Mathematics Department

Colloquium

Upcoming Seminar

March 23, 4:40 PM in P123
Diophantine and tropical geometry
David Zureick-Brown, Department of Mathematics and Computer Science, Emory University

Diophantine geometry is the study of integral solutions to a polynomial equation. For instance, for integers \(a,b,c \geq 2\) satisfying \(1/a + 1/b + 1/c < 1\), Darmon and Granville proved that the individual generalized Fermat equation \(x^a + y^b = z^c\) has only finitely many coprime integer solutions. Conjecturally something stronger is true: for \(a,b,c \geq 3\) there are no non-trivial solutions. 
I'll discuss various other Diophantine problems, with a focus on the underlying intuition and conjectural framework. I will especially focus on the uniformity conjecture, and will explain new ideas from tropical geometry and our recent partial proof of the uniformity conjecture.

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.

2016-17 Schedule

Fall

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

Sept 1Meeting with majors.
Sept 8Private Data Analysis
Adam Groce, Mathematics Department, Reed College
Today governments, universities, hospitals, and companies all maintain huge databases of private information. That information, if it could be analyzed, holds the key to a remarkable array of potential discoveries in medicine, social science, and other areas. However, the private nature of the data limits much of this analysis. The field of private data analysis seeks to provide tools for analyzing such data while protecting privacy. However, before such tools can be created, it is necessary to formally define the desired notion of "privacy." We will begin by talking about what definition is appropriate, and then we will look at how certain analyses can be carried out under the constraint of protecting privacy.  Finally, I'll give a brief overview of recent developments in this field and what sort of challenges remain to be solved.
Sept 15Set Theory via the Polynomial Method
Joe Buhler, CCR/Reed College
Sept 22A Curious Monoid
Keith Pardue, NSA
I will describe a curious monoid* \(U\) whose elements are certain trees with colored leaves. My collaborator and I discovered this monoid through a speculation in category theory, and we were surprised and enchanted by the rich combinatorial structure that we found. I'll finish with a basic question about \(U\) that we have not been able to resolve.

* A monoid is a set \(M\) with a binary operation \(M\times M\to M\) that is associative and has an identity. Elements of a monoid need not have inverses, as they must have in a group.
Sept 29Student research projects
Chris Keane and Ricardo Rojas-Echenique, Reed College

Chris Keane: A Combinatorial Model for Quantum Matrices

Abstract: A combinatorial model for the algebra of quantum matrices was discovered by K. Casteels in 2014. After explaining the construction of the model, we will explain its relevance to the study of the prime spectrum of the algebra of quantum matrices. Specifically, we will explain how the model gives a way to interpret ideal membership for a certain class of primes, and also present how this interpretation makes it possible to compute generating sets for this class of primes.

Ricardo Rojas-Echenique: Bijections Between the Recurrent Sandpiles of an Eulerian Digraph and its Reverse

Abstract: An Eulerian digraph is a directed graph where the indegree of each vertex is equal to the outdegree.  After designating a single vertex of an Eulerian digraph as a sink, the set of recurrent sandpiles on the graph forms a finite abelian group isomorphic to the cokernel of the Laplacian matrix of the graph. Given a digraph, we can also construct its reverse graph by reversing the direction of each edge. In the case of Eulerian digraphs we have the salient property that the Laplacian of the reverse graph is equal to the transpose of the Laplacian of the original graph.  This allows us to conclude the two groups of recurrent sandpiles are isomorphic.  While this isomorphism shows that the number of recurrent sandpiles on an Eulerian digraph and its reverse are equal, the two sets of sandpiles can look quite different. We discuss a few possible natural bijections between the recurrent sandpiles on an Eulerian digraph and its reverse.

