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Number Theory Seminar
Spring, 2006
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Calendar
January 18
PSA 216, 2:40 pm.
Organizational Meeting
January 25
PSA 307, 2:40 pm.
David Roberts, University of Minnesota – Morris, Octic 2-adic fields
Abstract. John Jones and I have established a database cataloging low degree p-adic fields. One case very much harder than all the previous cases is octic 2-adic fields. I will discuss details associated with this case and sample applications to number fields.
January 26 - Colloquium
PSH 433, 3:40 pm.
David Roberts, University of Minnesota – Morris, Fractalized Cyclotomic Polynomials
Abstract. Cyclotomic polynomials play an important role in number theory. The main case is when the index is a prime power. I will show how the definition of a cyclotomic polynomial of index pm can be “fractalized” to give 3m different polynomials in Z[x]. On the one hand, good features of the cyclotomic polynomial transfer to the more general fractalized polynomials, as the new polynomials are also all irreducible over the integers and their discriminants are all still powers of p. On the other hand, the new polynomials are more complicated, in that Galois groups are no longer abelian and complex roots no longer lie on a circle, but rather are fractal in appearance. I will keep the talk accessible to a general mathematical audience, but indicate at the end connections with an open problem in number theory.
February 1
PSA 307, 2:40 pm.
Ahmed Matar, An Introduction to Galois Cohomology (Part I)
Abstract. Galois cohomology is the study of the group cohomology of Galois modules. For example, if L/K is Galois extension of fields and G=Gal(L/K), then L is a G-module in a natural way and one may talk about the cohomology of L with coefficients in G. This talk is meant to be an introduction to Galois cohomology and some of its applications to number theory. I will start this talk by reviewing the theory of derived functors and group cohomology. I will later talk about profinite groups and their cohomology, which I will follow with a short discussion of infinite Galois theory. Finally, I will discuss Galois cohomology and state a few of its applications to number theory.
February 8
PSA 307, 2:40 pm.
Ahmed Matar, An Introduction to Galois Cohomology (Part II)
February 15
PSA 307, 2:40 pm.
Ahmed Matar, An Introduction to Galois Cohomology (Part III)
February 22
PSA 307, 2:40 pm.
Jacob Shotwell, Infinite Galois Extensions
March 1
PSA 307, 2:40 pm.
Shawn Elledge, Adeles and Ideles (Part I)
Abstract. We begin by reviewing everything (valuations, completions, etc.). We then cover the restricted direct product of topological groups. Adeles and ideles will then be defined. Additional stuff will follow.
March 8
PSA 307, 2:40 pm.
Shawn Elledge, Adeles and Ideles (Part II)
March 15
SPRING BREAK
March 22
PSA 307, 2:40 pm.
Francisco Portillo Bobadilla, National University of Mexico, Computational Evidence of Mazur-Tate Conjecture for Elliptic Curves
Abstract. On their article "Refined Conjectures of the Birch Swinnerton-Dyer type", B. Mazur and J. Tate presented a series of conjectures of BSD type over finite layers. These conjectures were presented on the general setting of Abelian Varieties. Also, in the same article, a few examples of elliptic curves were included as empirical evidence in favor of the conjectures. We will explain in more detail the form of the conjectures in the case of elliptic curves, and we will present a broad amount of examples in different cases of reduction.
March 23- Colloquium
PSA 309, 3:40 pm.
Daniel Bump, Stanford University, Weyl Group Multiple Dirichlet Series
Abstract. Weyl Group Multiple Dirichlet Series are Dirichlet series that can be constructed out of Gauss sums that have groups of functional equations isomorphic to finite reflection groups. Progress has been recently made on understanding the combinatorics of their coefficients. Joint work with Brubaker, Chinta, Friedberg and Hoffstein.
March 24
PSA 307, 2:40 pm.
Daniel Bump, Stanford University, Small Representations of Odd Orthogonal Groups
Abstract. We will discuss certain analogs of theta functions living on odd orthogonal groups, and some of their possible applications to number theory. Joint work with Friedberg and Ginzburg.
March 29
PSA 307, 2:40 pm.
Alejandra Alvarado, The Mordell-Weil Theorem over Q, Part I
Abstract. The Mordell Weil Theorem states that for a number field K and an elliptic curve E defined over K, the group E(K) is finitely generated. I will prove the case for when K=Q.
April 5
PSA 307, 2:40 pm.
Alejandra Alvarado, The Mordell-Weil Theorem over Q, Part II
April 12
PSA 307, 2:40 pm.
Annette Spyker, Effectiveness of the Rabin-Miller compositeness test
Abstract. The Rabin-Miller test is a probabilistic primality test, meaning that it will either prove a number composite or show that the number is likely to be prime. The test begins by choosing a random number, and the success of the test depends upon that choice of number. If we know our number is composite, a successful choice of random number is one that allows us to prove the number is composite. We consider the chance of choosing a successful random number given that we know the prime factorization of the number that is being tested.
April 19
PSA 307, 2:40 pm.
Helene Nehrebecki, The Golden Ratio (Part I)
Abstract. Identities of The Golden Ratio in Number Theory and Geometry will be discussed. Included will be how The Golden Ratio has been used to design monuments, how it is related to five-fold symmetry and Platonic solids by considering the golden triangle and the golden gnomon, and finding formulas which explain why it has certain properties.
April 20 - Colloquium
PSA 309, 3:40 pm.
