Mathematics Department


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.

2017-18 Schedule


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

Aug 31
Please note change in location.
Meeting with majors
Location: Psychology 105
Sept 7Lattices, finite subsets of the circle, and the trefoil knot
Kyle Ormsby, Reed College
I will explore a remarkable correspondence between
  • lattices in the complex plane (up to homothety),
  • nonempty subsets of the circle with at most three elements, and
  • the complement of the trefoil knot in the 3-sphere.
Along the way, we will encounter modular forms, Voronoi cells, and open book decompositions.  The talk should be accessible to anyone who knows how to multiply complex numbers.
Sept 14How to Win the Lottery
David Roe, Department of Mathematics, University of Pittsburgh
From 2005 through 2012, three groups in Massachusetts earned millions of dollars playing a lottery game called Cash Winfall.  I'll describe how the game worked, explain why it was exploitable, give connections with projective geometry and share stories from my participation in one of the pools.  You'll also learn why almost every other lottery is not worth playing.
Sept 21 A New Approach to Euler Calculus for Continuous Integrands
Carl McTague, University of Rochester

The Euler characteristic satisfies an inclusion-exclusion principle χ(X∪Y)=χ(X)+χ(Y)-χ(X∩Y), which lets one regard it as a measure – a peculiar one where a point has measure 1 while a circle has measure 0. One can use it to integrate simple functions (in the sense of measure theory), and the resulting integral calculus has deep roots in algebraic geometry and has recently found surprising applications to data analysis. However, since it is only finitely – not countably – additive, it does not fit into the framework of Lebesgue integration, and there are problems integrating even the most elementary non-simple functions. I will describe a new approach to this calculus, based on differential geometry, which makes it possible to integrate a large class of non-simple functions, and which hints at new ways to apply differential geometry to data analysis.

Sept 28
Bicolored Trees, Shabat Polynomials and Monodromy Groups Uniquely Determined by Passport
Naiomi Cameron, Lewis & Clark College
Location: Physics 123
In this talk, I will discuss the results of an undergraduate research project focused on so-called dessin d'enfants.  Roughly speaking, a dessin d'enfant (or dessin for short) is a bicolored graph embedded into a Riemann surface.  Dessins which are trees can be associated analytically to pre-images of certain polynomials, called Shabat polynomials, and also algebraically to their monodromy group, which is the group generated from rotations of edges about its vertices.  One known invariant for dessins is called the passport, which is the multiset of the degrees of its vertices and faces.  The focus of this project was to determine the Shabat polynomials and monodromy groups for trees uniquely determined by their passport.
Oct 5 Homotopy Types as a Foundation for Mathematics
Emily Riehl, Johns Hopkins University

The Curry-Howard correspondence formalizes an analogy between computer programs and mathematical proofs. This talk will introduce alternative foundations for mathematics animated by this analogy. The basic object is called a type, which can be simultaneously interpreted as something like a set or as something like a mathematical proposition. Homotopy type theory refers to the recent discovery that a type can also be interpreted as something like a topological space. We will discuss the implications of this homotopy theoretic interpretation for the so-called univalent foundations of mathematics.

Oct 12Classifying Manifolds: Applications of Topology to Geometry and Physics
Christine Escher, Oregon State University
A fundamental and deep problem in mathematics is a classification of the objects of study: which objects are the same, which are different.  The tools for the classification of this talk come from algebraic topology but the interest and motivation for the classifications come from differential geometry and theoretical physics.   I will give an overview over which objects we study and what we mean by "the same" and "different".   In particular, I will discuss the classification of Euclidean spaces and spheres.
Oct 19Fall Break
Oct 26Azumaya Algebras: Bridging Algebra and Topology
Ben Williams, University of British Columbia
Since the discovery of Quaternions by Hamilton in 1843, noncommutative division algebras have been a topic of study in algebra. In the mid-20th century, they were generalized considerably by several people, culminating in an abstract and general definition of "Azumaya algebra" by Grothendieck in 1968. In this formulation, certain algebraic problems may admit solutions by geometric, or strictly speaking, topological, methods. I will explain what the algebraic terms mean, how Grothendieck's idea works and why it is ingenious, and how you can use it to solve problems.
Nov 2Melting of Ice, Obstacles and Walls: An Introduction to Free Boundary Problems
Mariana Smit Vega Garcia, University of Washington
Free boundary problems arise naturally in a number of physical phenomena and deal with solving partial differential equations (PDEs) in a domain Ω, a part of whose boundary is unknown in advance. A famous example of a free boundary problem is the Stefan problem, which describes the temperature distribution in a medium undergoing a phase change, for example melting of ice. In this talk we will see examples of free boundary problems, questions people are interested in answering, and some techniques used in the field to address those questions. No prior knowledge of Partial Differential Equations will be assumed.
Nov 9
Please note change in location.
Bringing Data Science to Government Surveys: Incorporating New Data Sources into Survey Estimation
Kelly McConville, Swarthmore College
Location: Physics 123