Oct 6Paramodularity computations for squarefree level
Jerry Shurman, Department of Mathematics, Reed College
The Paramodular Conjecture of Brumer and Kramer is a degree \(2\) modularity conjecture.  The analytic space in the conjecture is \({\mathcal S}_2(\operatorname{K}(N))\), the space of weight \(2\), level \(N\) Siegel paramodular cusp forms.  Earlier computational work confirmed the conjecture for prime \(N\) up to \(600\), and now new work has confirmed it for nonprime squarefree \(N\) up to \(300\). The interesting new cases are \( N=249, 295\), where one form exists beyond the additive (Gritsenko) lift space of the Jacobi form space \(\operatorname{J}_{2,N}^{\rm cusp}\), as predicted by the existence of suitable abelian surfaces for those \(N\).
Oct 13
Cancelled
MNS/STEM
TBA
Oct 20Fall break
Oct 27Tableaux, Hives, and Integer Triangles: Playing with puzzles to solve difficult problems
Natalie Hobson, Department of Mathematics, University of Georgia
Do you think solving number puzzles like Sudoku and Kenken is done just for fun? Well mathematicians have used three types of puzzle games similar to these to solve many difficult problems. Some of these problems have been open for over fifty years and still intrigue many. In this talk, we will learn how to play these puzzles, discover their beauty and structure, and investigate remaining open questions they might help us solve. 
Nov 3Introduction to algebraic K-theory
Mona Merling, Department of Mathematics, Johns Hopkins University
Algebraic K-groups are deep invariants of rings, which  hide beautiful patterns and connections to problems in number theory. Lower K-groups have explicit algebraic descriptions, but higher algebraic K-groups require sophisticated topological and categorical machinery to define, and their introduction by Quillen was the culminating point of a long search for a definition that would meaningfully generalize the existing definitions of lower K-groups. In this talk, I will give a flavor of algebraic K-theory and its history, touching on current research.
Nov 10Hopping Particles And Rhombic Tableaux
Olya Mandelshtam, Department of Mathematics, UCLA

The asymmetric simple exclusion process (ASEP) is a model from statistical physics that describes the dynamics of particles hopping right and left on a finite 1-dimensional lattice. Particles can enter and exit at the left and right boundaries, and at most one particle can occupy each site. The ASEP has been well-studied, for instance as a model for processes such as protein synthesis, molecular and cellular transport, and traffic flow. The ASEP is remarkable in that it is one of very few non-equilibrium processes for which there exist exact formulae for its stationary distribution. Moreover, it displays rich combinatorial structure: one can compute the stationary probabilities for the ASEP using fillings of certain tableaux. I will discuss some of the combinatorial results from the past decade, including recent developments.

Nov 17Modeling perceptual invariances in biological sensory processing
Alexander Dimitrov, Department of Mathematics and Statistics, Washington State University, Vancouver
The sense of hearing is an elaborate perceptual process. Sounds reaching our ears vary in multiple features: pitch, intensity, rate. Yet when we parse speech, our comprehension is little affected by the vast variety of ways in which a single phrase can be uttered. This amazing ability to extract relevant information from wildly varying sensory signals is also ubiquitous in other sensory modalities, and is by no means restricted only to human speech. Even though the effect itself is well characterized, we do not understand the approaches used by different neural systems to achieve such performance.
 
In an ongoing project, we are testing the hypothesis that broadly invariant signal processing is achieved through various combinations of locally invariant elements. The main questions we would like to address are: 1. What are the characteristics of locally-invariant units in auditory pathways? 2. How are biological locally-invariant units combined to form globally invariant processors? 3. What are the appropriate mathematical structures with which to address and model these sensory processes?

The mathematical aspects of the research involve an interesting combination of probability theory (a must in the study of biological sensory systems) and group theory, needed to characterize invariants and symmetries. The combination defines the concepts of a probabilistic symmetry, and expands the scope of probabilities on group structures, originally introduced by Grenander. 
Nov 24Thanksgiving
Dec 1Inverse problems: when what we measure is not what we want
Doug Nychka, National Center for Atmospheric Research
Location: Psychology 105
What will the weather be like tomorrow?  What is the density of the Sun's atmosphere?  How much carbon dioxide was in the atmosphere 500 years ago?  Answers to these questions rely on using measurements that are not direct observations of the quantities of interest – a common problem in science and engineering. A powerful way to solve these problems is to formulate a statistical model that relates the unknown variables to what we observe and then find the solution that is most likely given the observations at hand. As a specific example this lecture will explain how to reconstruct the annual record of CO2 in the atmosphere from an Antarctic ice core. An important result of this analysis is not only being able to estimate these concentrations but also attach error estimates to them. This example also serves as a gentle introduction to Bayesian hierarchical models and their connection with regularization for solving mathematical inverse problems.

