# 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.

## 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

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**October 23, 2023**

Jackson Goodman

Colby College

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**October 30, 2023**

Laura Storch

Bates

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**November 6, 2023**

Math Mentor for Students Event

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**November 13, 2023**

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**November 20, 2023**

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**November 27, 2023**

Meredith Greer

Bates College

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**December 4, 2023**

Rebecca Hardenbrook

Dartmouth College

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## Spring 2024

**February 12, 2024**

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**February 19, 2024**

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**February 26, 2024 (Runnals Dinner)
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Karamatou Yacoubou Djima

Wellesley College

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**March 4, 2024**

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**March 11, 2024**

Ayo Adeniran

Colby College

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**March 18, 2024**

Oscar Fernandez

Wellesley College

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**April 1, 2024**

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**April 8, 2024**

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**April 15, 2024**

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**April 22, 2024**

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**April 29, 2024**

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**May 6, 2024**

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## 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
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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.