Government agencies across the world are estimating important population quantities using data collected under a complex sample design.  With the advent of big data and advancements in technology, a wealth of additional data sources, such as remotely sensed data and administrative data, are available.  In this talk, I present a class of survey estimators that seek to combine existing survey data with these new sources of data, often using machine learning methods to do so.  I then examine the utility of these modern techniques in estimating official statistics.

Nov 14
Please note change in date.
Please note change in location.
The Social and Demographic Dimensions of an Online Fitness Community
Zack Almquist , University of Minnesota
Location: Physics 123

Increased attention is being paid to the promotion of healthy habits via mobile applications, which foster online fitness communities and offer users the ability to track their physical fitness. At the intersection of social media and activity tracking, these platforms represent affordable, scalable technologies for capturing information about both physical activity and peer-to-peer interactions. Importantly, the large-scale behavioral trace data archived by these platforms offers researchers a novel and powerful opportunity to examine health-related behaviors. In this study, we explore the characteristics and dynamics of social exercise (i.e. fitness activities with at least one peer physically co-present), using data collected from an online fitness community popular with cyclists and runners. Our analysis of the personal networks of the online fitness users uncovers a strong gender-based homophily between users, as well as distinct generative processes for network formation across genders. We further examine how performance and social feedback vary by the number/gender of people participating in an activity; our results indicate that when peers are physically co-present for fitness activities (i.e. when users participate in group workouts) (i) exercise tends to be of higher performance (e.g., longer distance, higher heart rate), and (ii) users receive more online feedback from other users. These effects of co-present activity partners are true for both men and women, though we do find that women are proportionately more likely to engage in group activities. Altogether, these results improve our understanding of how technological solutions such as mobile apps and online communities may be utilized in developing affordable and large-scale health intervention strategies.

