The department colloquium is a series of talks given by mathematicians and statisticians aimed at an undergraduate audience. Everyone is welcome to attend!

Fall 2024 Schedule

Date	Speaker	Affiliation
Friday, September 20th	Ophelia Adams	University of Rochester
Friday, October 4th	Jeff Goldsmith	Columbia University
Tuesday, October 22nd	Dean Spyropoulos	Michigan State University
Monday, October 28th	Zheng Bian	Clarkson University
Thursday, November 14th	Joshua Snoke	RAND Corporation
Friday, November 22nd	Ivan Cheltsov	University of Edinburgh
Wednesday, December 4th	Nikolas Schonskeck	Rockefeller University

Over the last 10-20 years, applications of topology to fields such as neuroscience, genomics, and data analysis have exploded in number. The most recent conference on applied topology in Oxford drew hundreds of participants from all over the world. In this talk, we will provide an introduction to how algebraic topology can be used to find the “shape of data” and how that shape can encode information that would otherwise be inaccessible. We will begin by introducing some of the main tools from algebraic topology used in modern topological data analysis such as simplicial and persistent homology. Time permitting, we will then apply these tools in the context of neuroscience and explore how neural circuits learn to represent topologically complex coding patterns. This talk is aimed at a broad undergraduate audience. No familiarity with topology will be assumed.

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Every day we deal with geometric objects defined by algebraic equations (circles, parabolas, hyperbolas, splines, spheres, hyperboloids, etc). In many applications, we have to parametrize them using the simplest possible functions: rational functions in several variables. To find such a parametrization may be tricky. This is a classical problem; Pythagoras found a rational parametrization of the circle when he found Pythagorean triples, and the same approach gives explicit rational parametrizations of the sphere or any geometric object given by one quadratic equation. In more complicated cases, the problem can be difficult. Quite often rational parametrizations do not exist. For example, most planar cubic curves cannot be parametrized by rational functions (the proof for the Fermat cubic curves follows from Euler’s proof of Fermat’s Last Theorem for exponent 3). In this talk, I will focus on the existence of rational parametrizations of cubics–geometric objects defined by one equation of degree 3.

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Researchers and agencies who collect data often want to share the data for purposes of education, training, or expanding research opportunities. The creation of synthetic data, i.e., simulated data generated to contain similar properties to the confidential data, has become a popular approach for sharing versions of the data while preserving privacy. This talk will cover some of the common methods used for generating and evaluating synthetic data, highlighting the benefits and drawbacks. I will then present recent work on the development of synthetic data methods for survey data, which contain additional complexities such as multiple levels of observations, non-proportional sampling, and survey weights. These methods are used in an application to synthesize the Longitudinal Aging Study in India (LASI), a panel study of key economic, social, and health characteristics of India’s older population.

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As a leaf falls slowly but almost surely, numerous microscopic collisions with air particles compose their erratic movements into the macroscopic symphony of the wind. This is a vivid example of the mean-field—an effective description of high-dimensional dynamics by a low-dimensional system. In a large network, where units evolve and interact with their neighbors, we describe the behavior of a hub—a very well connected unit, in terms of the mean-field, subject to statistically controlled fluctuations. More detailed insights will be presented in this talk, including a scaling relation between network size and frequency of large fluctuations, the network size induced desynchronization, and the Gaussian statistics of the fluctuations. Time permitting, we will also state the Theorems behind these phenomena and give hints to the proofs.

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A 3-manifold is a topological space which, at any given point, looks a lot like the space in which we live. One reason to care about knots is that we can produce (many!) 3-manifolds from them. After getting comfortable with this correspondence, I’ll describe a situation in which we can promote a knot invariant to a 3-manifold invariant. The key ingredients are particular objects which unite representation theory and low-dimensional topology, called Jones-Wenzl projectors.

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In the last ten years, technological advances have made many activity- and physiology-monitoring wearable devices available for use in both clinical trials and large-scale epidemiological studies. This trend will continue and even expand as devices become cheaper and more reliable. These developments open up a tremendous opportunity for clinical and public health researchers to collect critical data at an unprecedented level of detail, while posing new challenges for statistical analysis of rich, complex data. This talk will present a collection of examples and analysis approaches that use accelerometer data, including activity classification; identifying and interpreting variability in activity trajectories; building regression models in which activity trajectories are the response; and understanding shifts in the circadian rhythms that underlie the timing of activity. We’ll draw on several applications, including the Baltimore Longitudinal Study of Aging and data collected through the Columbia Center for Children’s Environmental Health.

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In casual conversation, “numbers” usually mean the real numbers, which have a familiar
feel as the numbers of measurement. The mathematical constructions–first appearing
in the 19th century–that put them on a more rigorous foundation situate them in a
particular way, as completions, relative to the rational numbers. In this talk, we will
explore the p-adic numbers, a central object in modern number theory. These are a
similar-but-stranger kind of numbers which may also be constructed as a completion
of the rational numbers, but were not discovered until the early 20th century.
Curiously, centuries and millennia before the developments above, mathematicians (in
many dierent times, cultures, and places, for reasons both pure and applied) came
close to discovering the p-adic numbers. We will trace one of these imagined
developments to motivate and construct the p-adic numbers ourselves, study their
properties, and indicate how they, and the considerations which could have led to
their earlier discover, anticipate other modern developments in mathematics and
nearby fields.