Spring

4:40-5:30pm in P123 (unless marked otherwise). Directions to Reed.

Jan 26Equivocating Yao: Constant-Rounds Adaptively Secure Multiparty Computation in the Plain Model
Muthuramakrishnan Venkitasubramaniam, Department of Computer Science, University of Rochester
Yao’s circuit garbling scheme is one of the basic building blocks of crytographic protocol design. Originally designed to enable two-message, two-party secure computation, the scheme has been extended in many ways and has innumerable applications. Still, a basic question has remained open throughout the years: Can the scheme be extended to guarantee security in the face of an adversary that corrupts both parties, adaptively, as the computation proceeds?

We provide a positive answer to this question. We define a new type of encryption, called functionally equivocal encryption (FEE), and show that when Yao’s scheme is implemented with an FEE as the underlying encryption mechanism, it becomes secure against such adaptive adversaries. We then show how to implement FEE from any one way function.

Combining our scheme with non-committing encryption, we obtain the first two-message, two-party computation protocol, and the first constant-rounds multiparty computation protocol, in the plain model, that are secure against semi-honest adversaries who can adaptively corrupt all parties. We also provide extensions to the multiparty setting (with UC-security) and applications to leakage resilience.
Feb 2An Introduction to the Geometric Group Theory
Mark Sapir, Department of Mathematics, Vanderbilt University
Geometric group theory studies relationships between group
theory and geometry. I am going to talk about how to use continuous objects (weird metric spaces) to get information about discreet objects (finitely generated groups).
Feb 9Accessing Hidden Populations Using Respondent-Driven Sampling
Katherine McLaughlin, Statistics Department, Oregon State University

Respondent-driven sampling (RDS) is a network sampling methodology used worldwide to sample key populations at high risk for HIV/AIDS who often practice stigmatized/illegal behaviors and are not typically reachable by conventional sampling techniques. In RDS, study participants recruit their peers to enroll, resulting in a sampling mechanism that is unknown to researchers and not ignorable. Additionally, units in the sample are not independent because of potential homophily in both the underlying social network and the recruitment process. In this talk, I provide an overview of the RDS methodology, discuss and analyze current methods to compute the sample weights, and provide examples from key populations in Morocco. I critique assumptions that are required about both the population and sampling process for current methods, and introduce the two-sided rational-choice preferential recruitment (RCPR) model, which relaxes the assumption that people recruit uniformly at random from their social network. I develop inference for this model within a Bayesian framework using a form of constrained Metropolis-Hastings. This framework results in a tractable generative model for the RDS sampling mechanism, enhancing both design-based and model-based inference.

Feb 16Partial differential equations in General Relativity
Mihai Tohaneanu , Department of Mathematics, University of Kentucky
This talk is meant to be a gentle introduction to the theory of partial differential equations, and some of its applications to General Relativity. In particular we will discuss Einstein’s Equations and the (still open) problem of stability of black holes.
Feb 23Magic Squares
Anurag Singh, Department of Mathematics, University of Utah

A magic square, for us, is a matrix with nonnegative integer entries such that each row and column has the same sum, called the line sum.  Set \(H_n(r)\) to be the number of \(n\times n\) magic squares with line sum \(r\).  A conjecture of Anand, Dumir, and Gupta, proved by Stanley, states that \(H_n(r)\) is a polynomial in \(r\) of degree \((n-1)^2\).  We will explain how results such as this are proved using commutative algebra.

