Computing Numbers of Bounded Height - David Krumm, 8/3/2011. Notes

Abstract: I will present recent work of John Doyle and myself, giving a fast algorithm for computing all numbers of bounded height in a given number field.

The 15-Theorem - Allan Lacy, 7/27/2011.

Abstract: The 15-Theorem (Conway-Schneeberger) characterizes when a positive definite quadratic form with integer-matrix is universal. In this talk I will present a proof of this result, due to Manjul Bhargava. I will try to motivate this proof by looking an old result of Ramanujan on the universality of the quaternary forms ax^2 + by^2 + cy^2 + du^2, which captures the main ideas of Bhargava's proof. Finally, I will mention some generalizations of the 15-Theorem, including the 290-Theorem, a joint work of Bhargava and J.Hanke.

Supersingular Elliptic Curves - Jim Stankewicz, 7/20/2011.

Abstract: Over number fields, the theory of complex multiplication provides an interface between the theory of elliptic curves (generally hard) and quadratic forms (generally easier). Is there a similar sort of special elliptic curve over finite fields? The answer is yes, and we show how to make use of these supersingular elliptic curves in computation.

(Over-Convergent) Modular Symbols - Kate Thompson, 7/13/2011.

Abstract: This talk is not lacking in discussion topics. We will begin by talking about the traditional space of modular symbols. Specifically, we will concentrate on weight two modular symbols. We will introduce Manin relations and the Manin continued fraction track, giving an explicit example on beginning computations with modular symbols. Then--time permitting--we will move on to higher weights and discuss over-convergent modular symbols. Ideally we would like to end with a theorem of Buzzard and Calegari involving slopes of 2-adic modular symbols, and the work that came from a student group at the 2011 Arizona Winter School. Some familiarity with modular forms and distribution spaces would be preferable, but is not wholly necessary.

Rational preperiodic points of quadratic polynomials over the rational numbers - John Doyle, 7/6/2011.

Abstract: It has been shown unconditionally that a quadratic polynomial with quadratic coefficients cannot admit a rational periodic cycle of length 4 (Morton) or 5 (Flynn-Poonen-Schaefer), and conditionally (assuming the BSD conjecture) that the same holds for 6 (Stoll). It has been conjectured that such a polynomial cannot admit a rational periodic cycle of length greater than 3. If one assumes this conjecture, one can give a complete classification of rational preperiodic points for quadratic polynomials over the rational numbers. I will present this classification and describe the key ideas of the proof, which is due to Poonen.

Approximation by Polynomials with Integer Coefficients, and When a Power Series Represents a Rational Function - Nathan Walters, 6/15/2011.

Abstract: This talk is in several parts. The first is a statement of two elementary theorems: first, a result of Pal regarding when it is possible to approximate a continuous function on an interval by means of polynomials with integer coefficients; second, a result of Polya which governs when a power series with integer coefficients represents a rational function. The second part of the talk is an introduction to the concept of capacity, and the role it plays in answering generalizations of these questions. The third part is a discussion of the generalizations of this notion of capacity, and how they allow even more general questions to be resolved. The talk is in preparation for a talk at an REU program, and should be accessible to undergraduate math majors.

Many Versions of Hensel's Lemma - David Krumm, 10/7/2010. Notes

Abstract: Hensel's Lemma is a fundamental result in the theory of valued fields, and can also be stated in the context of rings with I-adic topologies. In this talk I will present several results in both of these contexts that are known as Hensel's Lemma, and I'll discuss some implication relations between them. I will mention the important notion of a Henselian local ring, and the existence of a Henselization for any Noetherian local ring.

The Method of Chabauty and Coleman - Jim Stankewicz, 9/16/2010.

Abstract: After the resolution of Mordell's Conjecture by Gerd Faltings, the question about points on curves over number fields was no longer "are there finitely many?" but "how many are there?" Robert Coleman found an explicit bound for some curves in 1985 based on a partial solution to Mordell's conjecture by Claude Chabauty in 1941. Further developments by Lorenzini-Tucker, Stoll and Brown will also be discussed. Although there are no prerequisites for this talk, a familiarity with p-adic analysis and the Jacobian variety of a curve will be useful.

The Big Ostrowski Theorem - David Krumm, 7/7/2010. Notes

Abstract: Ostrowski's theorem determines all archimedean normed fields up to isomorphism, and is used repeteadly in the development of the basic theory of absolute values on fields. For example, it is essential in the classification of absolute values on number fields, and it's an important tool in the study of extensions of absolute values. In this talk I will present an elementary proof of Ostrowski's theorem.

Introduction to Quaternion Algebras - Kate Thompson, 6/30/2010

Abstract: Simply put, the quaternions are a non-commutative extension of the complex numbers with applications ranging from physics to number theory. After introducing some basic constructions, theorems and definitions, we will concentrate on the relationship between quaternion algebras and quadratic forms. Examples over the reals and rationals will be given. If time permits, we will also discuss the differences in constructions over fields of characteristic 2. (Note: for this talk no prior knowledge of quaternion algebras is assumed.)

The Polya-Carlson Theorem on Rational Functions - Nathan Walters, 6/16/2010

Abstract: It is well known that any rational function can be expanded into a Laurent series centered at infinity. A natural next question to ask is whether or not a given Laurent series actually represents a rational function. The Polya-Carlson theorem answers this question in the case that the coefficients of the expansion are integers: such a function is rational if and only if it can be analytically continued to the complement of a "small" set, with size calculated with respect to a concept known as the capacity of that set. This talk endeavors to give a (very basic) introduction to capacity theory while also sketching the remarkable proof of the theorem.

Geometry of Numbers Approach to Finding Solutions to the Extended Legendre Equation - Laura Nunley, 3/31/2010

Abstract: I will present some research done under the advisement of Pete Clark, in which we attempted to apply a geometry of numbers approach to find solutions of diagonal quadratic forms in 4 or more variables.

Vinogradov's Theorem - John Doyle, 3/24/2010

Abstract: Perhaps one of the most famous unsolved problems in number theory is Goldbach's conjecture that every even number can be written as the sum of two primes. A weaker version, known as the ternary Goldbach conjecture, states that every odd number can be written as the sum of three primes. In 1937, Vinogradov gave an asymptotic for the number of representations of an odd number as the sum of three primes, and one corollary of his result is that the ternary Goldbach conjecture is true for sufficiently large odd numbers. I will present the main ideas of the proof of Vinogradov's Theorem.

The Hasse Principle - Jim Stankewicz, 1/20/2010

Abstract: Time permitting, I will give the Who, What, When, Where, Why and How of the Hasse Principle and its myriad formulations.