# 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.

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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 – Diamond 141**

Fatou Sanogo

Bates College

## Fall 2022

**September 19**

Fernando Gouvêa

Colby College

Mr. Euler, you can’t do that!

Abstract: As we all know, in mathematics you have to do things carefully and correctly. But nobody seems to have told Leonhard Euler, the greatest mathematician of the 18th century, certain things that all calculus students know. This talk is about Euler doing the wrong things and still getting the right answer. Everything in the talk is about infinite series and figuring out what their sums are, so if you know about series and convergence you’ll enjoy following Euler’s steps.

**October 24**

Stephanie Dobson

Colby College

Animal communication enables collective migration in a dynamic ocean

Abstract: Blue whales (Balaenoptera musculus) in the eastern North Pacific ocean migrate annually between southern breeding and northern foraging, yet the precise mechanics and cues of migration remain unknown. The population is known to communicate with a variety of call types over distances of hundreds of kilometers. Empirical evidence from hydrophones proposes that these calls are a main driver of the breeding migration, but it is believed that environmental cues and events, social communications, and breeding patterns all factor into migration decisions. Here, we utilize an individual-based model to examine the role of long-range social communication in driving the timings of the collective migrations. We measure and compare population-level outcomes of migration decisions based on combinations of individual foraging behaviors, prey intake, and social communications against a yearday-driven migration mechanism and a null (non-migratory) model. Our results help extract the important role of social communication in blue whale migrations and find that social communications are necessary to reproduce realistic and observed migration patterns. Moreover, the socially informed strategies enable flexible migration timings that maximize prey intake in dynamic conditions and anomalous years.

**October 31 Diamond 122**

Leo Livshits

Colby College

“What Is … ?” Colloquium: Lengthy Adventures

Abstract: I was once told to imagine “unbending a curve without stretching it”. My mind’s eye produced images of a thin piece of wire being unbent, and of wire molecules being rearranged in the process. “Isn’t the outer length of a semi-circular wire longer than the inner length?” teased the Devil’s advocate. “Think of a bend in a road; isn’t this why we use the differential in cars?”

I tried to quiet the little Devil by imagining thinner and thinner wires, and was gaining an upper hand until the pesky objector noted that “at some point you have to go so sub-atomic that there is no discernible wire to speak of.” But if I abandon the idea of “unbending without stretching”, how exactly do I make sense of a length of a curve? When the inner voice piped in with “… and what is a curve, anyway?”, I decided it was time to get serious about the issue.

In this “What is … ?” colloquium aimed at students, we will discuss common and less common notions of length for a curve in multiple dimensions and go on a brisk jaunt through some subtle but fundamental ideas of Analysis. The talk will serve as a shameless advertisement of the value of MA338 & MA439 to one’s intellectual development and peace of mind.

**November 7 (Runnals Dinner Guest Speaker) Olin 001**

Talea Mayo

Emory University

Climate Change Impacts on Hurricane Storm Surge Risk

**November 28 Diamond 122**

Changningphaabi Namoijam

Colby College

Number Theory and Function Fields

Abstract: In classical number theory, we study integers and objects related to integers. We can consider polynomials as analogues of integers because there are many results in the world of polynomials that complement ones from the classical case. For example, there is an analogue of the irrational number π. In this talk, we will explore this analogous π and some other number theoretic results relating to polynomials.