Colloquium

September 16, 2024

Joseph Hoisington
Colby College

Abstract: The isoperimetric inequality states that, among all figures in the plane with the same perimeter, the circle has the largest area. It answers one of the oldest questions in geometry and has fascinated mathematicians and writers for centuries. We will talk about the isoperimetric inequality, its importance in mathematics, and how it has appeared in areas as diverse as biology, politics, and literature.

September 23, 2024

Rob Benedetto
Amherst College
Polyhedra: Plato, Archimedes, Euler

Abstract: A polyhedron is a three-dimensional analog of a polygon, such as a cube, pyramid, or prism. This talk is about two special classes of polyhedra with elegant structures, and their study over the past 25 centuries. Regular polyhedra were known to Pythagoras and Plato, and semiregular polyhedra were introduced later by Archimedes. We will present all these regular and semiregular polyhedra, and discuss their history. There will be some simple counting arguments to go with the many pretty pictures, but otherwise, no prior mathematical background is required to understand this talk.

September 30, 2024

Evan Randles
Colby College
A random walker stumbles onto a proof of Lagrange’s theorem

Abstract: The study of random walks on groups lives at the intersection of three primary branches of mathematics: probability, algebra, and analysis. Following its initial investigation by G. Pólya in 1921, the modern theory of random walks on groups was born out of the pioneering work of H. Kesten and has since reached a high level of sophistication, seeing applications to biology, chemistry, physics, economics, and mathematics itself. For example, the theory of random walks can tell you how many shuffles it takes to
sufficiently randomize a deck of cards; this is an application of analysis and algebra to probability (and gambling). Similarly, you can use random walks to learn things about analysis from algebra (e.g., underlying group properties can tell you how solutions to PDEs behave) and, conversely, you can learn about algebra from analysis (e.g., you can use inequalities to decide if a group is nilpotent). Recently, in my study of random walks on finitely-generated Abelian groups with Tony Yan ’25, I stumbled onto a proof of
a central result in algebra: Lagrange’s theorem (for finite Abelian groups). In this talk, I will present my proof. I will also discuss a potential direction for future research, likely suitable for a Colby student.

October 7, 2024

Ray Maresca
Bowdoin College
Frieze patterns, triangulations of polygons, and representations of quivers

Abstract: Frieze patterns have been around for thousands of years, dating back at least to Greco-Roman architecture. They are patterns that follow different types of symmetries and are often found on buildings, rugs, pottery, coasters, and other types of artwork. In the 1970’s Coxeter developed frieze grids that share some of the symmetric properties found in frieze patterns. Quivers, or directed graphs, and their representations on the other hand, have origins dating back to the 1950’s. In the 2000’s, mathematicians realized that these frieze grids are deeply connected with quivers and their representations. In this talk, we will explore this connection through triangulations of polygons. Some knowledge of linear algebra (vector spaces and linear transformations) is assumed, but most of the talk presumes no prior knowledge. 

October 21, 2024 – Olin 001

Becca Thomases
Smith College
Runnals Women in Mathematics Dinner
How to swim through goo

Abstract: Non-Newtonian or complex fluids describe a wide class of materials from biological fluids like mucus and blood to everyday household products like shampoo and paint. There are many problems in physics and biology where understanding motion of (or in) complex fluids is essential for understanding natural phenomena. Tools from mathematical analysis and computational simulations can shed light on these complex problems that are significant in many biological, environmental, and industrial applications. I will describe some recent work on modeling micro-organism swimming in viscoelastic fluids, and understanding the mechanisms that lead to speed changes for swimmers in complex fluids.

October 28, 2024

Sam Lin
Colby College
Spheres all of whose geodesics are closed

Abstract: Have you ever wondered why flight paths are often arcs rather than straight lines? The reason lies in the concept of geodesics. In simple terms, geodesics are the shortest paths between two points within a given space, generalizing straight lines in “flat” (Euclidean) geometry to “curved” (non-Euclidean) geometries such as spherical geometry.

On a perfectly round sphere, geodesics are given by great circles—the largest possible circles that can be drawn on the sphere. Notably, all geodesics on a perfectly round sphere are closed loops. This raises a natural question: are there other spaces on which all geodesics are closed loops?

