# Mathematics Department

## Colloquium

#### Upcoming Seminar

November 25, 4:10 PM in Physics 123
Markov Chain Monte Carlo Estimation for Bayesian Disease Cluster Detection
Albert Kim, Mathematics Department, Reed College

We consider the problem of detecting clusters of non-infectious and rare diseases.  A class of cluster detection methods known as scan statistics superimpose a large number of circles onto a study region.  For each of these circles, the significance of any excess of observed cases over what is expected is determined under the null hypothesis of equal risk of disease inside and outside the circle.  These circles form the basis of our Bayesian model which identifies individual areas with high posterior probability of being in a disease cluster.  The method incorporates prior information about disease clusters and accounts for multiple clusters simultaneously. After presenting an application of the method to brain and breast cancer in Western Washington state, we discuss the Markov Chain Monte Carlo method used to estimate all discrete posterior quantities.  In particular we focus on the issue of improving the efficiency of the algorithm, which is paramount when considering large study regions and when further extending the methodology to detect spatio-temporal clusters.

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. Secure Multiparty ComputationAdam Groce, Mathematics Department, Reed CollegeImagine two companies are trying to decide the merit of merging. Doing so requires comparing the books and client lists of the two companies, but both are reluctant to show this proprietary information to each other. Is there a way for them to do these complicated financial calculations without either seeing the other’s books? Secure multiparty computation (MPC) gives us a way to do exactly this sort of joint computation. In this talk we will see how it’s done. On the way, we’ll see how modern cryptography defines security for a problem like this and we’ll look at encryption, oblivious transfer, and other tools that are used in solving it. We’ll also take a quick step away from the theoretical work to look at how MPC results are being used in the real world. Please note change in date.Random multigraphs and a method for finding hidden relationships in networksJohn Rañola, Department of Biomathematics, University of California Los AngelesNetworks are often modeled by random graphs, a model where every pair of nodes is either connected by an edge or not. In Facebook and other social networks, random graphs are often formed by joining a pair of nodes, in this case people, if they are friends on the site. Although this is useful for some purposes, a lot of information is lost to the model. Attributes such as the number of messages or the number of “likes” exchanged can give more information than simply friends or not friends. In this talk, I will present a random multigraph model where every pair of nodes can be connected by multiple edges such as the number of messages exchanged. Under certain assumptions, I will show that the model can be used to find pairs of nodes that have relationships unlike the others. In the Facebook example, the special pairs of nodes may be best friends, family members, sweethearts, or something other than friends. I will present results from applying this model to various real data including neuronal connections, interacting genes in radiation hybrids, letter and word pairs in seven Shakespearean plays, and Fortune 500 companies connected by shared board members. Please note change in date.Predicting First Flowering Events Using Survival ModelsMaria Terres, Department of Statistical Science, Duke UniversityFirst flowering events in cherry trees are believed to be closely related to temperature patterns during the winter and spring months. Earlier works have incorporated the idea of temperature thresholds, defining chill and heat functions based on these thresholds. However, selection of the thresholds is often arbitrary and shared across species and locations. We propose a survival model with spatially and temporally varying covariates having functional forms representing chill and heat accumulation leading up to first flowering events. Thresholds are chosen utlizing the ranked probability scores, selecting the threshold pair that minimizes the difference between the predicted and observed cumulative probability curves. We first apply the model using temporally varying covariates to analyze 29 years of flowering data for four cherry species (Cerasus spp.) grown in Hachioji, Japan. This allows us to investigate whether relationship with temperature may vary between earlier and later flowering species. Next, the model is applied to 52 years of flowering data for 45 Cerasus spachiana × C. speciosa trees grown across Japan's Honshu Island using spatially and temporally varying covariates and spatial random effects. By exploring flowering dates across locations, we can explore how the relationship between temperature and first flowering events varies through space. Finally, these models are used to predict shifts in flowering due to climate change based on output from an ensemble of three regional climate models. Complications of Compliance in Causal InferenceDavid Watson, Department of Statistics, Harvard UniversityRandomized experiments are the gold standard for estimating causal effects of an active treatment compared to a control treatment.  However, complications of compliance often arise when subjects do not take the treatment assigned to them.  We consider the case of one-way compliance in which some subjects decline to take the active treatment when assigned.  In such cases, the standard "intention-to-treat" analysis is still justified, but results must be interpreted as a causal effect of assignment to treatment as opposed to receipt of treatment.  Of course, investigators primarily care about the effect of the treatment for subjects who would actually receive the treatment.  This idea motivates estimating the causal effect for a latent, or partially observed, subgroup of subjects called the compliers.  We describe how to estimate the complier average causal effect under certain assumptions and show how naive estimators based on the observed treatment received do not have a causal interpretation.  We use a clinical trial of an intervention for high cholesterol patients as a motivating example. Please note change in date.Markov Chain Monte Carlo Estimation for Bayesian Disease Cluster DetectionAlbert Kim, Mathematics Department, Reed CollegeWe consider the problem of detecting clusters of non-infectious and rare diseases.  A class of cluster detection methods known as scan statistics superimpose a large number of circles onto a study region.  For each of these circles, the significance of any excess of observed cases over what is expected is determined under the null hypothesis of equal risk of disease inside and outside the circle.  These circles form the basis of our Bayesian model which identifies individual areas with high posterior probability of being in a disease cluster.  The method incorporates prior information about disease clusters and accounts for multiple clusters simultaneously. After presenting an application of the method to brain and breast cancer in Western Washington state, we discuss the Markov Chain Monte Carlo method used to estimate all discrete posterior quantities.  In particular we focus on the issue of improving the efficiency of the algorithm, which is paramount when considering large study regions and when further extending the methodology to detect spatio-temporal clusters. Thanksgiving. No talk this week. Euler characteristic: an introduction to thinking topologicallyAnna Marie Bohmann, Mathematics Department, Northwestern UniversityYou may have encountered the concept of Euler characteristic when thinking about soccer balls or Euclidean solids. We'll discuss counting and Euler characteristic: counting points, counting potatoes, and counting holes.

#### Spring

4:10-5:00 in Physics 123 (unless marked otherwise). Directions to Reed.

Jan 29 TBAMartina Morris, Departments of Sociology and Statistics, University of Washington Design of 3D printed mathematical artHenry Segerman, Department of Mathematics, Oklahoma State University TBA TBA TBAWill Rosenbaum, Graduate Student and Teaching Assistant at UCLA TBA TBAMohamed Omar, Department of Mathematics, Harvey Mudd College Spring Break TBA TBAAna Rita Pires, Department of Mathematics, Fordham University TBA TBAGlencora Borradaile, School of Electrical Engineering and Computer Science, Oregon State University TBA TBA