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.
Evan Randles, Colby
On the range and periodic structure of random walks on the integer lattice
In the study of random walks on the d-dimensional integer lattice, it is common to assume that a given random walk is both irreducible and aperiodic. Though these common hypotheses make the analysis particularly simple, they rule out interesting behavior about the periodic nature of generic random walks – a subject first approached by G. Pólya from a combinatorial perspective. In this talk, I will take a harmonic analysis viewpoint at the range and periodic of random walks on the lattice. With the help of Fourier transform, I will investigate a function which, for a given random walk, describes the set of possible lattice sites for the random walk at step n. This function will then be seen to appear naturally in a statement of local theorems for random walks on the integer lattice where one makes no assumptions concerning aperiodicity or irreducibility.
A panel of math and math sci majors will discuss the joys and challenges of being a major.
Lucia Petito, Harvard School of Public Health
An Exploration into Misclassified Group Tested Current Status Data
Group testing, first introduced by a military doctor in 1943, has been used as a method to reduce costs when estimating the prevalence of a binary characteristic based on a screening test of m groups that include n independent individuals in total. If the unknown prevalence in question is low, and the screening test suffers from misclassification, more precise prevalence estimates can be obtained from group testing than from testing all n samples separately. In some applications, the individual binary response corresponds to whether an underlying “time to incidence” variable T is less than an observed screening time C. This data structure at the individual level is known as current status data. Given sufficient variation in the observed Cs, it is possible to estimate the distribution function F of T non-parametrically using the pool-adjacent-violators algorithm. Here, we develop a nonparametric estimator of F based on group tested current status data for groups of size k = n/m where the group tests “positive” if and only if any individual unobserved T is less than its corresponding observed C. We will investigate the performance of the group-based estimator as compared to the individual test nonparametric maximum likelihood estimator, and show that the former can be more precise in the presence of misclassification for low values of F(t). We then apply this estimator to the age-at- incidence curve for hepatitis C infection in a sample of U.S. women who gave birth to a child in 2014, where group assignment is done at random and based on maternal age.
Starting at 4:00 pm in Olin 1. Refreshments at 3:30 pm outside of Davis 216
Xiao-Li Meng, Dean of the Harvard Graduate School of Arts and Sciences
The Law of Large Populations: The return of the long-ignored N and how it can affect our 2020 vision
For over a century now, we statisticians have successfully convinced ourselves and almost everyone else, that in statistical inference the size of the population N can be ignored, especially when it is large. Instead, we focused on the size of the sample, n, the key driving force for both the Law of Large Numbers and the Central Limit Theorem. We were thus taught that the statistical error (standard error) goes down with n typically at the rate of 1/√n. However, all these rely on the presumption that our data have perfect quality, in the sense of being equivalent to a probabilistic sample. A largely overlooked statistical identity, a potential counterpart to the Euler identity in
mathematics, reveals a Law of Large Populations (LLP), a law that we should be all afraid of. That is, once we lose control over data quality, the systematic error (bias) in the usual estimators, relative to the benchmarking standard error from simple random sampling, goes up with N at the rate of √N. The coefficient in front of √N can be viewed as a data defect index, which is the simple Pearson correlation between the reporting/recording indicator and the value reported/recorded. Because of the multiplier√N, a seemingly tiny correlation, say, 0.005, can have detrimental effect on the quality of inference. Without understanding of this LLP, “big data” can do more harm than good because of the drastically inflated precision assessment hence a gross overconfidence, setting us up to be caught by surprise when the reality unfolds, as we all experienced during the 2016 US presidential election. Data from Cooperative Congressional Election Study (CCES, conducted by Stephen Ansolabehere, Douglas River and others, and analyzed by Shiro Kuriwaki), are used to estimate the data defect index for the 2016 US election, with the aim to gain a clearer vision for the 2020 election and beyond.
Robert Benedetto, Amherst
An Introduction to Complex Dynamics and the Mandelbrot Set
Suppose we fix a constant c
and consider the polynomial function f(z)=z2+c
. What happens if we consider f
composed with itself many times: f∘f
, then f∘f∘f
, and so on? Complex Dynamics is the study of this kind of repeated iteration, and it leads naturally to the analysis of an intricate region known as the Mandelbrot Set. In this talk, we will give a very basic introduction to the subject, including some beautiful fractal pictures.
