Talks (unless otherwise indicated) are in Davis 301 from 4–5 PM on Mondays. Refreshments begin at 3:30 on the second floor of Davis.
To make sure you get email updates, add yourself to the mathstu (if a student) or mathothers (if not) email groups. Or check the Colby Math&Stats facebook page.
You can see last semester’s schedule.
Wes Viles and Evan Randles
Meet Our New Faculty
We welcome you to join us for our first colloquium of the year. Come meet two of our new faculty members and also learn about the mathematics and statistics electives we will be offering this year. This is also a great chance to catch up with your fellow students and professors. Hope to see you there!
We encourage you to go to the talks on mathematics and gerrymandering at Bowdoin.
Fernando Gouvêa, Colby
Polynomial Endoscopy: What can you know if you can’t find the roots?
This talk is about polynomial equations and their roots, a topic with a long history. We will focus on the period after the 16th century discovery of how to solve equations of degree 3 and 4. It quickly became clear that degree 5 was much more difficult, leading to the question of what might be said about the roots without actually finding them. We will pay particular attention to Descartes’s Rule of Signs and the polynomial discriminant.
Rosa Orellana, Dartmouth
Graphs, Symmetry and Coloring
A graph consists of a set of vertices and a set of edges between the vertices. Graphs are useful to visualize relationships between different objects. In this talk, I will talk about coloring the set of vertices of a graph. In particular, I will define a polynomial that encodes all the possible ways to color the vertices of a graph if we do not allow two vertices to have the same color if there is an edge between them. This type of coloring is called proper coloring. One application of proper colorings is in solving scheduling problems.
The polynomial that we obtain which encodes all possible ways to properly color a graph is called the symmetric chromatic polynomial. The number of variables in the polynomial is the number of vertices. It is called symmetric because it doesn’t change when we permute the variables. In this talk, I will describe several properties of this polynomial and some unsolved problems related to this polynomial.
This talk will not assume any graph theory knowledge.
October 6 (Friday!)
Pamela Harris, Williams
Invisible Lattice Points
Patricia Cahn, Smith
How can we describe all spaces of a given dimension? We’ll start by describing 2-dimensional spaces, which look like donuts with any number of holes. Then we’ll learn how to describe 3-dimensional spaces, by using knots to build portals in our familiar 3-dimensional space.
Chris Chong, Bowdoin
The nonlinear mass-spring system: A simple homework problem, or the start of a scientific revolution?
The mass-spring system serves as a basic model for a plethora of natural and engineered systems, such earthquakes, sound, neurons, electrical circuits and more. I will demonstrate that the linear mass-spring system is straight-forward to analyze. But what about the nonlinear mass-spring system? I will talk about the history and implications of its study and I will discuss a few applications related to my own research. The talk will be accessible to students who have taken differential calculus.
Andrew Sage, Iowa State
A Walk Through a Random Forest
Since its creation in 2001, random forest methodology has emerged as a powerful technique for making predictions in classification and regression problems. This nonparametric approach is especially popular when working with complex datasets that contain many predictor variables. In this talk, we examine how random forests are grown and discuss their use in predicting student retention in science, technology, engineering, and mathematics (STEM) majors at Iowa State University. We then explore the impact of outliers on random forest predictions and describe a new method for improving the robustness of random forest regression.
Talk at 3:30 pm, Davis 201.
Refreshments at 3:00 pm, outside of Davis 216.
