Student Seminars will be held on Thursdays in Bailey 207 (unless otherwise noted) 12:50pm, during common hour throughout fall term. We will have a pizza lunch at 12:30pm in Bailey 204.
See you in the Fall '25 Semester!
Previous Seminars Spring '25:
May 22nd 2025- Union College Professor of Mathematics, Brenda Johnson Presents:
Title: Fixed Points and Fermat
Abstract: If you pick an integer a and a prime number p, the difference a^p-a will be divisible by p. This fact is known as Fermat's Little Theorem and is a standard result in number theory courses. There are many ways to prove this theorem, often involving ideas from number theory and abstract algebra. A less standard approach uses functions one might see in a first-year calculus course together with basic results from the field of dynamical systems. In this talk, we will see how to prove Fermat's Little Theorem and some related results in number theory from this dynamical systems perspective.
May 15th- Special Guest Speaker: Professor Moon Duchin (Cornell University, Math)
Thursday, May 1st- Professor Matthew Lamoureux, University of Connecticut's College of Engineering's School of Computing
Presents: A Gentle Introduction to Text Mining
Abstract: Many of the tools we use to analyze information rely on structure, so we often see data provided neatly in rows and columns (think Excel spreadsheets). That said, lots of data is unstructured yet still valuable; specifically, there is no shortage of written text in our world—books in our libraries, message boards on Reddit, customer feedback on products. This presentation will provide a quick overview of common techniques used in processing unstructured text data.
Thursday, April 24th- Assistant Professor of Mathematics, Junqing Qian presents: An Introduction to complex numbers and conformal maps
Abstract: Conformal mappings - angle-preserving mappings - have many real-life applications. To name a few, it has applications in Cartography (map-making), Aerodynamics, Electrostatics & Fluid Dynamics, Medical Imaging, etc. In today's talk, we will start from the basics of complex numbers to complex differentiation. Then we will present how being differentiable in the complex sense is equivalent to being an angle-preserving mapping. A concrete example of the conformal map cos(z) will be presented at the end.
Thursday, April 10th-Union College Assistant Professor of Mathematics, Phanuel Mariano presents: The isoperimetric inequality and its connection to probability
Abstract: The isoperimetric inequality is a geometric result relating the square of the circumference of a closed plane curve to the area it encloses, along with various generalizations of this relationship. There is a connection between inequalities that come up in probability and the classical isoperimetric inequality. The connection is through Brownian motion, which is a mathematical model for the random movement of a particle. It was first observed by Robert Brown in 1827 while looking at pollen grains through a microscope. I will discuss some classical isoperimetric problems related to the expected lifetime of Brownian motion from a domain.
Student Seminars for Winter '25:
Thursday, March 6th-Union College Math Professor Leila Khatami presents: A Game of Commuting Matrices
Abstract: One of the first things we learn about matrix multiplication is that it is not commutative; that is, if A and B are two square matrices of the same size, in general AB is not equal to BA. However, some pairs of matrices do commute. For example, if one of the matrices is the identity or zero matrix, or if A and B are the same, then AB = BA. This leads to a natural question: which pairs of matrices commute? More specifically, given a matrix, can we characterize matrices that commute with it? In this talk, we explore this question through the lens of a mathematical game, which can help us understand some key tools used in studying this problem.
Thursday, February 13th-Union College Emeritus Professor, Julius Barbanel Ph.D. Presents: The Euclidean Algorithm and Irrational Numbers
Abstract: The Euclidean Algorithm is a procedure for determining the greatest common divisor of two positive integers. Irrational numbers are real numbers that cannot be expressed as the ratio of two integers. These two ideas certainly do not seem to be related, but we shall explore a rather surprising historical connection between them. This exploration will include a quick tour of ancient Greek mathematics.
Thursday, February 6th- It is Movie Time!
