# 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.

### 2012-13 Schedule

#### Fall

4:40-5:30pm in Physics 123 (unless marked otherwise). Directions to Reed.

Aug 30 Meeting with majors. No talk this week.Time: 4:10 PM Centrally symmetric products of spheres with a few verticesIsabella Novik, Department of Mathematics, University of WashingtonTime: 4:10 PMWhat is the minimum number of vertices needed to triangulate a surface of a bagel, i.e. represent it as a union of triangles where every two triangles must either be disjoint or intersect along a vertex or a common edge? What if we also require our triangulation to have a certain symmetry? What about higher-dimensional analogs of this question? More specifically, what is the minimum number of vertices needed to triangulate Si x Sj --- the product of two spheres? To triangulate it in a way that the resulting simplicial complex is centrally symmetric"? While the first question is still open (at least when i and j are sufficiently large), the second question turns out to have a surprisingly simple solution for all values of i and j. In this talk we will discuss this answer and the combinatorial construction that achieves it. This is joint work with Steve Klee. Reed Student Summer ResearchSam Hopkins, Eddie Maldonado, and Marcus Robinson, Reed CollegeTime: 4:10 PMReed mathematics majors discuss their summer research. I. Pattern Avoidance in Permutations on Partially Ordered Sets, Sam Hopkins Abstract: We extend the concept of pattern avoidance in permutations on a totally ordered set to pattern avoidance in permutations on partially ordered sets. The number of permutations on P that avoid the pattern a is denoted $$Av_P(a)$$. We extend a proof of Simion and Schmidt to show that $$Av_P(123) <= Av_P(132)$$ for any poset P, and we exactly classify the posets for which equality holds. We also give asymptotically close bounds on the number of permutations on the Boolean lattice that avoid the pattern $$\{1\}\{1,2\}\{2\}$$. II. Density Structure of Graphs with Applications to Random Graph Models, Eddie Maldonado Abstract: How can we analyze the structure of graphs in relation to their dense subgraphs? We present an algorithm for orienting an undirected graph and a corresponding decomposition on directed graphs that gives certain information about the location of the densest subgraph of a given graph. We discuss conjectures and further work with regards to the properties and application of this decomposition, and use it to motivate a random graph model that appears to mimic the structure of certain real-world networks better than existing models. III. The Hilbert-Kunz Function, Marcus Robinson Reed Student Summer ResearchKelsey Houston-Edwards, Laura Lyman, Wyatt Alt, Mikhail Lepilov, Reed CollegeTime: 4:10 PMI. Coverings of the Integers, Kelsey Houston-Edwards and Laura Lyman Abstract: Our research explores two distinct types of coverings of the integers. First, we look at incongruent disjoint restricted covering systems (or simply, IRDCS) and their properties. We focus on an interesting example, called the 9-6-3 construction." The second type of covering is inspired by Erdös' minimum modulus problem: Given any natural number $$c$$, can one construct a covering of the integers using only moduli greater than $$c$$? II. Searching for combinations of transcription factors associated with differential gene expression in cichlids, Wyatt Alt III. Numerical Semigroups and the Frobenius Number, Mikhail Lepilov What is algebraic statistics?Thomas Kahle, ETH Zurich, Switzerland, and MSRI, Berkeley, California Time: 4:10 PMWhen Muriel Bristol claimed that she could taste whether tea or milk were poured first into a cup, Sir Ronald Fischer invented what is now known as Fischer's exact test.  It is used to determine whether, given the outcome of a tea tasting, there is evidence against the hypothesis of independence of Mrs. Bristol's answers and the truth. Surprisingly, testing statistical hypothesis is connected to toric ideals in commutative algebra.  We will present the Diaconis-Sturmfels theorem, a cornerstone of algebraic statistics. A GPS-Based Bicycle Route Choice Model for San Francisco, CaliforniaJeffrey Hood, Parsons BrinckerhoffTime: 4:10 PMRecognizing the environmental and health benefits of cycling, cities around the world are promoting use of the bicycle for everyday transportation, but with limited information about the preferences of cyclists and the effectiveness of investments in bicycle infrastructure.  To better understand the decision-making of cyclists, we estimated a route choice model with GPS data collected from smartphone users in San Francisco.  Traces were automatically filtered for activities and mode transfers, and matched to a network model.  Alternatives were extracted using repeated shortest path searches in which both link attributes and generalized cost coefficients were randomized.  The prior distribution for the coefficients was calibrated automatically using only the network.  A Path Size Multinomial Logit model revealed that bicycle lanes were preferred to other facility types, especially by infrequent cyclists.  Steep slopes were disfavored, especially by women and during commutes.  Other negative attributes included length and turns. Traffic volume, traffic speed, number of lanes, crime rates, and nightfall had no effect.  Marginal rates of substitution imply a user benefit of bike lanes of \$0.61 USD per km per trip.  