No background beyond Calculus 2 is required for this talk, but we’ll make occasional use of Taylor series and extensive use of polar coordinates.
Joe Chen, Colgate
Sandpiles in a Fractal Labyrinth
A sandpile on a graph is a configuration of indistinguishable particles (“chips”) occupying the vertices of the graph.
Whenever the number of chips at vertex $x$ s greater than or equal to the degree of $x$, we topple the vertex by emitting one chip to each neighboring vertex of $x$. The procedure continues until every vertex has fewer number of chips than its degree, in which case the configuration is said to be stable.
A classic problem in this area is to describe the stable sandpile cluster when there are initially $N$ chips at a single vertex.
As $N$ grows, what can we say about the shape and patterns of the cluster?
On the square lattice, it has been recently proved that the cluster contains many copies of the Apollonian gasket, a famous self-similar fractal. So in this sense we see a “fractal in a sandpile.”
On the other hand, I will describe a growing “sandpile on a fractal,” the fractal being the Sierpinski gasket.
It turns out that results on the cluster shapes and patterns are unusually sharp.
I will explain how these results arise from a combination of geometric and algebraic origins.
PS: There will be many pictures and animations in my talk.
Ernst Linder, University of New Hampshire
Uncertainty Quantification for Climate Downscaling: From Time Series to Machine Learning
As the climate continues to change, it is important to understand how rising temperatures and frequent intense storm events affect the built infrastructure and societal resources in the future. Global and regional climate models provide projections of future climate but these do not represent future weather because of their coarse spatial resolution. Yet scientific models for hydraulic structures for example require weather inputs at the daily to subdaily temporal scale and from a single location to a watershed scale. Statistical Downscaling (SD) provides the translation between projected climate and potential future weather scenario at the local level. Numerous methods for SD have been proposed, but the variability of estimation is often not accounted for, nor are most methods capable of replicating the statistical structure of local weather. By embedding the SD in a statistical estimation framework, such as time series modeling, and generalized linear modeling, we are able to account for the statistical uncertainty. Lastly, by extending the more traditional models using “machine learning” models, we are able to further reduce the uncertainty of estimating future weather scenarios. We illustrate the methods with examples from the freeze-thaw cycle and from regional precipitation analysis.
Tom Bellsky, University of Maine
Data assimilation, chaos, and weather forecasting
The atmosphere behaves as a nonlinear, chaotic dynamical system, where small differences in initial conditions can lead to great differences in future outcomes. Thus, accurately determining the current atmospheric state is a key step in short-term numerical weather forecasting. Data assimilation is a technique that combines model forecasts with observational data to produce a best guess of the current state. This talk will introduce Kalman filter data assimilation techniques under the context of weather forecasting and describe results on a low-dimensional dynamical system.
Padraig O. Cathain, Worcester Polytechnic Institute
Gowers-Host-Kra norms and Gowers structure on Euclidean spaces
Sam Wagstaff, Purdue
Complexity of Factoring Integers and Recognizing Primes
These two topics are essential for public key cryptography: Factoring integers should be hard but recognizing primes easy. We survey the history of these topics and describe some of the fastest known algorithms for them. Some of these algorithms are rigorous and some rely on unproved hypotheses. Some are deterministic and some choose random numbers. Some are practical and some are theoretical. Your computer uses one of these algorithms every time you send a credit card over the Internet.
May 14, 2018, 3:30pm, Davis 201.
Lattices, Paths, and Polynomials
The philosopher Plato argued that there were two realms of existence: one consisting of perfected constructions of unseen but significant objects called Ideals and another consisting of the imperfect experiences of those objects. Within the realm of Ideals, the argument goes, exist numbers. While we all have an understanding of a number, like “eight,” none of us have actually experienced Plato’s true interpretation of “eight.” We have only experienced various represenations and agreements about what “eight” means. This leads us to the possibility of manipulating various representations of numbers for various effects. One classic representation of a number involves expressing an integer n as a sum of positive integers called a partition. This too, however, can be manipulated. To differentiate between distinct partitions of n, diagrams called Young tableaux are often used as representations. We will investigate unique properties of these representations to see that a special class of symmetric polynomial is housed within lattices of Young tableaux. Finally, we will consider unknown polynomials possessing multiple symmetries as a result of restricted and altered Young lattices.