Union College Mathematics Department Presents: The Proof
The Proof: This documentary produced by Nova uses the search for a proof of Fermat's famous theorem to
provide a better understanding of the quest for knowledge. It is this quest that apparently motivates
mathematicians to attempt to solve what many feel is the unsolvable. For those of you unfamiliar with Fermat's Last Theorem, it states that for all natural numbers n greater than 2, 𝑥^𝑛 + 𝑦^𝑛 = 𝑧^𝑛 has no solutions for natural numbers x, y, and z. Over 300 years ago, Fermat stated that he had a proof of this result, but the margin of the paper was too small to include it. Because this problem had an obvious link with the Pythagorean Theorem (and because Fermat stated that he had a proof) mathematicians felt the argument would be found immediately. But for centuries people struggled to uncover the proof, and it was not until the mid 1990s that Andrew Wiles of Princeton University finally succeeded. For further information on the documentary, see https://www.pbs.org/wgbh/nova/proof/
Spring term 2024
May 30th-Professor of Mathematics, Emeritus- Karl Zimmermann
Odd Numbers, Square Numbers, and Pythagorean Triples
In this talk, we explore the relationship between odd numbers, square numbers, and Pythagorean triples (positive integer solutions to the equation x^2 + y^2 = z^2). We begin by using the sequence of odd numbers to explain certain patterns in the sequence of square numbers. Along the way, we see examples of Pythagorean triples arising naturally in the sequence of squares and use some of these triples to provide intuition as to why every odd integer greater than one is the leg of a Pythagorean triangle. After a hint at how to classify Pythagorean triples using the sequence of odd numbers, we look at how these ideas might be applied to the study of positive integer solutions, or lack of solutions, to the Fermat equation, x^n + y^n = z^n, for n = 3.
May 23rd-Maryam Mirzakhani
Movie presented: Secrets of the Surface
Secrets of the Surface introduces Maryam Mirzakhani to a wider public, telling the story
of her life as recounted by friends and colleagues, looking at her mathematical work
through the words of collaborators, and gauging her impact on future generations.
The film opens with a scene filmed at the Kherad Institute in Tehran in March 2019.
A group of high school girls are hard at work on a blackboard crowded with
geometric shapes and equations. One of them, Delara Jandaghi, explains that “there
is a very good feeling behind solving the problem. You know, when you try hard and
you can’t find the solution, and suddenly… ‘Ah! That’s it.’ And I think Maryam
Mirzakhani could show this passion to everyone.”
May 16th - Professor Greg Malen, Skidmore College
Graph Homomorphisms and Topological Bounds on Chromatic Number
A common theme in mathematics is to try to learn about an object by mapping it, or encoding it, into another object. In graph theory, we do this using graph homomorphisms, which are maps from a graph G to a graph H that preserve adjacencies. For two fixed graphs G and H, you can then consider all of the different possible ways to map G into H. This set of all such maps, called Hom(G,H), can be viewed as a topological object, and it turns out that we can then use some basic tools in topology to study important properties of the graphs we started with. In this talk, I will give an introduction to graph homomorphisms and the space Hom(G,H) (with lots of images and diagrams!), and explore its connection to the chromatic number of a graph.
May 2nd - Professor Emeritus Julius Barbanel, Union College
A Fun Probability Problem
Consider the following:
100 prisoners are numbered 1, 2, …, 100.
100 boxes are labeled B1, B2, …, B100.
100 pieces of paper have the numbers P1, P2,…, P100 written on them.
The pieces of paper are randomly distributed in the boxes, one paper in each box. Each box is closed so no one can see which paper it contains.
One by one, each of the prisoners gets to pick 50 boxes. Each prisoner’s goal is to find the box that has their number in it. The prisoners all go free if and only if every prisoner finds their number. If even one prisoner fails to find their number, then no prisoners go free.
The prisoners go one at a time. After each prisoner’s turn, the boxes and papers are returned to their original state.
The prisoners may discuss strategy before the experiment starts, but no communication is allowed once the experiment begins.
If each prisoner picks 50 boxes randomly, it is not hard to show that the probability that the players all succeed is approximately zero! We will show how this radically improves if the prisoners all agree to follow a certain strategy.