In this talk, I will take you through the fascinating history of this question, highlighting the intriguing examples by Tannery (1892) and Zoll (1903). I will also present a related theorem that my Ph.D. advisor and I proved in 2017. This talk is designed to be accessible. No prior knowledge of non-Euclidean geometry is needed, and there will be many pictures!

November 4, 2024

Jack Petok
Colby College
Rational points on curves

Abstract: Given a polynomial equation in two variables, can we exactly find all of the rational number solutions to this equation? In other words, can we give a nice description of all of the (perhaps infinitely many) solutions? We’ll start with finding all rational number solutions for a quadratic equation in two variables. Then we will discuss the much harder problem of describing all rational number solutions to a cubic equation in two variables, called an elliptic curve.  Geometry will play a starring role in our discussions of both problems. These problems were studied in ancient times, but are also an active area of modern math research: we’ll tell the story of some surprising results proved this century (and some from this year!) about the number of parameters (rank) needed to describe the rational number solutions to cubic equations.

November 21, 2024 (Thursday)

Haoyu Song
Purdue University
Mathematics of Participatory Budgeting

Abstract: Since the Enlightenment, mathematics has been central in designing and evaluating public choice mechanisms. This talk focuses on participatory budgeting (PB), a democratic process adopted by cities worldwide, which enables citizens to make direct decisions on allocating public funds to community projects. In a PB problem, there is a predetermined budget B and a list of m projects, each with an associated cost. The objective is to select a combination of projects that best reflects the preferences of all voters while staying within budget. First, We will introduce the concept of the proportional core, a classical fairness criterion requiring no subgroup of voters would collectively prefer an alternative set of projects with a total cost proportional to their size. We argue that the core is an ideal fairness notion for PB, as it gives each subgroup a justified level of influence over the final decision. Unfortunately, a core allocation may not exist. To address this, we explore two alternative fairness concepts: the approximate core and the randomized core. In both scenarios, there is a metric called approximation ratio that measures how far an allocation is from meeting core fairness. We will show that by adapting the Lindahl equilibrium to our discrete PB setting, we can significantly lower the approximation ratio from 32 (Jiang et al 2020) to 6.24. 

February 10, 2025

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February 17, 2025

Carrie Diaz Eaton
Bates College
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February 24, 2025

Colin Adams
Williams College
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March 3, 2025

Tamar Friedmann
Colby College
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March 10, 2025

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March 17, 2025

Mathematics Mentoring Event
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March 31, 2025

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April 7, 2025

Naomi Tanabe
Bowdoin College
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April 14, 2025

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April 21, 2025

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April 28, 2025

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May 5, 2025

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February 19, 2024, Olin 001 (Followed by the Runnals Dinner, Parker Reed, SSWAC, 6:00 pm)

Karamatou Yacoubou Djima
Wellesley College
Classifying medical images using tools from mathematical data science

Abstract: Mathematical data science has permeated all areas of modern medicine, from processing medical images to diagnosing medical conditions. In this talk, I will discuss how I use applied mathematics for my ongoing interest in detecting autism using placental images. In recent studies, differences in the morphology of the placental surface network have been associated with developmental disorders. This suggests that the placental surface network could potentially serve as a biomarker for the early diagnosis and treatment of autism. In this talk, I will describe the automated extraction of vascular networks using applied harmonic analysis techniques, which express data into simple, efficient building blocks. Then, I will survey other tools that I am exploring to tackle this problem, including deep learning and network theory.

March 4, 2024

Matthew Hernandez
University of Maine

Splashing curves in fluid models and magnetic field squeezing outside plasma

Abstract: For a model of a surface of a body of water (or some other fluid) in motion, a splash singularity is said to occur when the surface starts as a simple “sheet,” before two parts of the surface rise up, move towards one another, and eventually crash into each other. Though theoretical mathematical methods fail to predict exactly what happens afterwards, they can be used to prove various fluids can form splash singularities, which are easily observed at the beach.

It is harder to observe a body of plasma up close, such as the sun, which creates a magnetic field just outside. For a hypothetical splash singularity in this case, the two parts of the surface must approach each other and squeeze the magnetic field lines completely together into a pinch in the outside region. Is this possible? Or must the magnetic field “push back” when it gets squeezed?

In this talk we will explain the ideas behind constructing splash singularities for models of water and plasma, and the behavior of the magnetic field squeezing in the plasma case.