Gregory Dungan, USMA
Introduction to Topological Data Analysis and Applications
Geometry and Topology are similar areas of math in that they both rely on a notion of “closeness”, but geometry is much more rigid while topology is more relaxed. In data analysis where there is a lot of noise, geometrical spaces are much too rigid to use as models; however, topology is conducive to such conditions. We will introduce a fairly recent area of data analysis that models point-clouds with spaces in topology. The topological features (or “shape”) of a point-cloud can be qualitatively and quantitatively analyzed via topological techniques even when the data is high-dimensional or noisy. By characterizing the shape of point-clouds, we may be able to distinguish them. For example, if we have point-clouds representing biological information of cancerous tumors, we may be able to identify different subtypes
Wes Viles, Colby
Network Characterization through Composite and High-order Interactions
Network data analysis fundamentally rests upon the collection of basic measurements relevant to interactions among elements within a system. A network representation of that system can be constructed whereby the quantities of interest are of a relational and combinatorial nature. Network analysis of biological systems, such as the human-associated microbiome, is inherently challenging since it frequently relies on high-dimensional, statistically-dependent measurements. This presentation will describe the analytical framework and computational methods used in my research on problems of network characterization. On the topic of subgraph counts in large networks, I will provide insight on the probabilistic approach my coauthors and I developed for quantifying the propagation of low-rate measurement error through the process of network construction and summary. On an applied matter, I will subsequently describe our computational approaches to the estimation and statistical evaluation of mesoscopic complex network structures, including dynamic communities of multi-slice networks and genuine high-order ecological interactions of the human microbiota from which vital synergistic biological processes emerge. Our quantification of the complexity, as measured by total correlation, is one aspect of our aim to develop computational methods for predicting high-order microbial interactions that are pertinent to clinical endpoints.
Tuesday, November 14, starting at 3:30 pm, Davis 201
Refreshments at 3:00 pm, outside of Davis 216
Jerzy Wieczorek, Carnegie Mellon
Model-Selection Properties of Forward Selection with Sequential Cross-Validation
Forward selection (FS) is a popular variable-selection method for linear regression, valued for its low computational costs. However, its model-selection properties have long been unclear: When can we expect FS to choose the correct variable set, if one exists? Working in a high-dimensional setting, we derive sufficient conditions for FS to select the right model with probability going to 1, at first assuming the true model size is known. Since the true model size is rarely known in practice, we also derive sufficient conditions for model-selection consistency of FS with a data-driven stopping rule, based on a sequential variant of cross-validation (CV). We also discuss the practical problem of setting the data-splitting ratio for CV. We illustrate our methods on the Million Song Dataset to predict a song’s release year from audio features.
Thursday, November 16, starting at 3:30 pm, Davis 301
Refreshments at 3:00 pm, outside of Davis 216
Jianing Yang ’18, Colby
Jianing will be giving two 20 minute talks about research she did at Williams this past summer
Talk 1: Limiting Distributions in Generalized Zeckendorf Decompositions
Zeckendorf’s Theorem states that every positive integer can be uniquely decomposed into a sum of nonadjacent Fibonacci numbers. Interestingly this property can be used as a definition of the Fibonacci numbers, and thus it is natural to consider extensions to other recurrence sequences and decomposition laws. In addition to unique decomposition, these sequences led to Gaussian behavior in the distribution of number of summands in decompositions, and geometric decay in the probability of a gap of length $g$. We extend these arguments to allow the bin sizes $b_n$ and the legal number of summands chosen from the $n$th bin to vary. We further generalize by examining cases where we place adjacency conditions on the bins, where we use a Central Limit Theorem type result to study the limiting behavior.
Talk 2: Biases in Fourier coefficients of elliptic curve and cuspidal newform families
Random Matrix Theory, originally developed to model energy levels of heavy nuclei, has had remarkable success in predicting the main term of the behavior of zeros of $L$-functions. Unfortunately these models cannot see the lower order terms, where the arithmetic of the family lives. We prove results on lower order terms for many families of $L$-functions. In some one-parameter families of elliptic curves, where these terms are related to excess rank, we observe that the second moment of the Fourier coefficients has a negative bias. We extend these techniques to more general families of elliptic curves as well as higher moments and prove the bias is present there as well. We also compute all higher moments of Fourier coefficients from cuspidal newform families.
Ben Mathes, Colby
I will show how certain matrices are related to the theory of “orthogonal polynomials”.
A little Linear Algebra is the only prerequisite you need for this talk.