# Colloquium

Talks (unless otherwise indicated) are in Davis 301 from 4:00–5:00 PM on Mondays. Refreshments begin at 3:30 PM 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 Facebook page.

## Spring 2024

**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

**c**= {1,2,3,…,n}\

**t**as a list of parking preferences where car i attempts to park in spot c_i and if not available, then proceeds up the street to find the next available spot, if any. A parking completion corresponds to a set of preferences

**c**where all cars park. In this talk, we will explore how parking completions are related to restricted lattice paths in the plane. We will also present results for both the ordered and unordered variations of the problem by use of a pair of operations (termed

**Join**and

**Split**). A nice consequence of our results is a new volume formula for most Pitman-Stanley polytopes. This is joint work with H. Nam, P.E. Harris, G. Dorpalen-Barry, S. Butler, J.L. Martin, C. Hettle, and Q. Liang.

**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 22, 2024**

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**April 29, 2024**

Lisa Naples

Fairfield University

Title

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**May 6, 2024**

Caitlyn Parmelee

Keene State College

Title

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## Fall 2023

**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 20, 2023**

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**November 27, 2023**

Meredith Greer

Bates College

Mathematical Epidemiology on a Small College Campus

Abstract:

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!

## Spring 2023

**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

**Sum Camp: Putting Sums back into Summer**

**Scott Taylor & Thom Klepach**

**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.