Nov 16
Please note change in location.
Genotyping Polyploids
David Gerard, University of Chicago
Location: Physics 123
Modern genomics has revolutionized how we answer questions about evolution, population dynamics, medicine, and plant and animal breeding. To answer these questions, we must first be able to detect and quantify (or "genotype") differences in individual genomes. Many scientists have used next generation sequencing technologies to genotype diploid individuals (those with two copies of their genomes). However, methods to genotype polyploids (those with more than two copies of their genomes) are just emerging. We present two main contributions: (1) many datasets feature related individuals, and so we use the structure of Mendelian segregation to borrow strength between polyploid siblings to improve genotyping; (2) we additionally draw attention to and then model common aspects of next generation sequencing data: sequencing error, allelic bias, overdispersion, and outlying observations. We verify our method in simulations and apply it to a dataset of hexaploid sweet potatoes.
Nov 21
Please note change in date.
Please note change in location.
Too many parameters! Covariance, cancer, and cells.
Shannon McCurdy, University of California, Berkeley
Location: Physics 123
Covariance, a measure of the joint variability of random variables, is an important quantity in statistics.  Estimating covariance is challenging in the high dimensional setting, where the number of features is much greater than the number of samples.  I will introduce approaches for estimating covariance and discuss applications to biomedical and biological data: cancer genomics and mouse brain cell gene expression.
Nov 23Thanksgiving Break
Nov 28
Please note change in date.
Please note change in location.
Personalism and Dempster-Shafer Analysis for the 21st Century
Paul Edlefsen, Fred Hutchinson Cancer Research Center
Location: Physics 123
Personalism, like its more familiar cousins objectivism and subjectivism, is a perspective on the role of the statistician (or more generally, the scientist: "you") in the conduct of science. The differences among these perspectives usually impact little on the daily conduct of statisticians, to our great relief. We are already sufficiently aware of our expected role on a paper or grant to ensure that the science is reproducible. However we are also not unaware of the several difficult issues that we face in statistical science including our contribution to the too-often disappointing performance of phase 3 trials, which results in a high cost for the biomedical enterprise and possibly in lost opportunity for quality human living. There are a host of thorny issues we must face, including how to maximize information yield from research investments while accounting for multiple testing and post-hoc inference. Furthermore, "p>>n" problems (high dimensional covariates with low numbers of observations) arise increasingly often in clinical trials. Can "Fisher's Greatest Blunder" (which is how Brad Efron described Fisher's "Fiducial" methodology) possibly offer an insight into a solution to all of these problems? I will begin by introducing the personalist perspective on the role that "you" play in the scientific process and I will briefly describe Dempster-Shafer (DS) methodology for statistical inference and prediction. I will argue that DS potentially offers a path toward a truly cohesive statistical inference framework for the 21st century.
Nov 30Garbled Circuits: New Results for Arithmetic and High Fan-In Computations
Mike Rosulek, Oregon State University
Secure computation is a cryptographic technique in which parties can perform a computation on private data to learn only the outcome of the computation while revealing nothing about the inputs. One of the most promising approaches for secure computation is a technique called *garbled circuits*, which can be thought of as a way to "encrypt" a boolean circuit in a particular way.
In the first half of this talk I will give an introduction to garbled circuits. Standard constructions for garbled circuits are only feasible for computations expressed in terms of low fan-in, boolean (bit-level) operations. However, boolean circuits are often a poor model in which to express many computations of interest. I will spend the second half of the talk presenting new garbled circuit constructions that natively support arithmetic operations directly over the integers and high-fan-in boolean operations. These new constructions offer exponential cost improvements over standard boolean circuits, for some natural kinds of computations. These results are joint work with Marshall Ball & Tal Malkin.


