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.

Sept. 10
Scott Taylor
Colby College
The mathematics of knotted objects

Title

Abstract

How can mathematics be used to study knotted objects, like strings, loops, and molecules? In this talk I’ll discuss a variety of ways mathematicians quantify what it means for an object to be knotted and how knotted objects themselves (sometimes) act like numbers. I’ll highlight work I’ve done with Colby students in constructing a new class of knotted spatial graphs which are almost unknotted.

 

Sept. 17
Stephanie Eaneff
Talus Analytics
What does the Biological Weapons Convention have to do with statistics?
Case studies in data science and public policy

Title

Abstract

Policy makers tasked with developing processes, agreements, and regulations require relevant, accurate, and transparent information to inform their decisions. However, in many cases, decision makers do not have access to the data they need, when they need it, in formats that clearly communicate results. In some cases, for instance, during the early days of a rapidly unfolding infectious disease outbreak, data may simply not yet be available. In other cases, data may exist but have not yet been structured, analyzed, and visualized to support decision-making. This talk will explore three case studies in the use of data science to inform public health policy decisions, including prioritizing efforts to combat infectious diseases and tracking hospital capacity during a large-scale influenza outbreak. We will discuss considerations, challenges, and lessons learned in collecting, structuring, analyzing, and reporting on data to inform public policy.

 

Sept. 24
Colby Student Summer Research

Title

Abstract

Summer research assistants in the Math/Stats department will present short talks on their projects.

Shabab Ahmed ’19: Solvability of Polynomials

Simon Xu ’20 and Alice Gao ’20: Convex Neural Code

Charles Parham ’20 and Qidong He ’21: Knot Invariants

 

October 1
Meet our new faculty!

Title

Abstract

We are fortunate to have several new faculty in our department. Come meet them and find out something about their research and teaching.
Ariel Keller
Gabi Bontea
George Melvin
Jerzy Wieczorek

 

Oct. 8
Thiago Serra
Mitsubishi Electric Research Labs
Bounding and Counting Linear Regions of Deep Neural Networks

Title

Abstract

We investigate the complexity of deep neural networks (DNN) that represent piecewise linear (PWL) functions. In particular, we study the number of linear regions, i.e. pieces, that a PWL function represented by a DNN can attain, both theoretically and empirically. We present (i) tighter upper and lower bounds for the maximum number of linear regions on rectifier networks, which are exact for inputs of dimension one; (ii) a first upper bound for multi-layer maxout networks; and (iii) a first method to perform exact enumeration or counting of the number of regions by modeling the DNN with a mixed-integer linear formulation. These bounds come from leveraging the dimension of the space defining each linear region. The results also indicate that a deep rectifier network can only have more linear regions than any shallow counterpart with same number of neurons if that number exceeds the dimension of the input.

Joint work with Christian Tjandraatmadja (Carnegie Mellon University, now at Google) and Srikumar Ramalingam (The University of Utah), presented at the 35th International Conference on Machine Learning (ICML 2018).

 

Oct. 11 (Thurs.)
Danny Rorabaugh
University of Tennessee
Integer Sequences

Title

Abstract

The powers of 2: {1, 2, 4, 8, 16, 32, 64, …}. The maximum number of regions obtained when slicing a circle with n lines: {1, 2, 4, 7, 11, 16, 22, 29, …}. Integer sequences are found throughout mathematics, but especially in number theory and combinatorics. They might count a class of mathematical objects or just have fascinating numerical properties. What is the rule for constructing the following sequence: {1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, …}?

The number of neutrons in the most abundant isotope of each element on the periodic table: {0, 2, 4, 5, 6, 6, 7, 8, 10, 10, 12, …}. The year in which the population of earth reached n billion: {1804, 1927, 1959, 1974, 1987, 1999, 2011(, …?)}. Integer sequences are everywhere. Can you figure out what this sequence represents: {3, 3, 5, 4, 4, 3, 5, 5, 4, 3, 6, …}?

In this talk, we share the joys of studying integer sequences and tools available for investigation. With sequences, you can practice programming and make connections between disciplines; they are the source of both silly puzzles and deep, unsolved mathematical questions.

 

Oct. 22
Helen Wong
Claremont McKenna
It’s not algebra, it’s knot algebra

Title

Abstract

Knots are an everyday part of life. They appear in shoelaces, computer cords, and fishing wire. But they also appear in less likely places; for example in bacterial DNA, in debunked atomic theories of Lord Kelvin, and in current schemes for creating a quantum computer by braiding anyons. Come to this talk to find out more about the mathematical theory of knots, and in particular, how mathematicians use techniques from algebra to understand them better!

