Mathematics Department

Colloquium

Upcoming Seminar

January 26, 4:40 PM in P123
TBA
Muthuramakrishnan Venkitasubramaniam, Department of Computer Science, University of Rochester

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 26TBA
Muthuramakrishnan Venkitasubramaniam, Department of Computer Science, University of Rochester
Feb 2TBA
Mark Sapir, Department of Mathematics, Vanderbilt University
Feb 9TBA
Feb 16TBA
Mihai Tohaneanu , Department of Mathematics, University of Kentucky
Feb 23TBA
Anurag Singh, Department of Mathematics, University of Utah
Mar 2TBA
Mar 9TBA
Lyudmila Korobenko, UPenn/Reed College
Mar 16Spring break
Mar 23TBA
David Zureick-Brown, Department of Mathematics and Computer Science, Emory University
Mar 30TBA
David Ayala, Montana State University
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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