Mar 2Security of Feistel Networks
Aishwarya Thiruvengadam, University of Maryland, Department of Computer Science
Block ciphers are cryptographic primitives used extensively in many security applications. A well-known approach to building practical block ciphers is to use a Feistel network, an iterated structure built on key-dependent, “random-looking” round functions. In this talk, I will outline a (strong) security notion for block ciphers known as indifferentiability and show that a 10-round Feistel network using random, independent keyed round functions satisfies this notion. This is joint work with Dana Dachman-Soled and Jonathan Katz at the University of Maryland.
Mar 9Unbounded Solutions to Degenerate Elliptic Equations
Lyudmila Korobenko, UPenn/Reed College

Subject of elliptic equations is well studied and has a long history. Of particular interest are properties of solutions to such equations. Roughly speaking there are two approaches: to define solutions a priori as “nice” regular functions (e.g. twice differentiable), or to try to make sense of the equation in the weakest possible sense which leads to the definition of weak solutions.

The advantage of the latter approach is that it is relatively easy to establish existence and uniqueness of weak solutions, but then one has to struggle to show regularity of these functions. When the operator is strictly elliptic, i.e. the eigenvalues are strictly positive, it is known that weak solutions are Hölder continuous \((C^{\alpha}\)), and in general this regularity cannot be improved. If on the other hand the operator degenerates at some point, i.e. the eigenvalues vanish, weak solutions may loose Hölder continuity and even become unbounded.

I will discuss some known regularity results for degenerate elliptic equations and present our latest local boundedness result which is sharp in 3 dimensions.
Mar 16Spring break
Mar 22The Six Types of Data Journalism Stories
Andrew Flowers, fivethirtyeight.com
Time: 6:30 PM
Location: Vollum Lecture Hall

Data journalists write stories with data and find stories in data. At FiveThirtyEight, data journalists' preferred tools are R and the tidyverse packages. R is used to find stories through exploratory analysis, web scraping, and statistics; then, to tell stories, we leverage R’s powerful chart and interactive packages (like ggplot2). This talk will profile the six types of data journalism stories, motivated by novelty, outlier, archetype, trend, debunking, and forecast. As examples, several FiveThirtyEight stories or projects that used R extensively will be profiled, covering topics such as p-hacking in nutrition studies, the history of police officer shootings, how Uber is affecting New York City taxis, and what local characteristics explained Donald Trump winning the Presidency.

Andrew Flowers is a journalist and data scientist. He was previously the quantitative editor and economics writer for FiveThirtyEight.com, Nate Silver's data-driven news website. Before that, he was an economic analyst at the Federal Reserve Bank of Atlanta. He graduated with a B.A. in economics from the University of Chicago. He is passionate about R, open source software, politics, and basketball.

Sponsored by the Math Department and the Portland R User Group.

Mar 23Diophantine and tropical geometry
David Zureick-Brown, Department of Mathematics and Computer Science, Emory University
Diophantine geometry is the study of integral solutions to a polynomial equation. For instance, for integers \(a,b,c \geq 2\) satisfying \(1/a + 1/b + 1/c < 1\), Darmon and Granville proved that the individual generalized Fermat equation \(x^a + y^b = z^c\) has only finitely many coprime integer solutions. Conjecturally something stronger is true: for \(a,b,c \geq 3\) there are no non-trivial solutions. 
I'll discuss various other Diophantine problems, with a focus on the underlying intuition and conjectural framework. I will especially focus on the uniformity conjecture, and will explain new ideas from tropical geometry and our recent partial proof of the uniformity conjecture.
Mar 30Airport Bags and Contact Geometry
David Ayala, Montana State University
The set of all arrangements of an airport bag (you know, those bags with two wheels) in an airport is simple enough to describe.  However, the mechanics of how its wheels move put a restriction on the allowable paths in this set.  This infinitesimal constraint is an instance of a contact structure.  In this talk we'll conjure a mathematical articulation of this structure.  We'll go on to discuss other natural examples coming from familiar situations and from robotics.  We'll go farther and witness how, in the presence of such a structure, spaces are effectively connected graphs with uncountably many vertices and edges.  
Apr 6TBA
Apr 13TBA
Richard Moy, Department of Mathematics, Willamette University
Apr 20TBA
Jayadev Athreya, Director Washington Experimental Mathematics Lab, Department of Mathematics, University of Washington
Apr 27TBA
Clark Barwick, Department of Mathematics, MIT

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