 

Oct. 29
Chris Moore
Colby College (Biology)
Tempering positive feedback in population models of mutualism

Title

Abstract

Ecology is the study of the abundance and distributions of organisms. Among the multitude of factors that affect the abundance and distributions of organisms is the interactions between species. One type of interaction, mutualism, describes where two species increase each other’s survival or reproduction such that fitness is increased in each other’s presence. Despite mutualism being omnipresent in ecological communities, it’s been ignored by theoreticians because the standard dynamic model of mutualism (known as the Lotka-Volterra mutualism model) inconveniently results in positive feedback where populations grow exponentially without bound. For my postdoctoral work I burdened myself with trying to find biologically-relevant ways of preventing the positive feedback in the Lotka-Volterra mutualism model. I will present two ways that I added realistic aspects of biology into the Lotka-Volterra mutualism model: (i) by adding non-linear within-species density dependence and (ii) by explicitly considering the density of the species receiving mutualistic benefits. Adding non-linear within-species density dependence revealed that for species where per-capita growth rates are decreased at high densities, mutualistic interactions were always stabilized. Furthermore, separating birth and death rates, as long as one decreased at high densities, I found that mutualistic interactions were always stable. Considering how mutualistic benefits are received when explicitly considering the receiving species’ density was driven by dissecting results from empirical studies. My meta-analysis found that most mutualistic benefits received by a species only increases fitness when the receiving species is at low densities. I subsequently developed a model to incorporate species benefitting from mutualism at low densities, high densities, and independent of density. When species receive benefits at low densities, mutualistic interactions never result in positive feedback. By adding realistic biology into the Lotka-Volterra mutualism model of mutualism, I’ve described mechanisms that can temper positive feedback, and I hope to have have opened up some provocative lines of investigation about implicit assumptions of commonly-used population models and the nature of modeling species interactions.

 

Nov. 5
Statistics Candidate
Regression Methods for Network Indexed Data: Modeling Occurrences of Burglary in Boston, Massachusetts

Title

Abstract

After defining and giving examples of network data, we introduce our motivating example: modeling occurrences of residential burglary in Boston, MA. Viewing the city of Boston as a network of streets and intersections, we discuss a generalized linear model composed of vertex indexed predictors. We introduce the graph Laplacian as a regularization tool and briefly discuss the interpretations of our proposed model.

 

Nov. 9 (Friday) Davis 201
Statistics Candidate
A casual approach to real-world data: emulating randomized trials in SEER-Medicare

Title

Abstract

Researchers are often interested in evaluating the comparative effectiveness of cancer therapies. Most commonly, these questions are addressed by conducting randomized trials, where the treatment assignment is randomized. However, large randomized trials can be prohibitively expensive or logistically infeasible, so researchers have turned to observational data sources to generate evidence to answer these questions. Unfortunately, naïve analyses of observational data easily lend themselves to misleading and biased results. Here, we will discuss best practices when applying the target trial framework developed by Hernán & Robins (2016) in the SEER-Medicare linked database to emulate randomized trials. As a case study, we will consider the emulation of a trial to evaluate adjuvant fluorouracil-based chemotherapy to treat stage II colorectal cancer.

Hernán, M. A., & Robins, J. M. (2016). Using big data to emulate a target trial when a randomized trial is not available. American journal of epidemiology183(8), 758-764.

Warren, J. L., Klabunde, C. N., Schrag, D., Bach, P. B., & Riley, G. F. (2002). Overview of the SEER-Medicare data: content, research applications, and generalizability to the United States elderly population. Medical care, IV3-IV18.

 

Nov. 15 (Thursday)
Statistics Candidate
ARGO2: Accurate, real-time flu tracking with Internet search data

Title

Abstract

In today’s digital age, people leave traces about nearly every aspect of their lives on the Internet. Such “big data” from Internet search engines offer great potential for real-time tracking of public health and social events, including influenza (flu) epidemics. Accurate, high-resolution tracking of flu activities at the regional level helps public health agencies make informed and proactive decisions, especially in the face of outbreaks. However, due to the complex data structure and reduced quality of localized Internet data, few established methods provide satisfactory performance. Our newly proposed method, ARGO2 (2-step Augmented Regression with GOogle data), efficiently combines publicly available Google search data with traditional flu surveillance data from the US Centers for Disease Control and Prevention (CDC); it effectively incorporates cross-regional, inter-resolution (national and regional) dependencies in flu activities. ARGO2 gives leading performance in flu tracking across all US regions compared with currently available, Internet-data-based methods. It also has the potential and flexibility to further include information from additional Internet sources, such as Twitter or Instagram, and to track other social, economic and public health events at local levels.

 

Nov. 19
Statistics Candidate
Modeling Forensic Fingerprint Decisions with Item Response Theory

Title

Abstract

Fingerprint examination is perhaps the most well-known of the forensic science methods, yet remains one of the most subjective. Even as automated systems become more prolific and accurate, final comparison decisions are left to individual forensic examiners. Given the same comparison task, different fingerprint examiners may come to different conclusions and/or use different decision criteria to come to those conclusions. There has been an influx of studies attempting to estimate overall error rates as well as understand decision-making processes in forensic analyses. Methods from educational testing, particularly item response theory (IRT), can provide valuable insight into these studies by accounting for both individual differences among examiners as well as comparison tasks of varying difficulty.

There are, however, fundamental differences between forensic science applications and traditional IRT settings. In this talk, I will discuss these differences and demonstrate the use of IRTrees, a tree-based IRT framework, to account for these differences. I will also discuss the utility of such an approach in the forensic science domain.

 

Dec. 3
Milja Poe
On S-unit Equations

Title

Abstract

We will be studying S-unit equations, in particular the equation:

X + Y = 1

 

We fix a finite set of prime numbers, for example 2,3, and 5. We assume that X and Y are both S-units, in rational numbers.   This means that X and Y are rational numbers such that their numerators and denominators factor into primes only in the set S.  So if S is the set 2,3, and 5, then 1/4 + 3/4 =1, and 1/5 + 4/5 =1 would be solutions to the S-unit equation above.

We want to study the number of solutions, and their distribution, of the above S-unit equation. And how these solutions might depend on the set S.