April 18th - Amalia Jerison
An Introduction to the Period Conjecture
Periods are complex numbers whose real and imaginary parts can be expressed as integrals of rational functions with rational coefficients, over domains defined by polynomial inequalities with rational coefficients. Kontsevich (1999) and Kontsevich and Zagier (2001) conjectured that all linear algebraic relations among periods can be derived from the basic operations on integrals - linearity, change of variables and Stokes' theorem. In this talk I will introduce the conjecture in its simplest form and in an equivalent form in terms of cohomology groups.
April 11th - Professor Kim Plofker - Union College Mathematics
Cycles, Spheres, and Demons: Understanding Eclipses in Pre-modern Mathematics
For thousands of years, lunar and solar eclipses have been very significant events in human beings' understanding of the cosmos. They also played a huge role across cultures in the shaping of mathematical practice, as a discipline and as a profession. This talk investigates how and why early mathematicians managed to turn these awe-inspiring phenomena into something familiar and predictable.
Winter term 2024
There will be two late-afternoon math seminars, with refreshments being served beforehand at 4:00pm in Bailey 204, the Math Common Room. We hope to see you there!
• Thursday, March 7th, 4:45 – 5:45pm in Bailey 207
• Monday, March 11th, 4:45 – 5:45pm in Bailey 207
- Please join us
NOTE: time change -- 2:00pm with light refreshments in Bailey 204 following the talk
Feb 29, 2024 - Mark Mortensen, Class of 1985
How To Prepare for an Actuarial Career
Actuaries are well-trained and highly respected risk management professionals who use a lot of math and statistics to solve problems in many industries and government. I will provide examples of the types of risk analysis an Actuary might perform and the tools an Actuary might use. I will give an overview of the education and examination process and talk about ways that large insurance companies help recent college graduates to complete the requirements. I will provide steps you can take now to prepare for your actuarial career.
This week, there will be two late-afternoon math seminars, with refreshments being served beforehand at 4:15pm in Bailey 204, the Math Common Room. We hope to see you there!
• Tuesday, January 16, 4:45 – 5:45pm in Bailey 207
• Friday, January 19, 4:45 – 5:45pm in Bailey 207
- Please join us Thursday, January 11th at 4:45pm - Bailey Hall 207
- Refreshments will be available at 4:15pm in Bailey Hall 204
- Please join us Tuesday, January 9th at 4:45pm - Bailey Hall 207
- Refreshments will be available at 4:15pm in Bailey Hall 204
Fall term 2023
Nov 9, 2023 - Math seminars are done for the term, but...
...Join us for a joint Mathematics and Economics Student Lunch Seminar at 1:00-1:30pm in Bailey 207.
This is an information session on economics graduate programs at Rensselaer Polytechnic Institute (RPI). Learn about RPI's brand new PhD program in Applied Economics and Policy and RPI's MS program in Economics.
There will be a Q&A session and food will be provided.
Nov 2, 2023 - Ralph Morrison, Williams College
Chip-firing on banana paths
In this talk, we'll explore chip-firing games on graphs. Start with a graph, which is a
collection of nodes connected by edges. Then, place some poker chips on the vertices of the graph. Finally, move the chips around the graph by selecting a node and having it donate chips to its neighboring nodes, one along each edge connected to it. We'll study what these games look like on a family of graphs called "banana paths", which have all their nodes in a row, with (possibly many!) edges connecting each node to the one or two nodes next to it. Our main question is this: what's the smallest number of chips we can place on a banana path so that we can move a chip to any node we like, without having a negative number of chips anywhere? This is joint work with undergraduates from Williams' SMALL REU in 2023: Marchelle Bougher, Nila Cibu, Cassie Ding, Steven DiSilvio, Sasha Kononova, Chan Lee, and Krish Singal.
Oct 26, 2023 - Nathaniel Whitaker, UMass Amherst
a mathematical journey through segregation and hidden figures
In this talk, I will describe my journey from segregation to becoming a research
Mathematician of African descent, a rarity in mathematics. This journey is in the
backdrop of Virginia’s massive resistance to integration and happened in the same
community with the characters in the movie and book Hidden Figures. My journey
continued in becoming the Department Head of a major research mathematics and
statistics department and the Dean of a major research university. I will also describe
briefly some of my research in fluid mechanics and math biology.