March 11, 2024

Ayo Adeniran
Colby College
Parking completions

Abstract: Parking functions are well-known objects in combinatorics. One interesting generalization of parking functions are parking completions. Given a strictly increasing sequence t with entries from {1,2,3,…,n}, we can think of t as a list of spots already taken in a street with n parking spots, and its complement

March 18, 2024

Colby Math Students
Colby College
Thinking about a Math(Sci) Major? Please join the math faculty and 4 seniors who will talk about why they chose a major in our department and their experiences. Please come prepared to learn and ask questions about majoring in Math!

April 1, 2024

Marissa Masden
ICERM
Hyperplane Arrangements, Polyhedral Geometry, and Topology in Machine Learning!

Abstract: Modern machine learning makes frequent use of a type of function called a ReLU Neural Network. ReLU neural networks are simple to describe with a list of matrices and vectors. However, they are often called “black boxes” because when they are “trained” to perform a particular task, it is very difficult to prove they will perform consistently on future data! To begin to understand these functions, we discuss how each matrix-vector combination gives us a structure known as a hyperplane arrangement, a collection of (n-1)-dimensional planes in ℝⁿ. When combining all the matrices and vectors of a ReLU neural network, we then obtain a cell-like structure called a polyhedral complex. I will share how hyperplane arrangements and polyhedral complexes come from ReLU neural networks, a little about how mathematicians study these objects (including a peek into the mathematics of topology, which studies the connectivity of objects), and finally a few experiments that show how we can use these tools to understand what a neural network has learned! 

April 15, 2024

David Zureick-Brown
Amherst College
Beyond Fermat’s Last Theorem

Abstract: What do we (number theorists) do with ourselves now that Fermat’s last theorem (FLT) has fallen?

I’ll discuss numerous generalizations of FLT — for instance, for fixed integers a,b,c >= 2 satisfying 1/a + 1/b + 1/c < 1, Darmon and Granville proved the single 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. More generally, I’ll discuss my subfield “arithmetic geometry”, and in particular the geometric intuitions that underlie the conjecture framework of modern number theory.

April 29, 2024

Lisa Naples
Fairfield University
Size and Structure

Abstract: Given a set, a basic question that one may ask is: “What is the size of the set?” The way that we answer this question may be dependent on the structure of the set. For example, if the set is a line or curve segment, we will likely calculate the length. On the other hand, if the set is a polygonal region, we will calculate the area. Of course, the structure of sets that we encounter in Euclidean space may be much more diverse than curves or polygons. In this talk we will follow a line of questioning to build up relationships between size and structure that persist even when the sets are of less familiar shape. Along the way, we will highlight some important concepts from calculus and real analysis that help us to answer our existing questions as well as develop new ones.

May 6, 2024

Caitlyn Parmelee
Keene State College
Math + Neuro: Modeling the Brain

Abstract: How does the structure of a neural network affect its behavior? To answer this question, we use a particular neural network model, Combinatorial Threshold-Linear Networks (CTLNs). CTLNs are a toy model that allow us to focus on the relationship between structure and function in neural networks. While on sabbatical in Fall 2023, I proved some new and exciting results about CTLNs while I was a research fellow at the Institute for Computational and Experimental Research in Mathematics (ICERM) at Brown University.  Come hear a bit about the intersection of mathematics and neural network modeling!

September 18, 2023

Scott Taylor
Colby College
Removing warts and wrinkles from companions or how to regroup when it all falls apart

Abstract: What to do when a 10-year long project suddenly falls apart? I’ll describe the trajectory of a recent project that started off promising, fell apart for 8 months, and then with a moment inspiration came back together. It concerns a way of measuring how complex a knot is and how that complexity might change when we tie one knot (the pattern knot) in the shape of another knot (the companion knot) to create a new knot (the satellite knot). Along the way we’ll explore other 3-dimensional spaces and extract some life lessons from the mathematical process.

October 2, 2023

Benjamin Weiss
Unum
Insurance Company Tail Risks

Abstract: Insurance companies are required to report on our best estimates, but we’re also required to maintain adequate reserves for tail risks. What is a risk? How do we measure it? What do we need to do for data cleaning, or lack of data?

In addition to providing some background for the context of the problem, I’ll be presenting some work my summer intern completed this past summer estimating Unum’s 1-in-200 risk stresses. No statistics background will be required (my intern had none when he started!).