Coefficients were applied to a trip assignment model that will be used to evaluate prospective investments in bicycle infrastructure in San Francisco. Statistics and Biological Measurements: Some Early History and Some Modern Tools for Gene ExpressionDaniel Schafer, Department of Statistics, Oregon State UniversityTime: 4:10 PMIn the second part of the talk I will sketch some current research on statistical tools for identifying patterns of gene expression from RNA sequencing (RNA-seq) technology, including specialized modeling of the negative binomial probability distribution, higher-order asymptotics for improved inference from small samples, and graphical tools for communicating complex results.  In the first part, I will introduce some relevant statistical issues by recounting the early history of modern statistics starting with Darwin’s publication of On the Origin of Species.  By covering both early history and a very modern application, I wish to clarify the continuing role of the classical mathematical foundations of statistics but also illustrate the evolution of the field induced by new data-producing technologies and by ever-increasing computational speed.  About a third of the talk should be understandable to everyone and another third to those who have taken a course in probability and statistics. Fall breakTime: 4:10 PM Planets, Polynomials, and PolytopesMarshall Hampton, Department of Mathematics and Statistics, University of Minnesota, Duluth Time: 4:10 PMThe Newtonian N-body problem has a long and interesting history that connects many branches of mathematics. This talk will survey some highlights of this history, with an emphasis on some special types of orbits.  The study of these orbits leads into a beautiful combination of mathematics involving polynomials, polytopes, and "tropical" algebraic geometry. The Archimedes PalimpsestTom WietingTime: 4:10 PMOn Thursday, October 29, 1998, a thirteenth century prayer book -- composed on parchment, bound by wood, compromised by mold, and abused by time -- was sold for more than two million dollars to an anonymous collector, Mr. B, at Christie's auction house in New York.  One may reasonably ask for motivation.  In fact, this prayer book is a palimpsest: a new book prepared from an old book by scraping from the parchment pages the text of the old to make space for the text of the new.  In this case, the old book proved to contain tenth century copies of seven essays (two of which, The Method and the Stomachion, were otherwise unpreserved) by the most celebrated mathematician of ancient Greece: Archimedes of Syracuse.  The object of this lecture is to describe the history and the significance of the Archimedes Palimpsest, to review the work at the Walters Museum in Baltimore on the restoration and imaging of the old book, but then to illustrate the "method" of Archimedes, an ingenious intertwining of mechanics and mathematics, by which he discovered many of his beautiful theorems.} Different patients get the same cancer in different ways: Can statistics help define a clinical problem?Megan Othus, Biostatistics and Biomathematics group, Hutchinson Cancer Research Center Time: 4:10 PMFor most cancers there is a standard of care therapy, which is determined from clinical trials demonstrating that patients live longer when receiving that therapy at diagnosis. But standard of care therapy does not work for all patients. Some patients' cancer will continue to grow even while receiving treatment. Other patients' cancer will respond to the therapy for some time, but then the cancer will start to grow again.  Intuitively, these groups seem distinct, but many of those patients would be classified under one category called resistant.  The general consensus is that "resistant" patients should receive experimental theraphy, but the definition of resistance can vary widely between research groups.  We evaluated whether acute myeloid leukemia (AML) resistance can be defined empirically from data and compared various existing definitions of resistance in terms of clinical utility. Additionally, we assessed whether data can predict at the time of diagnosis if a patient will likely be resistant. Applied LogicsChristopher Stone, Department of Computer Science, Harvey Mudd CollegeTime: 4:10 PMComputer Science was born from logic; Alan Turing, Alonzo Church, and other logicians provided the theoretical underpinnings for computation, while Boolean logic underlies the digital hardware. The connections continue to this day, with a variety of different logics having direct CS applications. In this talk, I will describe some applications of Computer Science where less common logics make an appearance, including temporal logic (time), linear logic (resources), and constructive logic (computability). ThanksgivingTime: 4:10 PM Finite Graphs and Riemann Surfaces: Hurwitz groups and graphsScott Corry, Department of Mathematics, Lawrence UniversityTime: 4:10 PMOver the past 20 years, tantalizing analogies have emerged between the world of Riemann surfaces and the world of finite graphs. In this talk, I will provide an overview focused on symmetry groups. In particular, I will present genus bounds for the maximal size of the relevant symmetry groups in both contexts, and discuss Hurwitz groups for surfaces and their analogue for graphs: these are the largest groups that arise for surfaces/graphs of a given genus. I will conclude by presenting a new result demonstrating an unexpected direct connection between finite graphs and Riemann surfaces -- the analogy is deeper than expected.

