Research Statement
I decided to solicit more money from the government and throw together an NDSEG application. Here's a draft of my research statement. Most of the time writing it was spent trying to reduce it to less than 3000 characters...
My professional goal is to become a top-rate research mathematician and educator. My fascination lies in arithmetic geometry - a field of mathematics which brings the many tools of modern mathematics to bear on problems in number theory.
As a graduate student, my role is to understand a large, central piece of mathematics and find a small, manageable piece to guide my thesis work and introduce me to enough machinery to start a mathematical career.
Here is a central problem that motivates a lot of interesting and useful mathematics. Let f(x_1,...,x_n) be a polynomial with integer coefficients in n-many variables x_1,...,x_n. An integer (or rational) solution is an n-tuple of numbers a_1,...,a_n such that f(a_1,...,a_n) = 0 (for example, 3,2,1 is a solution to x^2 + y^3 - z^11 = 0).
A Tantilizing Question is the following: Can we construct an algorithm with input a polynomial f, and output a simple answer of yes or no to the question "Does f have an integer solution?" The surprising conclusion is negative - not only has no such algorithm been found, but there is a proof that no such algorithm exists!
Any reasonable number theorist would now cite cryptography as the key application of number theory. And while this is true, the solution to our Tantilizing Question holds a lesser known significance. Its solution is a consequence of the following: nearly any problem with a mathematical formulation is equivalent to showing that a particularl polynomial has an integer solution!
Celebrated examples include the Four-Color Theorem, Poincare's Conjecture, the decidability of which theorems are true in a logical system, and any question of existence in discrete mathematics. This illustrates that Number Theory plays the same role in mathematics as particle physics in the physical sciences.
The particular (humble) piece of this larger problems that occupies my thoughts is the following: find all integer solutions to the equation x^2 + y^3 = z^11. This involves an extension of recent work by my advisor Bjorn Poonen and others - they solved the equation x^2 + y^3 = z^7. The appeal of this problem is that it brings some of the most sophisticated and recent tools of modern mathematics to bear on such a simple problem, and illustrates the difficulty of and fascination hidden in these simple problems.
The key is to first understand twists of the modular curve X(11); thus begins a foray into the recent work on modularity by Wiles and Ribet. Arguments in the difficult field of Non-Abelian Cohomology follow, and elements of classical representation theory of finite groups and invariant theory are necessary here too. Various methods from explicit number theory - Chaubaty's method, the Mordell Weil sieve, Descent, and a few new tricks - finish the problem.
The (2,3,7) case of Poonen et. all was cutting edge; curiosity to see if these techniques will extend to any other problems abounds.
My professional goal is to become a top-rate research mathematician and educator. My fascination lies in arithmetic geometry - a field of mathematics which brings the many tools of modern mathematics to bear on problems in number theory.
As a graduate student, my role is to understand a large, central piece of mathematics and find a small, manageable piece to guide my thesis work and introduce me to enough machinery to start a mathematical career.
Here is a central problem that motivates a lot of interesting and useful mathematics. Let f(x_1,...,x_n) be a polynomial with integer coefficients in n-many variables x_1,...,x_n. An integer (or rational) solution is an n-tuple of numbers a_1,...,a_n such that f(a_1,...,a_n) = 0 (for example, 3,2,1 is a solution to x^2 + y^3 - z^11 = 0).
A Tantilizing Question is the following: Can we construct an algorithm with input a polynomial f, and output a simple answer of yes or no to the question "Does f have an integer solution?" The surprising conclusion is negative - not only has no such algorithm been found, but there is a proof that no such algorithm exists!
Any reasonable number theorist would now cite cryptography as the key application of number theory. And while this is true, the solution to our Tantilizing Question holds a lesser known significance. Its solution is a consequence of the following: nearly any problem with a mathematical formulation is equivalent to showing that a particularl polynomial has an integer solution!
Celebrated examples include the Four-Color Theorem, Poincare's Conjecture, the decidability of which theorems are true in a logical system, and any question of existence in discrete mathematics. This illustrates that Number Theory plays the same role in mathematics as particle physics in the physical sciences.
The particular (humble) piece of this larger problems that occupies my thoughts is the following: find all integer solutions to the equation x^2 + y^3 = z^11. This involves an extension of recent work by my advisor Bjorn Poonen and others - they solved the equation x^2 + y^3 = z^7. The appeal of this problem is that it brings some of the most sophisticated and recent tools of modern mathematics to bear on such a simple problem, and illustrates the difficulty of and fascination hidden in these simple problems.
The key is to first understand twists of the modular curve X(11); thus begins a foray into the recent work on modularity by Wiles and Ribet. Arguments in the difficult field of Non-Abelian Cohomology follow, and elements of classical representation theory of finite groups and invariant theory are necessary here too. Various methods from explicit number theory - Chaubaty's method, the Mordell Weil sieve, Descent, and a few new tricks - finish the problem.
The (2,3,7) case of Poonen et. all was cutting edge; curiosity to see if these techniques will extend to any other problems abounds.