can be solved even before one's omelet has cooled.
Oct 7Student Presentations
Gaurav Venkataraman, Julia Porcino, Nick Salter, Tianyuan Xu
Time: 4:10 PM
##### A spike train metric for estimating the parameters of a model neuron

Gaurav Venkataraman

##### Symmetric sandpiles and dominoes on a Möbius strip

Julia Porcino, Nick Salter, Tianyuan Xu

Oct 14Student Presentations
Laura Florescu, David Krueger, Andrew Winterman
Time: 4:10 PM

Laura Florescu

David Krueger

##### Statistical Tests of Equivalence: An Application to cDNA Microarrays

Andrew Winterman

Oct 21Fall break
Time: 4:10 PM
Oct 28Automorphic Spectral Theory and Number Theoretic Applications
Amy Decelles, Department of Mathematics, University of Minnesota
Time: 4:10 PM

Innocent-sounding questions about numbers and geometry sometimes require serious mathematics to answer. Spectral theory, breaking down complex functions into fundamental “waves,” as a prism breaks down light into its constituent waves of red through violet, is one modern method for dealing with such questions. As a simple example, Fourier series of Bernoulli polynomials compute special values of Riemann’s zeta function, a function related to the distribution of prime numbers. More serious spectral theory, along with standard methods from complex analysis, gives a relationship between the number of lattice points in an expanding region in hyperbolic space and the “automorphic spectrum,” the fundamental waves in this context. As a final example, the recent work of Diaconu, Garrett, and Goldfeld uses spectral theory to produce identities involving moments of L-functions, from which they extract information about the behavior of the L-functions on the critical line, proving a subconvex bound, which, in the absence of the currently unattainable hypotheses of Lindelöf and Riemann, suffices for many number-theoretic applications.

Nov 4Inverse Problems for Distributions of Parameters in PDE Systems
Nathan Gibson, Department of Mathematics, Oregon State University
Time: 4:10 PM

We describe a general first order linear system of partial differential equations and discuss the formulation of an inverse problem for some parameters given measurements of some other quantities.  In many applications a simple constant parameter is not sufficient to represent the dynamics; thus we discuss what is meant by a distribution of parameters.  This approach can be effectively used in uncertainty quantification, but we will apply it in a more straight-forward manner to a particular problem where the resulting system can be efficiently simulated.  Lastly we will present numerical results for a sample inverse problem for a distribution of parameters. Aspects of this research were explored during the Math REU at OSU for the past few years.

Nov 11No talk this week
Time: 4:10 PM
Nov 18No talk this week
Time: 4:10 PM
Nov 25Thanksgiving break
Time: 4:10 PM
Dec 2
Please note change in location.
Accounting for cured patients in cancer clinical trials
Megan Othus, Biostatistics and Biomathematics group, Hutchinson Cancer Research Center
Time: 4:10 PM
Location: Psychology 105
Patients with some cancers, including leukemia and breast cancer, can be cured of their disease.  Current designs of adult cancer clinical trials do not incorporate the fact that some patients are cured and that the objective of some new therapies is to both increase the proportion of patients cured and to prolong survival among patients who are not cured.  This talk will discuss statistical issues associated with designing clinical trials in which some patients may be cured of their disease.