#### Spring

4:40-5:30pm in Physics 123 (unless marked otherwise). Directions to Reed.

Jan 31 An Invitation to Algebraic StatisticsBernd Sturmfels, Department of Mathematics, UC BerkeleyTime: 4:10 PMStatistical models for discrete data are subsets of a probability simplex. Most models that are used in practice can be represented by polynomials. This lecture focuses on this polynomial representation and its applications. Topics to be covered include joint distributions, contingency tables, independence, mixture models, Markov bases, tensor rank, Bayesian integrals, and the hyperdeterminant. Intersecting loops on surfaces and string topologyKate Poirier, University of California, BerkeleyTime: 4:10 PMString topology is the study of algebraic structures arising from intersecting loops in manifolds. These structures encode interesting topological and geometric information about the underlying manifold.  The Goldman bracket is an example of a string topology operation which is given by intersecting loops on surfaces—manifolds of dimension two. In this talk, I will define the Goldman bracket for surfaces and introduce generalizations of it to string topology operations for manifolds of dimension three and higher. I will also formulate conjectures and describe work in progress concerning the richer algebraic structures we expect to see from generalized string topology operations. The Brouwer fixed point theoremAngĂ©lica Osorno, University of ChicagoTime: 4:10 PMThe Brouwer fixed point theorem is one of the most well-known theorems in topology, in part for the number of applications it has across various fields of mathematics, and in part because it can be proved in many ways.  The theorem states that a continuous function from a ball to itself has a fixed point. In this talk I will explain the theorem with examples, present applications, and introduce some concepts in algebraic topology in order to prove it. No prior knowledge of topology is expected. Curvature of shapes of dimensions one, two, and threeJeff Jauregui, University of PennsylvaniaTime: 4:10 PMA plane is flat, but most other surfaces you might think of—for instance, a sphere, a torus, or a paraboloid—exhibit a degree of curvature. My goal is to survey the foundations of how mathematicians have thought about the idea of curvature, going back to the time of Newton and Leibniz.  We will see a number of ways of understanding the Gauss curvature and will conclude with a glimpse of how curvature appears in our own universe. Generalizing a correspondence between permutations and tableaux to complex reflection groups $$G(r,p,n)$$Aba Mbirika, Bowdoin CollegeTime: 4:10 PMThe classical Robinson-Schensted algorithm establishes a bijection between permutations in the symmetric group $$\mathfrak{S}_n$$ and ordered pairs of same-shape standard Young tableaux of size $$n$$.  This map has proven particularly well-suited to certain questions in the representation theory of both $$\mathfrak{S}_n$$ and the semisimple Lie groups of type $$A$$.  For instance, Kazhdan-Lusztig cells as well as the primitive spectra of semisimple Lie algebras can be readily described in terms of images of this correspondence. Other sometimes more elementary representation-theoretic information requires more work to extract from standard Young tableaux.  For instance, in independent work, Reifegerste and Sjöstrand developed a method for reading the value of the sign representation of a permutation in $$\mathfrak{S}_n$$.  In this talk, we extend this result to the imprimitive complex reflection groups $$G(r,p,n)$$ via  a generalized Robinson-Schensted algorithm. Intrinsic knotting and linking in graphsDanielle O'Donnol, Imperial College London, EnglandTime: 4:10 PMA graph is a set of points and a set of edges between them. A spatial graph is an embedding of the graph in $$\mathbb{R}^3$$. A graph can be put into three-space in all different ways. A graph $$G$$ is called intrinsically knotted if every embedding of $$G$$ in $$\mathbb{R}^3$$ contains a nontrivial knot. I will talk about intrinsically knotting graphs, as well as intrinsically linked graphs and intrinsically $$n$$-linked graphs. We will look at results finding the smallest graphs that have these properties. Algebraic deformations of rational functionsKyle Ormsby, Massachusetts Institute of TechnologyTime: 4:10 PMGiven two rational functions of a single variable, when is it possible to algebraically deform one into the other?  In this talk, I will explain the meaning of this question, describe its connection to homotopy groups and degree in topology, and finally phrase an answer in the language of quadratic forms.  