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Spring 2024 Schedule

Date	Speaker	Affiliation
Tuesday, March 26th	Phanuel de Andrade Mariano	Union College
Tuesday, April 9th	Anna Pun	CUNY Baruch College
Friday, April 26th	Leo Goldmakher	Williams College

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Are there infinitely many primes of the form n^2+1, where n is an integer? No one knows. In fact, there’s no example of any (single variable) polynomial of degree 2 or greater that’s been proved to output infinitely many primes. By contrast, the linear polynomial n+1 outputs infinitely many primes, a fact that’s been known for over 2000 years. Rather less trivially, Dirichlet proved in 1837 that any linear polynomial of the form an+b with a, b coprime must output infinitely many primes. To make his proof work, Dirichlet introduced certain nice functions called characters, which evolved (over the course of the next hundred years) into fundamental objects of study in algebra and number theory. I will discuss some of the history and mathematics of Dirichlet’s characters, including a very recent and simple characterization of them that seems to have been previously overlooked.

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Tableaux are one of the most fundamental and versatile objects in algebraic combinatorics, as they can encode and connect various concepts and structures in the field. In this talk, we will start with the definition and properties of Young tableaux, which are graphical representations of partitions of integers. We will then see how tableaux can be used in algebra: their connection to symmetric functions and partition algebras; how they can be related to various combinatorial operations, such as the RSK-algorithm and the Jeu-de-taquin procedure; and how they can give rise to various combinatorial structures, such as lattice paths, vacillating tableaux, and parking functions. We will also explore some variations of tableaux, such as composition tableaux and set-valued tableaux, and discuss some interesting problems and conjectures that arise from them. We will conclude with some open questions and directions for future research on tableaux and their applications in algebraic combinatorics.

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Consider a perfectly insulated 1-dimensional rod, or a 2-dimensional plate, or better yet, a 3-dimensional room. Perfectly insulated means that heat cannot escape this room. The Hot Spots Problem asks about what happens to the location of the “hot spots” and “cold spots” of this insulated body over a long period of time. To understand this problem we will introduce the equation that describes the evolution of heat over time. Moreover, we will discuss what is known (and not known) about this problem. We end the talk by discussing the connection between the Hot Spots Problem and Probability theory. In particular, this connection will be through a theory of random particles called Brownian motion.

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Fall 2023 Schedule

Date	Speaker	Affiliation
Tuesday, October 3rd	Colin Adams	Williams College
Wednesday, October 11th	Pablo Soberón	CUNY Baruch College
Thursday, October 26th	Joe Kraisler	Amherst College
Thursday, November 2nd	Karen Parshall	University of Virginia
Thursday, November 16th	Colby Kelln	Cornell University

Imagine we are hired to tile an infinitely large bathroom floor. What tile shapes could we use? We will use math to explore and refine this question to make sure that our client is happy with our plans before we start laying grout.

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World War I had marked a break in business as usual within the American mathematical research community. In its aftermath, there was a stirring sense of entering into “a new era in the development of our science.”  And then the stock market crashed. Would it be possible in such newly straitened times to sustain into the 1930s the momentum that American mathematicians had managed to build in the 1920s? This talk will explore the contours of an answer to that question.

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Electronic band theory was one of the early 20th century achievements of quantum mechanics and placed solids into three categories: conductors, semiconductors, and insulators. However, starting in the 1980s with the discovery of the Quantum Hall Effect, a new phase of matter known as Topological Insulators (TIs) were theorized and eventually realized. These materials act as insulators in the interior, or bulk, while allowing electrons to freely move along the boundary, or edge, of the material. Additionally, there is a relationship between the a) number of states which exist on the boundary and b) a property of the interior which is protected under small defects. This relationship is often referred to as the Bulk-Edge Correspondence. 

We will study the simplest example of a 1 dimensional topological insulator, the SSH (Su-Schrieffer-Heeger) model of polyacetylene, and prove the bulk edge correspondence for this particular system. No previous physics knowledge is required.

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The study of intersection patterns of convex sets is a central topic in combinatorial geometry.  In this talk, we will discuss the applications of this area to two different topics: art galleries and voting theory.  

In art gallery problems we seek conditions on the blueprint of an art gallery that guarantee that few guards can keep every painting safe.  In voting theory, given a group of people such that every person has an interval of tolerance in different topics, we seek conditions that guarantee that all such intervals overlap.  We focus on connections of these two topics with quantitative Helly theorems, which characterize finite families of convex sets whose intersection is not only non-empty, but quantifiably large.

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Being a tale of adventure on the high seas involving great risk to the tale teller, and how an understanding of the mathematical theory of knots saved his bacon. No nautical or mathematical background assumed!