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

Jan 25 Lambda Rings, Polytopes, K-theory: Surprising Structures in Mathematics
Jonathan Campbell, Vanderbilt University
In this talk I'll define define a lambda ring, a mathematical structure that encodes the sort of exterior powers one sees in, for example, differential forms. I'll also discuss the philosophy of algebraic K-theory which is a sort of way of remembering how various mathematical structures "break into pieces".  Finally, I'll venture in to how the structure of lambda rings popped up in my research in an unexpected place: the algebraic K-theory of polytopes, also known as the scissors congruence problem or Hilbert's Third Problem.
Feb 1Fixed point theory, the Euler characteristic, and the syntax of strings
John Lind, Reed College
A solution to the equation f(x) = x is called a fixed point.  At first, the task of determining fixed points seems like a purely algebraic problem, but if the function f is defined on an object with interesting geometry, then topological tools can be used.  I will explain how the Euler characteristic and its generalizations give insight into the study of fixed points.  Along the way, we will use diagrams of strings to encode calculations in linear algebra.
Feb 8The four color theorem and beyond!
Emily Peters, Loyola University Chicago
The famous four color theorem says that, if I need to color a map so that any two countries that share a boundary are different colors, I can do it using (at most) four colors.  The four color theorem was a conjecture for almost a hundred years, and was finally proved in 1976 by Appel and Haken.  Because their proof relied heavily on computer assistance, it was controversial at the time.  This controversy moved the focus away from the interesting mathematical steps that occur before a computer steps in.  In this talk, I plan to focus on that aspect of their proof, and recent applications of these ideas in other settings.
Feb 15
Please note change in location.
Color Space
Tom Wieting , Reed College
Location: Physics 123
What is Color: a property intrinsic to material bodies; a property inherent in light itself; a state of the brain; of the mind?  In any case, “Color” is a noun.  But, then: to what does it refer?  The object of this lecture is to apply the relentlessly precise syntax of Mathematics to produce, from the space L of Physical Lights, an abstraction: the space C of Perceptual Lights, that is, Color Space.  In due course, we will encounter the Color Cone, the linear and metric structures on the cone, coordinates for the cone, the International Commission on Color (CIE), the Color Cube, ……  We require only the methods of linear algebra. We offer nothing new, really, but we offer a tidy, easily remembered diagram, as a point of entry to a complex subject. Finally, Ben Salzberg and I will (attempt to) reconstruct, with slides and filters, the perplexing experiment on color vision by Edwin Land (1950s).  
Feb 22 Modeling and Solving Problems in Last Mile Delivery
Oliver Lum, University of Maryland
Whenever FedEx or Amazon deliver a package to you, they have to solve a vehicle routing problem (VRP) to facilitate last mile distribution.  VRPs are generalizations of the traveling salesperson problem, so they are known to be NP-hard, yet these companies have to solve very large instances of these problems.  We shall discuss some of the approximation algorithms and metaheuristics that are used in practice to address these problems, and conclude with a preview into what the research community predicts will be the future of delivery. 
Mar 1From Twisty Puzzles to Group Theory
Igor Kriz, University of Michigan
While discussing a "zero input" recipe for solving Rubik's cube (and just about any other "twisty puzzle" one can buy), we will learn about some fundamental concepts of group theory, the mathematical study of symmetry. We will then demonstrate a puzzle called the Number Planet which is different because its group theory is different. Following its lead, we will go on to talk about the amazing sporadic finite simple groups, including "the Monster."
Mar 8Symmetries in Algebra: The What? The How? And the Why Care?
Chelsea Walton, Temple University & University of Illinois Urbana-Champagne
Symmetry is one of the oldest notions in mathematics and one of the first mathematical concepts that we learn as children. The goals of this talk are to make the notion of symmetry precise in terms of abstract algebra, to relate this definition with various occurrences of symmetry in everyday life, and to touch upon the types of problems I like to think about in my research in noncommutative algebra. The talk will be down-to-earth. Questions and follow-up emails are strongly encouraged.
Mar 15Spring Break
Mar 22Conjectures on the local and global dynamics of rational functions
David Krumm, Colby College
In the field of Arithmetic Dynamics there is an important open problem known as the Uniform Boundedness Conjecture. Despite many efforts over the last two decades, even the simplest case of the conjecture has not been proved. In this talk we will introduce the general conjecture and then discuss an ongoing research program which aims to tackle a strong form of the simplest case of the conjecture by using tools from Galois theory and algebraic geometry.
Mar 26
Please note change in date.
Please note change in time.
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Counting Parameters: The Essentials of Essential Dimension
Rebecca Black , University of Maryland
Time: 4:30 PM
Location: Physics 123
Many motivating questions in algebraic geometry come from the quest to describe the complexity of geometric objects using algebraic invariants (otherwise known as…numbers). A fundamental measure of complexity is dimension, a number that captures how many independent parameters are needed to specify a point of a geometric object. The notion of dimension gets a bit subtler if we’re interested in the complexity of objects imbued with certain algebraic structures or specific types of symmetry. For example, how many parameters do I need to define a “generic” degree n polynomial, if what I care about is the algebraic structure of the solution set (so I’m allowed to make certain changes of variables without fundamentally changing the properties of the polynomial)? I’ll answer that question for small n (spoiler: in general we don’t know!) and describe how that question connects to the symmetric group Sn of permutations of n elements. This computation is an example of “essential dimension,” which provides one way to describe the inherent complexity of classes of objects that (like solutions to polynomials) sit at the intersection of algebra and geometry.
Mar 27
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Please note change in location.
Why I learned to Love Undefined Behavior in C++
Erich Keane , Intel
Location: Psychology 105
In this talk I’ll concentrate on some of the commonly criticized forms of Undefined Behavior in C++, and explain both the original justifications as well as the current benefits for optimization and machine flexibility.  Even though things like well-defined signed overflow seem to make sense, the performance implications of doing so are significant!  I’ll cover this and a few other examples in this talk.
Mar 29Noether’s problem
Asher Auel, Yale University
In 1913, Emmy Noether posed a problem relating the invariant theory of finite groups to Galois theory.  In the 105 years since, her problem has generated an incredible amount of diverse research activity, but still remains largely unsolved.  I’ll introduce Noether’s problem, review some of its history and relationships with other problems in number theory, topology, and algebraic geometry, and also highlight some of its interesting open cases, for example, the alternating group A_6.
Apr 5The underlying topology of data
José Perea, Michigan State Universtiy

Topology, and particularly algebraic topology, seeks to develop computable invariants that describe the overall shape of abstract spaces. This talk will be about how such invariants (e.g., the Euler Characteristic and the Homology of a space) can be used to analyze scientific data sets. I will use several examples to illustrate how these ideas have real applications.