#### Spring

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

Feb 10 From Euclidean triangles to algebraic curves (of algebraic curves)Thomas Schmidt, Department of Mathematics, Oregon State UniversityTime: 4:10 PMPlaying billiards on a Euclidean "rational-angled'' polygon can be viewed as following straight lines on an associated flat (real) surface.  When the flat surface has a large appropriate symmetry group, there is an associated algebraic (complex) curve in the space parametrizing the equivalence classes of complex curves of a fixed genus.    In the talk,  I will hint at some of these ideas, mainly with pictures. Building a New BiologyDrew Endy, Stanford Bioengineering, The BioBricks FoundationTime: 4:10 PMLocation: Psychology 105Natural organisms encode and sustain patterns over time as reproducing machines, with much of the information encoded via physical genetic material.  DNA sequencing and synthesis tools, when combined, make genetic material and sequence information fungible.  Puzzles and opportunities emerge at the intersection of the two previous statements.  For example, could we refactor the architecture of natural living systems to produce more understandable surrogates?  Can we realize and sustain geometric improvements in a capacity to engineer useful biologics?  Will we collectively explore and apply such capacities in ways that benefits all people and the planet? A uniform Artin-Rees property for infinite free resolutionsJanet Striuli, Fairfield University, ConnecticutTime: 4:10 PMFree resolutions of modules are generalizations of kernels and cokernels of linear transformations of vector spaces.  A free resolution of a finitely generated module M over a noetherian ring R is a sequence of free modules and R-linear maps, which after a choice of basis, are represented by matrices with entries in the ring. The study of free resolutions has flourished in recent years as many properties of the module and the ring can be detected with techniques borrowed from linear algebra.In most cases the free resolutions require an infinite sequence of maps and free modules, and therefore it is fundamental to study finite invariants of free resolutions.In this talk I will present invariants of Artin-Rees type and show their existence in certain cases. No talk this weekTime: 4:10 PM Simultaneous Confidence Intervals with more Power to Determine SignsPhilip B. Stark, Professor of Statistics, UC Berkeley Time: 4:10 PMJoint work with Y. Benjamini and V. MadarBy sacrificing equivariance and unbiasedness, we construct simultaneous confidence intervals for the components of a multivariate mean that determine the signs of the parameters more frequently than standard translation-equivariant intervals do.  When one or more estimated means are small, these new intervals sacrifice some length to avoid crossing zero. But when all the estimated means are large, the new intervals coincide with standard, equivariant, simultaneous confidence intervals, so there is no loss of precision in the "easy" case. The improvement in the ability to determine signs can be substantial. For example, if two means are to be estimated and the intervals are allowed to be at most 80% longer than standard intervals in the worst case, then when only one mean is small its sign is determined almost as well as by a one-sided test that ignores multiplicity and has a pre-specified direction. When both are small the sign is determined better than by two-sided tests that ignore multiplicity. The intervals are constructed by inverting level-α tests to form a 1-α confidence set, then projecting that set onto the coordinate axes to get confidence intervals. The tests have hyperrectangular acceptance regions that minimize the maximum amount by which the acceptance region protrudes from the orthant that contains the hypothesized parameter value, subject to a constraint on the maximum side length of the hyperrectangle. These tests are biased in general and the resulting confidence sets are equivariant under permutations of the coordinates and sign changes, but not under translation.In a trivariate example from the Women’s Health Initiative (WHI) randomized controlled clinical trial of Estrogen plus Progestin hormone therapy for postmenopausal women, the new intervals support the conclusion that hormone replacement therapy increases the risk of invasive breast cancer and coronary heart disease but standard simultaneous confidence intervals do not. Universal FractalsTom Wieting, Department of Mathematics, Reed CollegeTime: 4:10 PMThe Cantor Dust, the Sierpinski Gasket, the Koch Snowflake: these are some of the mythic images of modern Fractal Geometry. The object of this expository lecture is to describe mathematical methods by which such images, called Fractals, can be designed, indeed, to describe mathematical methods by which certain structures, called Universal Fractals, can be described from which all Fractals descend. Probability Theory and Discrete Dynamics provide the context. Contraction Mappings play the central role. Spring breakTime: 4:10 PM Complex Systems and Simple Explanations: Simulating networks for HIV prevention education in KenyaMartina Morris, Professor of Sociology and Statistics, University of WashingtonTime: 4:10 PMExplaining the persistent global disparities in the prevalence of HIV has been a major scientific challenge.  