The last 10 minutes of the talk will be about Unum specifically and our summer and full-time employment opportunities.

October 16, 2023

Neel Patel
University of Maine
Bubbles in Ground Water

Abstract: Underground reservoirs of water, called aquifers, are common in the state of Maine. Often, when breaking ground, gas or oil bubbles can contaminate the water. Partial differential equations (PDE) and Fourier analysis are powerful mathematical tools for describing and analyzing the behavior of fluid dynamics underground, i.e. porous media flows. Moreover, the mathematical theory developed using PDE can be applied to other porous media flow settings such as glaucoma in the eye, chromatography and petroleum extraction. In this talk, we will discuss why the dynamics of fluid bubbles is more challenging that an infinite interface system and how the Fourier transform can be used to prove stability of certain bubble geometries.

October 23, 2023

Jackson Goodman
Colby College
Curvature, Topology, and the Dirac Operator

Abstract: The Dirac operator is a differential operator which can be used to relate curvature- a local notion of shape-  to topology- a global notion of shape.   I will illustrate that story with concrete examples.  I will then discuss my research, in which I have used the Dirac operator to investigate the following questions: When can a given shape be deformed to have some notion of “positive curvature” (which we will define) everywhere? When can one shape with positive curvature be deformed into another, maintaining positive curvature during the deformation?

October 30, 2023

Laura Storch
Bates
Topological early warning signals of extinction: A role for algebraic topology in ecological forecasting

Abstract: Many populations and ecosystems are experiencing decline due to habitat degradation, global climate change, and increasing human pressures such as overfishing. Populations undergoing decline can experience a critical transition – an abrupt, irreversible shift in the dynamics of the population. It is particularly challenging to detect critical transitions in spatial populations, e.g., a grassland. Here, we use computational algebraic topology to quantify features in a landscape and observe how those features change over time during a critical transition such as an extinction event. In this way, we identify topological signatures of impending population collapse.

November 6, 2023

Math Mentoring Discussions

Davis 301

Are you thinking about being a math/mathsci major? Are you wondering what upper level math classes are like? Join us for informative discussions facilitated by upper-class math/mathsci majors who can share their experiences in topics such as*:
• Studying abroad as a math/mathsci major
• Beyond calculus – what higher level math courses are like
• Finding math research opportunities, REUs, or internships
• I think I want to be a math/mathsci major?
• Double majoring in math/mathsci and X
• Thriving in math classes as a student athlete
• Finding a mathematical community
• What it is like to be a Math teaching assistant (TA) or learning assistant (LA)
During the event, mentees can select which discussion they would like to join. Everyone will have the opportunity to switch topics/groups and learn from at least two mentors.

Interested in participating as a mentor or a mentee? Please register here: https://forms.gle/U3x59AswXXyJp4577/. Registration as a mentee is not required, but is encouraged to help us plan accordingly.

*Selection of final topics will be based on mentor availability.

November 13, 2023

Fernando Gouvea
Colby College
Factoring Primes

Abstract: Attempting to factor a prime number sounds like something Don Quixote might attempt. They are primes because they can’t be factored, after all. Sometimes, however, new settings can change what is possible. This talk will explain why nineteenth century number theorists decided to try to factor primes and discuss some of the resulting problems.

The only real pre-requisite is knowing what a prime number is; if you are familiar with modular arithmetic that will be helpful as well.

November 27, 2023

Meredith Greer
Bates College
Mathematical Epidemiology on a Small College Campus

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Mathematical epidemiology researchers study differential equation-based models, and other types of models, to build understanding of the dynamics of disease outbreaks. These models can show the rise and fall of infections, the importance of interactions between Infectious and Susceptible people, and the potential impact of different public health interventions. Careful modeling requires both conceptual approaches and meaningful connections with data. It turns out that a small, residential college campus is an ideal setting for building and using models. The students on such a campus typically form a consistent population within a semester; the population tends to be “well-mixed”, a feature that supports the use of differential equations for representing the outbreak; and students themselves are able to study outbreaks as they occur.

This talk is accessible to those who have studied just enough calculus to know that derivatives represent change. No formal background in differential equations is needed.

