The Trinity College Mathematics Colloquium meets periodically throughout the semester, usually on Thursdays from 12:15–1:15 pm in MECC 270. Talks are designed to be engaging and accessible to mathematicians across all fields. Talks are followed by lunch in the mathematics lounge.
Thursday February 12 - Felipe Ramírez (Wesleyan University)
Title: A Century Of Metric Dichotomies In Diophantine Approximation
Abstract:
The basic question of Diophantine approximation is: how well can you approximate irrational numbers by rational numbers? A classical theorem of Dirichlet (c. 1840) says that every real number x can be approximated by infinitely many rationals p/q to within a distance of 1/q^2. Since then, much of the research in Diophantine approximation has been animated by follow-up questions. Can 1/q^2 be replaced by some other function of q? Can we establish similar results if restrictions are placed on p/q, such as coprimality of p and q? Can we prove analogous results in higher-dimensional settings, like x in R^m? Very quickly, answers become metric/probabilistic. In 1924 (m=1) and 1926 (m>1), Khintchine proved the foundational result of metric Diophantine approximation, a zero-one law answering the aforementioned questions in a measure-theoretic way. I will discuss Khintchine's theorem and the follow-up questions to Khintchine's theorem that have animated much research over the last century, including some recent work in the setting where X is an n-by-m system of linear forms.
Thursday February 26 - Ann Trenk (Wellesley College).
Title: Neighborhood balanced graphs
Abstract:
We assign a color (red or blue) to each vertex of a graph, called a red-blue coloring, and it is permissible for adjacent vertices to get the same color. A graph is called a closed neighborhood balanced graph (CNBC) if it has a red-blue coloring in which for each vertex v there are an equal number of red and blue vertices in v's closed neighborhood. Analogously when open neighborhoods are considered the resulting graphs are called neighborhood balanced graphs (NBC). Both NBC and CNBC graphs can model situations where it is desirable to distribute two types of resources to vertices in a graph so that each vertex has a balance of resources among its neighbors.In this talk we discuss a variety of results about CNBC and NBC graphs and generalizations of them that allow for k colors and more flexibility on the number of vertices of each color in a neighborhood.
Thursday March 12 - Jonathan Lindbloom (Dartmouth College)
Title: Inverse problems through the lens of linear algebra
Abstract:
Many problems in science and engineering share a common structure: given noisy, indirect measurements of some unknown quantity, can we reliably reconstruct it?Problems of this type --- called inverse problems --- arise in a wide range of applications, including medical imaging, imaging of black holes, and real-time tsunami forecasting.
In this talk, we introduce some classical and modern solution strategies with particular emphasis on the underlying tools from linear algebra, culminating in recent research on hybrid projection methods and preconditioning techniques which accelerate computation for large-scale problems.
Thursday March 26 - Keith Conrad (UConn)
Title: Patterns that don't last
Abstract:
Conjectures in math are often based on finding patterns that last a while, but how much evidence is enough to make a conjecture plausible? For instance, if a pattern holds for the first hundred examples, is it likely to be true all the time? What if it holds for the first thousand examples? The goal of this talk is to survey various numerical patterns that hold for a while until they eventually stop being true, and the first breaking point will happen at increasingly larger numbers.
Thursday April 2 - Han-Bom Moon (Fordham University)
Title: Let's count points!
Abstract:
A fascinating fact in mathematics is that there are many interesting connections between seemingly different mathematical disciplines. In this talk, I will present a surprising formula counting integral points on polygons and sketch its proof. We will see a delightful interaction between algebra, geometry, and combinatorics. No prerequisite is assumed beyond single variable calculus. Everyone is welcome!
If you are interested in giving a talk, please get in touch!