At the end of the talk, we will arrive at the perimeter of motivic homotopy theory and will only be a stone's throw away from several open questions in that field. No talk this weekTime: 4:10 PM Using Presence-Absence Data to Model the Range of the Southern Ground Hornbill in South AfricaKristin Broms, Quantitative Ecology and Resource Management, University of WashingtonTime: 4:10 PMUnderstanding the ranges and movements of species are centralquestions in ecology. Occupancy models may be used to derive thisinformation along with species-habitat associations fromdetection-nondetection (presence-absence) data. I will describe what anoccupancy model is and introduce additional methods to make themspatially explicit. The models will be applied to the Southernground-hornbill (Bucorvus leadbeateri) using data collected from theSouthern African Bird Atlas Project 2, an atlas project running from2007 to present day. An atlas project uses checklists submitted by"citizen scientists" to create a database of detections for all birdspecies throughout South Africa. The results of the analysis create anaccurate map of the ground-hornbill’s range that may inform on its IUCNVulnerable status and may help improve re-introduction efforts. Seeing Genome Evolution Through Computer ScienceJulian Catchen, Institute of Ecology and Evolution, University of OregonTime: 4:10 PMOur increasing understanding of the complexity and richness of biological systems presents an opportunity for a new synthesis that uses principles of computer science to decode biological signals in massive data sets. Using this strategy, we can ask the question: how do genomes change over evolutionary time? Is the state of a genome defined by large numbers of infinitesimally small changes occurring over ages, or, are there large, disruptive changes that in short periods of time erect barriers to recombination and facilitate speciation? I will address this question at several different time scales, examining the effects of a whole genome duplication in teleost fish over millions of years as well as the discovery of large structural variation affecting phenotypic evolution in stickleback fish over thousands of years and in some cases even in mere decades. I will highlight several major computational systems and algorithms, including the Synteny Database and Stacks, erected to address these questions. Finally, I will discuss future plans for an informatics system that tracks objects within genome assemblies to identify and retain variation known to exist across individuals and populations. Squeak Squeaker Squeakem Squeak, or, The Mathematics of Deciphering the Mouse-ese Language.Adam Smith, Department of Mathematical Sciences, Lewis and Clark CollegeTime: 4:10 PMMice are one of the most common model organisms used in biomedical research. They are highly-social creatures that make complicated ultrasonic vocalizations. Understanding these vocalizations better will be a boon for researchers, especially those who study mental diseases, since it will allow them to better identify a mouse's affective state. However, current techniques to identify andextract mouse vocalizations are labor-intensive, requiring a large amount of human time (and money), and slowing the pace of research. We shortcut this process by using hidden Markov models: both to distinguish bona fide calls from background noise, and to follow the frequency of a call over time. We have found that our method offers an improvement on previous automated methods. Statistical Modeling of Communication NetworksAmber Tomas, Mathematica Policy ResearchTime: 4:10 PMWithin many populations there are frequent communications between pairs of individuals.  Such communications might be emails sent within a company, radio communications in a disaster zone or diplomatic communications between states.  Often it is of interest to understand the factors that drive the observed patterns of such communications, or to study how these factors are changing over over time.  Communications can be thought of as events occurring on the edges of a network which connects individuals in the population.  In this talk I'll present a model for such communications which uses ideas from social network theory to account for the complex correlation structure between events.  Applications to the Enron email corpus and the dynamics of hospital ward transfer patterns will be discussed. Homology search, a database of repeats, alignment, and trees.Travis Wheeler, Howard Hughes Medical Institute, Janelia FarmTime: 4:10 PMThe comparison of biological sequences lies at the heart of computational molecular biology. Most often, these comparisons aim to (1) recognize the signal of shared evolutionary history (homology search), (2) recover the tree-like structure of that history (phylogeny inference), or (3) identify conserved blocks among homologous sequences (sequence alignment). I will describe work on all three, with an emphasis on homology search. In particular, I will discuss the probabilistic underpinnings of homology search, and the benefits of addressing homology search with profile hidden Markov models (HMMs), especially as relates to DNA-DNA search with the program nhmmer. I will also present an application of nhmmer to the annotation of repetitive DNA using a database of transposable element family profile HMMs, called Dfam. Spring breakTime: 4:10 PM A Bayesian model for cluster detectionAlbert Y. Kim, GoogleTime: 4:10 PMWe consider the problem of detecting clusters of non-infectious and rare diseases. Cluster detection is the routine surveillance over a large expanse of small administrative regions to identify individual hot-spots of elevated residual spatial risk without any preconceptions about their locations.  