The Joint UNC and Duke Student Math Colloquium meets on Tuesdays, 4:30-5:30 pm, and is preceded by a special tea at 4 pm. To receive announcements about this program, send an email with no message and no subject to firstname.lastname@example.org. This program is supported by an award from the Kenan-Biddle Partnership, which is funded by the William R. Kenan Charitable Trust and The Mary Duke Biddle Foundation.
Student organizers: Humberto Diaz (Duke), Katrina Morgan (UNC), Dylan Muckerman (UNC).
Faculty advisors: Linda Green (UNC), Heekyoung Hahn (Duke).
|March 17, 2015 at UNC, Phillips Hall 332
||Jared Wunsch (Northwestern University)
||Trace formulae on smooth and singular manifolds
|April 7, 2015 at UNC, Phillips Hall 332
||Farshid Hajir (University of Massachusetts, Amherst)
||200 Years of Classifying Integer Solutions of d=b²-4ac
|April 14, 2015 at Duke University, Gross Hall 103
||John Voight (Dartmouth College)
||Triangles, permutations, and (covers of) surfaces
|October 21, 2015 at Duke, Physics 119
||Jeremy Rouse (Wake Forest University)
||Positive-definite quadratic forms representing all odd numbers
Trace formulae on smooth and singular manifolds
The Poisson summation formula tells us that the sum of the values of a function on the integers is the same as the sum of the values of its Fourier transform. I will explain how this beautiful formula can be generalized to any smooth manifold by thinking about solutions to the wave equation; this results in the celebrated Duistermaat-Guillemin trace formula. I will end by discussing a recent generalization of this formula to certain kinds of singular manifolds (those with cone points) obtained in joint work with G. Austin Ford. Most of the talk will be expository.
200 Years of Classifying Integer Solutions of d=b²-4ac
This survey talk for undergraduates is about an undergraduate student named Carl who collected together some of the mathematical thoughts of his teenage years and published them as a book when he turned 23. The book went viral, but nearly everyone who got a copy couldn’t really make head or tails of it, even though they could tell it was cool and important. One of Carl’s sharpest disciples, Pete, slept with a copy of the book under his pillow and wrote many articles over his lifetime trying to explain to himself and others what Carl was talking about. Even today, whether they know it or not, undergraduate math majors who take number theory or abstract algebra in college are indirectly studying Carl’s teenage musings. Despite thousands of papers and books written on the subject since then, verifying, amplifying, and contextualizing Carl’s ideas, including some recent work that garnered a 2014 Fields Medal, some of Carl’s most basic conjectures are still wide open. My goal for this talk is to explain enough about the classification of primitive integral binary quadratic forms of a given discriminant so that students will: (1) learn about the historical roots of some of their current curriculum and connect it with the current research of some of Carl’s disciples at UNC and Duke, (2) learn about an as yet unsolved problem in number theory, and (3) make a beeline for the library (or their laptops?) in search of a book that, in the view of some, is one of the greatest ever written.
Triangles, permutations, and (covers of) surfaces
There is a marvelous and deep connection between:
– surfaces obtained by gluing together copies of a triangle,
– triples of permutations whose product is the identity, and
– bicolored graphs equipped with a cyclic orientation, and
– sheeted covers of the sphere branched above at most 3 points.
The consequences of these equivalences are myriad for geometry, arithmetic, combinatorics, and group theory. In this talk, we will introduce these connections via examples and pictures.
Positive-definite quadratic forms representing all odd numbers.
Given a positive-definite quadratic form Q with integers coefficients it turns out
that Q represents every odd integer if and only if Q represents the numbers from 1 up to 451.
Rather than discussing the proof of this result, I will explain my mathematical journey in understanding
the problem (including a lot of wrong turns), and a bit about how it has shaped my future as a mathematician.
This talk will be accessible to mathematicians at all levels.