December 4, 2023

Rebecca Hardenbrook
Dartmouth College
A Beginner’s Guide to Topological Data Analysis and An Application to Arctic Sea Ice

Abstract: How do we find patterns in large, complex datasets, and how do we determine that those patterns are meaningful? In this talk, we will learn about one set of mathematical tools to help us answer this question: Topological Data Analysis, or TDA for short. After a journey through the fundamental concepts behind TDA, we will discuss an ongoing project focused on the application of TDA to the study of Arctic sea ice. No knowledge of topology or sea ice is required!

February 13, 2023

Lorelei Koss Yarnell
Colby College
Real and Complex Newton’s Method

Abstract: Perhaps you have seen the technique of iteration in a calculus or numerical methods class when using Newton’s method to approximate roots of equations. In 1879, Arthur Cayley posed a question about extending Newton’s Method to complex functions. In this talk, we discuss Cayley’s idea and its generalization to iterating complex functions. By the end, we will see how iteration gives rise to beautiful and complicated objects called Julia sets.

February 20, 2023

Evan Randles
Colby College
Local limit theorems on the integer lattice

Abstract: Random walk theory is concerned with understanding the probabilities associated to the position of a “random walker” who moves by taking
random steps, each independent of those before it. When this walk is done on the integer lattice (e.g., a city grid), the position of the random walker is predicted by studying the so-called convolution powers of a probability distribution. The famous local central limit theorem (discovered by De Moivre and proven by Laplace in 1795) states that these convolution powers (and hence the positions of a random walker) are well approximated by the Gaussian function (or heat kernel) as the number of steps becomes large. When these “probabilities” are allowed to take on complex values, convolution powers are seen to exhibit rich and striking behavior not seen in the probabilistic setting. In this talk, I will discuss some history on this topic dating back to its initial investigation by E. L. De Forest and subsequent study by a number of important mathematicians, including I. J. Schoenberg who was a professor at Colby in the 1930’s. I will then describe some recent progress in this study, including some joint work with H. Bui ’21 and L. Saloff-Coste. In particular, I will present a class of “generalized” local theorems which state that, under certain conditions, convolution powers of complex-valued functions are well approximated by sums of generalized “heat” kernels. Time permitting, I will also describe applications to data smoothing and numerical solution algorithms to partial differential equations.

March 6, 2023

Scott Taylor and Thom Klepach
Colby College

March 10, 2023 – Friday – Diamond 122

Timur Akhunov
Haverford College
Derivative gain and singularities for degenerate Laplace equation

Abstract: Many natural phenomena from oil exploration to weather prediction to finance are modeled with differential equations (DE). The Laplace equation plays a unifying role in the world of DE. Its solutions, famously, do not have singularities, which for a related heat equation means that spontaneous boiling of the water doesn’t happen without an external heat source. Singularities are fascinating. You can win $1M if you settle the question of singularities for fluid motion. I have been studying Laplace-type equations that can have singularities for more than a decade. What is different about them?

March 13, 2023

Westin King
Fordham University

We Know it is True, so Why Bother Proving it Again?

Demonstrating that a claim is true is not the only function that mathematical proofs serve. New proofs may incorporate (or even create) new techniques or interpretations and these may further be used to attack other problems. We will use several well-known combinatorial results to illustrate how finding an alternative proof can improve our understanding of related problems or mathematical objects.

March 27, 2023

Patricia Cahn
Smith College

Wallpaper patterns are infinite patterns in the plane.

It turns out that we can classify them using tools from topology, a field like geometry, but where shapes can stretch and bend. We’ll practice learning to instantly recognize these patterns (a fun party trick). If time permits, we’ll see how similar ideas can help us understand different three-dimensional spaces–think possible shapes of the universe.

April 10, 2023

Alumni Panel

April 17, 2023

Fatou Sanogo
Bates College

Tensor and its applications

Advances in modern computer technology has increased the number of available information which has given rise to the presence of multidimensional data. A tensor is a multidimensional array it also can be seen as a multidimensional matrix. In today’s world tensor decomposition faces some major challenges such as sensitivity to outliers and missing data.

In this talk I will introduce a tensor (matrix) completion algorithm using the CP decomposition and tensor (matrix) denoising. The tensor (matrix) completion problem is about finding the unknown tensor (matrix) from a given a tensor (matrix) with partially observed data. I will show some examples on how this algorithm is used for video or image completion and as recommendation systems.