Number Theory Talks

Spring, 2008

Receive credit for this seminar

Contact the seminar organizer

Calendar

February 1	Number Theory Seminar	1:40-2:30 p.m. PSA 308	Jing Long Hoelscher, University of Arizona	Galois extensions with prescribed ramification Abstract. With given places ramifying, which groups can occur as Galois groups over a field K? For function fields K of curves over algebraically closed fields k of characteristic p, this is answered by Abyhankar's Conjecture, which was proved by M. Raynaud and D. Harbater. For number fields K, I will generalize some results in Harbater's papers to give some answers to the problem above, especially when there is just one ramified prime. For function fields K over finite fields, I will give some constraints on the groups that can occur with given ramification, and take steps to develop Iwasawa theory for cyclotomic function fields.
February 22	Colloquium	1:30-2:30 p.m. PSF 101	Wayne Raskind, University of Southern California	Public Key Cryptography, Number Theory and Arithmetic Geometry Abstract. Many public key cryptographic systems are based on the discrete logarithm problem for finite abelian groups. In this talk, we will briefly review how this works and present some of our research (joint with Ming-Deh Huang of the USC Computer Science Department) on a general method for treating this problem for abelian algebraic groups over finite fields. We will then describe in some detail the cases of the multiplicative group and elliptic curves over finite fields, which reveal some surprising connections between the discrete logarithm problem and some fundamental problems in number theory and arithmetic geometry.
February 27	Number Theory Seminar	1:40-2:30 p.m. PSA 308	Helene Nehrebecki	Waring’s problem Abstract. Waring’s problem states that every positive integer is the sum of 9 cubes, 19 fourth powers, and so on. In this talk, we will discuss results leading to the solution of Waring’s problem. Finally, we will explore recent advancements in this area.
March 5	Number Theory Seminar	1:40-2:30 p.m. PSA 308	Alejandra Alvarado	Arithmetic progressions on curves Abstract. An arithmetic progression is a sequence of numbers such that the difference between any two consecutive numbers is constant. When we talk about an arithmetic progression on a curve F(x,y)=0, we mean an arithmetic progression in the x coordinates. In this talk we will discuss APs on curves of genus greater than zero. In particular, we will look at APs on certain types of elliptic curves.
March 19	Number Theory Seminar	1:40-2:30 p.m. PSA 308	David Roberts, University of Minnesota at Morris	Chebyshev covers and exceptional number fields Abstract. Click here to view.
March 26	Number Theory Seminar	1:40-2:30 p.m. PSA 308	Daniel Corral	A connected topology for the integers, I Abstract. We discuss a paper of Solomon W. Golomb, where arithmetic progressions are used to define a basis of open sets, yielding a topology on Z. We show how this topology can be used to prove the infinitude of primes.
April 2	Number Theory Seminar	1:40-2:30 p.m. PSA 308	Daniel Corral	A connected topology for the integers, II Abstract. We resume our exposition of the paper "A Connected Topology for the Integers" by Solomon W. Golomb. As in the previous talk, let D denote the Golomb topology generated by the empty set and a certain type of arithmetic progression in the integers. In this talk we will discuss a theorem of Golomb which states that the Dirchlet theorem on arithmetic progressions is equivalent to the density of the primes in (Z, D). It will also be shown that the set of primes has empty interior in (Z, D). The final result in the Golomb paper is that the set of positive integers n, where 6n-1 and 6n+1 are a prime pairing, is closed in a stronger topology than D. If time permits we will begin a discussion of a paper of A. M Kirch or present the related H. Furstenberg proof of the infinitude of primes.
April 9	Number Theory Seminar	1:40-2:30 p.m. PSA 308	Keenan Kidwell	The theory and computation of finite extensions of the p-adic numbers, I Abstract. The fields of real and p-adic numbers are examples of local fields, (locally compact topological fields), and are the only completions of Q, while number fields (finite extensions of Q) are examples of global fields. The number theoretic motivation for studying the properties of real and p-adic numbers is to obtain local information, which (hopefully) can be assembled into global information about number fields; an example of the success of this strategy is the Hasse-Minkowski Theorem, a local-global principle relating the existence of zeros of quadratic forms over the real and p-adic numbers to that of zeros over the rationals. One connection between number fields and p-adic fields is given by the so-called p-adic algebra of a number field F: F\otimes Q_p \cong\prod_{i=1}^g K_p,_i , where each K_p,_i is a finite extension of the p-adic numbers. Within a fixed algebraic closure, there are only finitely many extensions of the p-adic numbers of a given degree. In this two-part talk, we will discuss the theoretical aspects of p-adic fields, including residue degree, ramification index, unramified and totally ramified extensions, the continuity of roots of polynomial equations, and the aforementioned finiteness result, as well as some of the work of J. Jones and D. Roberts involving the determination and classification of the extensions of the p-adic numbers of a given degree.
April 16	Number Theory Seminar	1:40-2:30 p.m. PSA 308	Keenan Kidwell	The theory and computation of finite extensions of the p-adic numbers, II Abstract. The fields of real and p-adic numbers are examples of local fields, (locally compact topological fields), and the only completions of Q, while number fields (finite extensions of Q) are examples of global fields. The number theoretic motivation for studying the properties of real and p-adic numbers is to obtain local information, which (hopefully) can be assembled into global information about number fields; an example of the success of this strategy is the Hasse-Minkowski Theorem, a local-global principle relating the existence of zeros of quadratic forms over the real and p-adic numbers to that of zeros over the rationals. One connection between number fields and p-adic fields is given by the so-called p-adic algebra of a number field F: F\otimes Q_p \cong\prod_{i=1}^g K_p,_i , where each K_p,_i is a finite extension of the p-adic numbers. Within a fixed algebraic closure, there are only finitely many extensions of the p-adic numbers of a given degree. In this two-part talk, we will discuss the theoretical aspects of p-adic fields, including residue degree, ramification index, unramified and totally ramified extensions, the continuity of roots of polynomial equations, and the aforementioned finiteness result, as well as some of the work of J. Jones and D. Roberts involving the determination and classification of the extensions of the p-adic numbers of a given degree.
April 23	Number Theory Seminar	1:40-2:30 p.m. PSA 308	Dong Quan Nguyen	On the Duffin-Schaeffer conjecture, part I Abstract. We will prove the Duffin-Schaeffer theorem, which motivates the formulation of the Duffin-Schaeffer conjecture.
April 30	Number Theory Seminar	1:40-2:30 p.m. PSA 230	Dong Quan Nguyen	On the Duffin-Schaeffer conjecture, part II Abstract. We will prove the Duffin-Schaeffer theorem, which motivates the formulation of the Duffin-Schaeffer conjecture.