Starting from first principles, it is clear that the spread of HIV depends on the connectivity of the underlying partnership network.  Since most peopl have few sex or needle sharing partners, traditional epidemic theory has focused on the role of "core groups" and "superspreaders" for establishing the connectivity needed to sustain transmission.  This may be an appropriate model for curable infections in low prevalence settings, but it has turned out to be less successful for explaining a generalized epidemic, when the infection has become well established in a broader group of less active persons.  In low density networks, connectivity may instead be driven by concurrent partnerships—partnerships that overlap in time—rather than high partner acquisition rates.  Concurrency induces typical complex system dynamics in a network, with thresholds that provide opportunities for prevention.After the scientific challenge comes the public health challenge:  can these complex ideas be communicated effectively to a population with minimal mathematical background? This talk will trace one scientist's journey from mathematical theory to community based participatory research that led to a simple idea for HIV prevention. Topological Field TheoriesJohn Lind, Department of Mathematics, University of ChicagoTime: 4:10 PMA topological field theory (TFT) is a mathematical structure axiomatized from some ideas in modern physics.  The theory of TFTs is a nexus of translation between topology and algebra: the rules for glueing together manifolds (think of strings sweeping out surfaces within "space-time") give rise to the rules of multiplication on algebraic objects.  I will draw pictures and give examples.  The physicists are warned that anything I have to say about physics will be vague or even false. Integral closure and related operationsVeronica Crispin Quinonez, Department of Mathematics, Uppsala University/University of OregonTime: 4:10 PMIntegral closure of ideals has been studied for a long time. We describe how integral closure is related to the concept of reduction, which may be considered as a simplification of the original ideal preserving many important properties. If the ideal is monomial, its integral closure is defined as the convex hull of exponents of the generators. This fact is used to give a very simple algorithm for finding a minimal reduction of any monomial ideal in k[x,y]. We will also discuss other related operations, such as normality and core. Graded Ideals and Homological Dimension, or What is a Syzygy?Jason McCullough, Department of Mathematics, University of California at RiversideTime: 4:10 PMGiven a polynomial ring R = K[X1,X2,...,Xn] over a field K and an ideal I of R, there are various ways of studying I and the geometric properties it encodes. One way is to construct a graded free resolution.  We construct one by mapping a free module onto I, then mapping another free module onto the kernel of the previous map, and so on.  Hilbert's famous Syzygy Theorem tells us that this process stops in at most n steps.  It is easy to see that this bound is tight by considering the Koszul complex, which is a length n resolution of the maximal ideal (X1,...,Xn).  Can we achieve a maximal length resolution with fewer generators?  How few and in what degree?  Relatedly, we consider Stillman's Question: "Is there a bound on the length of the free resolution of I purely in terms of the degrees of the generators of I?"  We present some interesting examples and what they say about a possible answer to Stillman's Question. Spanning trees and the critical group of simplicial complexesArt Duval, Department of Mathematical Sciences, University of Texas at El Paso Time: 4:10 PMCayley's famous result that there are nn-2 spanning trees in a complete graph of n vertices has been proved, and generalized, in many ways.  One particularly nice way is the Matrix-Tree Theorem, that the number of spanning trees of an arbitrary graph can be enumerated by the determinant of a reduced Laplacian matrix.Building upon the work of Kalai and Adin, we extend the concept of a spanning tree from graphs to simplicial complexes, which are just higher-dimensional analogs of graphs. For all complexes Δ satisfying a mild technical condition, we show that the simplicial spanning trees of Δ can be enumerated using its Laplacian matrices, generalizing the Matrix-Tree theorem.We use these higher-dimensional spanning trees to extend the concept of a critical group of a graph (related to the sandpile model and the chip-firing game) to simplicial complexes.  As in the graphical case, the critical group of a simplicial complex (if its codimension 1 skeleton has a suitably nice spanning tree) can be computed directly from the reduced Laplacian, and its order is given by a weighted count of the spanning trees.This is joint work with Carly Klivans and Jeremy Martin. No talk this weekTime: 4:10 PM