Oct 19, 2023 - Harris Daniels, Amherst College
The congruent number problem
Much effort has gone into studying right triangles with rational side lengths. One
interesting question associated with these geometric objects is, what values occur as the area of such triangles? In fact, a 10th century manuscript asserts that this question is the "principal object of the theory of rational right triangles." We will survey what is known about this question, show how the question is related to elliptic curves, and end with a conjecture about the answer that is still open today.
Oct 2, 2023 - Leon Tatevossian, New York University
Modeling Stock-Price Evolution:
An Introduction to Diffusion Processes
The study of stock-price dynamics is formulated via the interplay between statistical
analysis of historical behavior (examining attributes like the potential stability of the running variability) and a “parametric” approach, where a specified probabilistic model is taken as the driving process. The model direction of these two themes leads to the
concept of stochastic differential equations (SDEs), a fundamental tool of “continuous time finance.” I’ll discuss the most-familiar SDE for stock-price propagation and explain the many ways in which observed behavior and economic arguments challenge the validity (and implications) of that model.
Oct 5, 2023 - Georgia Doing, Jackson Laboratory for Genomic Medicine
Wrangling microbial data
Of the trillions of microbial cells associated with a human body, only a handful have been studied in the laboratory. Understudied microbial genetic diversity, including that represented on the skin by staphylococci, is important for clinical outcomes of human hosts but too expansive to comprehensively study with molecular and biochemical techniques alone. Computational tools are a promising approach for investigating high dimensional microbiological data which are susceptible to machine learning. Novel gene annotations can be inferred by balancing the performance of black-box machine learning and the interpretability of linear correlations in algorithms with scalability to
match that of microbial diversity. In this talk I will present how such an approach can help analyze novel genes from two skin microbes: pathogen Staphylococcus aureus and
commensal Staphylococcus epidermidis.
Sep 28, 2023 - Kirsten Hogenson, Skidmore College
Rainbow static mastermind
The Rainbow Static Mastermind is a game where a player tries to guess a secret
sequence of n distinctly colored pegs. The player submits a list of guesses and they receive two
pieces of feedback per guess: the number of correct colors in the correct positions and the number
of correct colors in the wrong positions. To win, the player must correctly determine the secret code
based on this feedback. Mastermind is a well-studied game with applications to artificial intelligence,
data security, and bioinformatics. In the static version, researchers often seek a shortest list of
questions which is sufficient to win the game. In this talk, we will discuss optimal question lists in the
n=2 and n=3 cases. This research was completed during June 2023 and supported by Skidmore
College’s Faculty-Student Summer Collaborative Research grant program.
Sep 21, 2023 - Keith Conrad, University of Connecticut
APPLICATIONS OF DIVERGENCE OF THE HARMONIC SERIES
The harmonic series is the sum of all reciprocals 1 + 1/2 + 1/3 + 1/4 + ..., and a famous
counterintuitive result in calculus is that the harmonic series diverges even though its general term tends to 0. This role for the harmonic series is often the only way students see the harmonic series appear in math classes. However, the divergence of the harmonic series turns out to have applications to topics in math besides calculus and to events in your daily experience. By the end of this talk you will see several reasons that the divergence of the harmonic series should be intuitively reasonable.
Sep 14, 2023 - Mayah Teplitskiy, Union College
Our Quasi-Excellent Summer is Complete!
Have you ever heard of a quasi-excellent ring? No, not the one you wear on your finger!
Quasi-excellent local rings are really important algebraic structures used throughout
different fields of math. We are specifically interested in studying the relationship
between a quasi-excellent local ring and its completion. It is interesting to ask questions
like "When is a complete local ring T the completion of a quasi-excellent local integral
domain?" or "If T is the completion of a quasi-excellent local integral domain R, what is
the relationship between the prime ideals of T and R?" We provide answers to these
questions and take important steps towards being able to construct a local domain with
any desired prime ideal structure.