Apr 12
Please note change in location.
Exponential tails for the Boltzmann equation
Maja Taskovic, University of Pennsylvania
Location: Physics 123

The Boltzmann equation models the motion of a rarefied gas in which particles interact through binary collisions. Instead of tracking each particle separately, the Boltzmann equation describes the evolution of the particle density function. The focus of this talk will be the tail behaviour of such density functions (how they behave for large velocities). We will show that they have exponential decay. An interesting aspect of our result is in the use of Mittag-Leffler functions, which are a generalization of exponential functions. This is based on joint works with Alonso, Gamba and Pavlovic.

Apr 17
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Please note change in location.
A Quick Look at Speedy Software
Scott Meyers
Location: Psychology 105
Everybody wants fast computers, but that simple desire has complicated consequences. Hardware engineers bend over backwards to create fast machines, but getting the most out of them requires contortions by software developers. This engaging presentation jaunts through the Computer Science speed-related landscape, with featured stops at hardware caches, human behavior on the web, algorithmic complexity, the Spectre security vulnerability, and why the quest for faster systems will never end.
Apr 19Integer Points in Polytopes
Federico Castillo , University of California, Davis
The set of integer points in a polytope comes up in real life applications (scheduling, matching) and pure areas of math (Littlewood-Richardson coefficients). We study the question of enumeration. This is an algorithmically hard problem and the theory behind uses different areas of math. The main point of the talk is to show how abstract ideas from algebraic geometry can help us to understand this particular problem. We focus on a some local formulas that relates the number of integer points with the volumes of the faces. 
Apr 20
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Breaking the Code: Cryptography and Computing in World War II
Eric Roberts, Stanford University
At the height of the bombardment of Britain, a team of mathematicians and engineers at Bletchley Park was able to crack both the Enigma code used by the German military and the more sophisticated Lorenz cipher used by the diplomatic corps. Winston Churchill later wrote that the work at Bletchley was the key to victory in the Battle of the Atlantic and had shortened the war in Europe by two years. In my talk, I will describe the challenges the code breakers faced and show how electromechanical devices, designed in part by the pioneering computer scientist Alan Turing, were able to overcome those challenges to decrypt messages that the Germans regarded as unbreakable. I will also describe other projects undertaken by the Bletchley team, both during and after the war, that anticipated important ideas in modern computing, such as public-key encryption and large-scale statistical analysis of data.
Apr 26Atoms in Mathematics: A Brief Introduction to Factorization Theory
Ranthony A.C. Edmonds, University of Iowa
In the same way that atoms are the building blocks of ordinary matter, prime numbers are the building blocks of the integers. In fact, the fundamental theorem of arithmetic tells us that any integer can be factored as the product of prime numbers, and most importantly this factorization is unique. In this sense, the prime numbers could also be called the irreducible elements, or atoms, of the integers. In this talk, we extend the notion of an atom and unique factorization from the integers to an abstract structure called a ring. We will also discuss visualizing atoms with what is called an irreducible divisor graph, a concept from a booming new area of research that lies at the intersection of abstract algebra and graph theory.
Apr 30
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Formal verification of compilers in the presence of shared memory
Santiago Cuellar , Princeton
Time: 5:00 PM
In a world that is increasingly controlled by code, software should be verifiably secure. Even safe programs, however, can be dangerous if the compiler introduces bugs into the code. This talk will introduce the ideas behind formal software verification: the act of proving that software does what it is supposed to do using formal mathematics. It will then present key principles to reason formally about concurrent programs, and a strategy to prove that a compiler transforms those programs correctly.
May 4
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Automated Question Answering
Mark Hopkins, Allen Institute for Artificial Intelligence
Time: 5:00 PM
I will briefly discuss recent progress in the field of automated question answering and how Project Euclid at the Allen Institute of Artificial Intelligence fits into the bigger picture. Then, in order to provide an example of the kind of research problem that might arise in the field, I will take an in-depth look at a subproject about using machine learning techniques to improve the performance of statistical English parsers on technical language. Finally, I will give an overview of open research questions that interested students might work on.