A class of cluster detection procedures known as moving-window methods superimpose a large number of circular regions onto the study area.  For each of these circles, the significance of the observed data are determined with respect to the hypothesis of no unmeasured risk inside the circle, as compared to outside the circle.  We propose a Bayesian model that incorporates prior knowledge while accounting for multiple clusters.  All posterior probabilities are estimated using Markov chain Monte Carlo and identify individual areas with high posterior probability of being in a cluster.  After defining the model, applications of the model to epidemiological datasets will be presented. Time: 4:10 PM Set Bands: Looking for an AxiomatizationLawrence Valby, UC BerkeleyTime: 4:10 PMMaybe you can help me. I'm looking for an axiomatization of a certain class of binary operations I've been calling "set bands". The elements of set bands are pairs $$(s,s')$$ of sets, and the operation is given by $$(s,s')(t,t'):=(s\cap t,(s'\cap t)\cup t').$$  Yalcin and Rothschild pointed out recently that the idempotent and commutative actions $$A\times B\to A$$ are exactly the ones that can be expressed using set intersection.  This was such a tidy characterization that I wondered what happened if you allowed not just intersection but also union into the mix. In the talk I'll explain the previous result, how this new question leads to set bands, and partial progress toward an axiomatization. The Three-Body Problem: Open problems, new solutions and the shape sphere of triangles.Richard Montgomery, UC Santa CruzTime: 4:10 PMNewton's $$N$$-body problem can be viewed as the positing of a dynamics on the space of congruence classes of $$N$$-gons.  We begin with a tour of some of its solutions, focusing on $$N= 3$$.  We then describe the shape sphere which is the moduli space of oriented similarity classes of triangles as a new tool for formulating and solving problems within the three-body problem.  This sphere can be viewed as sitting inside the shape space , a Euclidean $$3$$-space representing the moduli of oriented congruence classes of triangles.  We will illustrate how the shape sphere helps us to formulate good questions and occasionally suggests solutions to some of the open problems within the planar three-body problem. Finding patterns in randomness: A new perspective on the longest increasing subsequence problemAaron Abrams, Washington and Lee UniversityTime: 4:10 PMGiven a random permutation of $$1, \dots, N$$, how long is the longest subsequence that's monotonically increasing?  This turns out to be a hard problem, and decades of work went into solving and understanding it.  By now the answer is well-known:  the expected length is $$2\sqrt{N}$$, and the length is distributed according to something called the Tracy-Widom distribution, which first arose as the distribution of the largest eigenvalue of a random matrix.Now, given a random permutation of $$1, \dots, N$$, how long is the longest subsequence that alternates between increasing and decreasing?  This problem turns out to be much easier.  Recently, Stanley showed that this length is normally distributed around a mean of $$2N/3$$.What is the difference between these two problems?  And what happens when you look for other patterns inside random permutations?  This is what I will talk about. Tracing Global-Scale Information Flow using Internet Chain-Letter DataDavid Liben-Nowell, Department of Computer Science, Carleton CollegeTime: 4:10 PMAlthough information, news, jokes, and opinions circulate continuously in the worldwide social network, the actual mechanics of how a single particular "meme" spreads on a global scale have largely remained mysterious. A major challenge lies in the difficulty of acquiring large-scale data that records the diffusion of a single piece of information. In this talk, I will survey some recent computational research that studies information propagation through the digital traces of online activity. I will focus on joint work with Jon Kleinberg and Flavio Chierichetti that traces such information-spreading processes via the reconstruction of the propagation of two massively circulated Internet chain letters, one protesting the beginning of the Iraq war and one protesting budget cuts in U.S. governmental funding of public radio.