Autumn, 2007 Calendar

October 24	Number Theory Seminar	10:40-11:30 a.m. PSH 552	Keenan Kidwell	Quadratic Forms Over the Field of Rationals and its Completions and the Hasse-Minkowski Theorem Abstract. We will define quadratic forms and quadratic modules, focusing on forms over the fields of rational, real, and p-adic numbers. The Hilbert symbol will be introduced, as well as the rank and discriminant of a quadratic form, and these tools will be used to give a necessary and sufficient condition for the equivalence of two forms over a field of characteristic different from 2. We will conclude with a statement and discussion of the proof of the Hasse-Minkowski Theorem, which relates the existence of zeros of a quadratic form over the rationals to that of zeros of the form over the real and p-adic fields.
October 31	Number Theory Seminar	10:40-11:30 a.m. PSH 552	Dong Quan Nguyen	Algebraic Points on Cubic Hypersurfaces I Abstract. We prove the Cassels Swinnerton-Dyer conjecture for the case n=3 following a paper of Coray.
November 7	Number Theory Seminar	10:40-11:30 a.m. PSH 552	Dong Quan Nguyen	Algebraic Points on Cubic Hypersurfaces II Abstract. We prove the Cassels Swinnerton-Dyer conjecture for the case n=3 following a paper of Coray.
November 14	Number Theory Seminar	10:40-11:30 a.m. PSH 552	Alejandra Alvarado	Integral Arithmetic Progressions on y²= x³+ k Abstract. Consider elliptic curves of the form y² = x³ + k over the rationals. We will investigate arithmetic progressions on these curves, of length three, four and five.
November 28	Number Theory Seminar	10:40-11:30 a.m. PSH 552	Chase Franks	On the mu-invariant of the Gamma-transform of a rational function Abstract. p-adic L-functions arise as Gamma-transforms of p-adic measures that are constructed from rational functions. Following Sinnott, we will compute the mu-invariant of the Gamma-transform of a rational function measure. This will lead to a proof of the Ferrero-Washington theorem, which says that mu = 0 for the basic Z_p-extension of an abelian number field.

Number Theory Talks (2006-2007)

Number Theory Talks (2005-2006)