Alice Silverberg, University of California at Irvine, Some Applications of Number Theory and Algebraic Geometry to Cryptography
Abstract. We will discuss cryptography based on the discrete logarithm problem, including an introduction to the Diffie-Hellman, ElGamal, and XTR cryptosystems. We will show that studying the underlying mathematics of these systems leads to interesting questions about algebraic tori, which in turn lead to new cryptosystems (such as the CEILIDH cryptosystem).
April 21
PSA 307, 2:40 pm.
Alice Silverberg, University of California at Irvine, Pairings in Cryptography
Abstract. We will discuss some open questions relating to the Weil pairing for elliptic curves and abelian varieties over finite fields, and give some applications of abelian varieties to pairing-based cryptography.
April 26
PSA 307, 2:40 pm.
Helene Nehrebecki, The Golden Ratio (Part II)
Calendar for Autumn, 2005
August 31
NUR 306, 12:40-1:30 p.m.
Ahmed Matar, Dimension Theory, Part I
Abstract. I begin this talk by defining the (Krull) dimension of an R-module M. I will later prove a theorem by Hilbert and Serre's fundamental theorem of dimension theory, which both relate the Krull dimension of M under certain conditions to degrees of polynomials associated to M. As corollaries, I will prove Krull's generalized principal ideal theorem as well as an interesting result stating that a Noetherian ring satisfies the descending chain condition with respect to prime ideals. I will end this talk by stating a few applications of both Serre's and Hilbert's theorems to algebraic geometry. This talk will be in 3 parts.
September 7
NUR 306, 12:40-1:30 p.m.
Ahmed Matar, Dimension Theory, Part II
September 14
NUR 306, 12:40-1:30 p.m.
Ahmed Matar, Dimension Theory, Part III
September 21
NUR 306, 12:40-1:30 p.m.
Jacob Shotwell, Theory of Automorphic Forms, Part I
Abstract. In this series of three talks, the theory of automorphic forms will be described and applied to central topics in number theory. During the first talk, automorphic forms will be motivated and defined after reviewing some results from algebra, number theory, and complex analysis. The second and third talks will generate the core results of this theory as it relates to number theory.
September 28
NUR 306, 12:40-1:30 p.m.
Jacob Shotwell, Theory of Automorphic Forms, Part II
October 5
NUR 306, 12:40-1:30 p.m.
Jacob Shotwell, Theory of Automorphic Forms, Part III
October 12
NUR 306, 12:40-1:30 p.m.
Rachel Wallington, Computing Ray Class Fields Using Hecke's Theorem
Abstract. One interesting application of class field theory is to find field extensions which are unramified outside of a certain set of primes. Let K be a number field, p be a prime number and S be a set of finite primes of K. In this talk, I will discuss how we can use Hecke's Theorem to find extensions of K of degree p which are unramified outside of S. To do this we will have to construct ray class fields that correspond to some modulus m of K. I will discuss in some detail the case when K contains a p-th root of unity and give an example.
October 19
NUR 306, 12:40-1:30 p.m.
Shawn Elledge, Groebner Bases
Abstract. We will discuss Groebner Bases. The main motivation will be ideal membership in polynomial rings; that is, given f in K[X1,...,Xn] and an ideal I of K[X1,...,Xn], exactly when is f in I? By computing a Groebner basis of I we will be able to sort of generalize the division algorithm of K[X] to K[X1,...,Xn] and then answer this question. We will also intersect varieties, answer membership in the radical of an ideal of K[X1,...,Xn], intersect two ideals in K[X1,...,Xn], and discover what the letter S stands for. This talk will be accessible to anyone who knows basic ring theory. All other oddities will be defined.
November 9
NUR 306, 12:40-1:30 p.m.
Helene Nehrebecki, Cases of Fermat's Last Theorem
Abstract. We will discuss Fermat's Last Theorem, and proofs for the cases of n=4 and n=3. We will also discuss Germain primes, two cases of Fermat's Last Theorem brought about in the 1820's, and Sophie Germain's proof relating to one of those cases. This talk is accessible to anyone who knows basic number theory.
November 16
NUR 306, 12:40-1:30 p.m.
Erwin Suazo, Some applications of K-Theory
Abstract. Algebraic K-theory plays an important role in different areas of mathematics: algebraic topology, number theory, algebraic geometry and even including operator theory. According to Jonathan Rosenberg, "Algebraic K-theory is the branch of algebra dealing with linear algebra over a general ring R instead of over a field. It associates to any ring R a sequence of abelian groups Ki(R). The first two of these, K0 and K1, are easy to describe. The others, rather mysterious." In this talk I will present how K-theory is applied to C*-algebras. No prerequisites are needed. I will present the Bott Periodicity Theorem, which shows that for C*-algebras the sequence of K-groups reduces to the first two: K0 and K1, and Elliot's Theorem, which shows that Approximately Finite-Dimensional C*-algebras are completely classified by their K0 groups.
November 23
NUR 306, 12:40-1:30 p.m.
Juan Manuel Garcia Celaya, Continued Fractions
Abstract. Continued Fractions are a powerful mixture of Analysis and Algebra, and they don't seem to receive the attention they deserve. Mathematicians often avoid them, and their study, in modern days, has been limited to a chapter in elementary Number Theory textbooks. Some classical and important results in Number Theory are much easier to prove with the use of Continued Fractions than using any more algebraically "pure" method. Nowadays, Continued Fractions remain an active area of research in a large number of fields ranging from pure mathematics to mathematical physics and approximation theory. I will try to show what we are missing, and we may figure out why the modern mathematical teaching has dropped this tool.
November 30
NUR 306, 12:40-1:30 p.m.
Juan Manuel Garcia Celaya, Continued Fractions, Part II
