Faculty Profile

Amy DeCelles

Assistant Professor of Mathematics

Education

Research

My research is in the field of number theory, arguably the oldest branch of mathematics.

Prime numbers are the most elementary numbers after the whole numbers, yet primes are subtle. Mathematicians have been seeking to understand the patterns in primes for centuries. The celebrated Riemann Hypothesis is the archetype of an unanswered question involving primes. Riemann’s brilliant observation was the precise connection between primes and zeroes of zeta. Questions regarding prime numbers can thus be addressed by studying zeta functions and L-functions.

My research applies the spectral theory of automorphic forms on higher rank groups to number theory. While proof of the Riemann and Lindelof hypotheses and their analogues for arbitrary L-functions remain elusive, understanding asymptotics or obtaining subconvex bounds for L-functions is sufficient for some number-theoretic applications. Diaconu and Garrett have extracted subconvex bounds from spectral identities involving second moments of automorphic L-functions for GL(2) over an arbitrary number field, and my thesis develops techniques for treating higher rank groups. An interesting sample application is lattice-point counting in symmetric spaces. The long term goal is to obtain asymptotics and subconvex bounds for automorphic L-functions attached to higher rank groups.

Courses

During the Spring 2012 semester, I taught Precalculus Mathematics, Calculus II, and Modern Geometry.

Precalculus Mathematics: modeling the dependency of one quantity upon another with linear, polynomial, power, rational, exponential, logisitic, logarithmic, and sinusoidal functions.

Calculus II:  applying the techniques of calculus to the distribution of the prime numbers and writing more formal mathematical arguments.  Along the way we develop the necessary calculus techniques, treating more carefully some of the topics of first semester calculus including the limit, the derivative, the integral, and the  fundamental theorem of calculus, as well as new topics including advanced integration techniques, sequences, and series.  The question of the distribution of primes is an important question, both because of its significance in the history of mathematics and because of its connections to open questions in mathematics today, including the Riemann Hypothesis.

Modern Geometry: following Stillwell’s Four Pillars of Geometry, we study Euclidean geometry: compass and straightedge constructions, axioms; analytic geometry: coordinates, vectors, transformations, geometry over fields; projective geometry: perspective drawing, axioms; transformation groups: isometries, non-Euclidean geometry; axiom systems and finite geometries; fractal geometry.  In addition, we will have two research projects: a short historical project, to be done individually, and a more extended exploration project, to be done in groups.

During the Fall 2011 semester I taught Calculus I and Linear Algebra.

Calculus I: studying the mathematics of continuously changing phenomena. In this course we will learn to describe such phenomena mathematically, identify which methods are appropriate to answer the question, answer the question using the methods of calculus, and present conclusions in plain English.

Linear Algebra: studying the mathematics of phenomena that depend linearly on several parameters. As a first example, to develop geometric intuition, we consider linear transformations of vectors in the plane. Studying systems of linear equations motivates the study of solving matrix equations, along with matrix decompositions. Finally we study abstract vector spaces and inner product spaces.

Lectures & Presentations

“Automorphic spectral identities, integral moments of L-functions, and lattice points in a symmetric space.”  University of Michigan Group, Lie, and Number Theory Seminar, Ann Arbor, MI, January 2012.

“Integral moments for Rankin-Selberg  convolutions for GL(n) × GL(n) over a totally complex number field.”  Purdue University Automorphic Forms Seminar, West Lafayette, IN, January 2012.

“Explicit fundamental solution for (Δ- λ)^ν on G/K.”  University of Notre Dame Lie Theory Seminar, Notre Dame, IN, December 2011.

“Geometry, Arithmetic, and Questions about Numbers.” Purdue University Calumet Mathematics Colloquium, Calumet, IN, November 2011.

“Number theoretic applications of the automorphic spectral theory of higher rank groups.” Ohio State Number Theory Seminar, Columbus, OH, October 2011.

“What is a number?” Wabash College Mathematics Colloquium, Crawfordsville, IN, October 2011.

“Pythagorean Triples and Fermat’s Last Theorem.” Goshen College Science Speakers Series, Goshen, IN, May 2011.

“Automorphic Spectral Theory and Number Theoretic Applications.” Reed College Mathematics Colloquium, Portland, OR, October 2010.

“Spectral identities and exact formulas for counting lattice points in symmetric spaces.” Midwest Number Theory Conference for Graduate Students, University of Wisconsin–Madison, November 2009.

“Relative trace formulas for GL(3), an alternate prescription for spectral identities.” Report from AIM Workshop on GL(3), University of Minnesota, November 2008.

“Instrinsically linkable graphs,” with Alice Wilson. MathFest, the summer meeting of the Mathematical Association of America, Providence, RI, August 2004.