Working Number Theory Abstracts




October 27, 2004
Speaker:    Matt Papanikolas (Texas A&M)
Title:          On Differential Galois Theory, Part I

Abstract:

We start out by reviewing the basic setting and objects of the classical Galois theory for field extensions generated by solutions of polynomial equations over a fixed ground field, e.g. Q. Then we introduce the analogous setting for field extensions generated by solutions of ordinary differential equations over a fixed ground field, e.g. C(t). In this context we introduce notions of derivation, differential extension, and differential automorphism, as well as show some examples. We finish up by stating the Galois correspondence for differential extensions, and the differential inverse Galois problem, which is a theorem of C. Tretkoff and M. Tretkoff.



November 3, 2004
Speaker:    Matt Papanikolas (Texas A&M)
Title:          On Differential Galois Theory, Part II

Abstract:

We start by reviewing the differential Galois correspondence over C(t). This leads to a natural isomorphism between Galois groups of differential field extensions and algebraic subgroups of GL_N(C). Dimension as a variety over C of such an algebraic group is equal to the trancendence degree of the corresponding field extension over C(t). We demonstrate a few examples of this, including Airy equation. Next, as a generalization of our approach starting with a homogeneous differential equation, we introduce the Picard-Vessiot theory of matrix differential equations. Then, in the opposite direction, we can start with an arbitrary matrix over C(t) and construct an N-th order homogeneous differential equation whose solutions are the same as those of the first order system with this coefficient matrix. This leads to a formal construction of differential rings and corresponding algebraic groups.



November 10, 2004
Speaker:    Ahmad El-Guindy (Texas A&M)
Title:          L-functions in Number Theory, Part I

Abstract:

L-functions play a central role in Number Theory, many of the well known results and conjectures are stated in terms of them. In this talk we start by recalling definitions and examples of some of the L-functions arising in different settings, and discuss the connections they have among themselves and to other arithmetic objects.



November 17, 2004
Speaker:    Matt Papanikolas (Texas A&M)
Title:          On Differential Galois Theory, Part III

Abstract:

We review the construction of a differential ring over C(t), its maximal differential ideal, and the corresponding differential Galois group viewed as an algebraic subroup of GL_N(C). The fraction field of such a ring is a differential extension of C(t) whose trancendence degree is again equal to the dimension of the Galois group. Next we review some applications of this machinery to Number Theory. These include well known trancendence results, such as Siegel-Shidlovsky theorem. We conclude with a brief review of Deligne's construction of the category of differential modules, which he proved to be a Tannakian category, i.e. a category of representations of an affine group scheme over C. It is also a result of Deligne that the corresponding group can be realized as the Galois group of a field extension L/C(t) which arises in the same manner as above.



December 1, 2004
Speaker:    Lenny Fukshansky (Texas A&M)
Title:          On Ihara Zeta Function, Part I

Abstract:
Ihara zeta function is zeta function of a graph. I will give its definition and state some of its important properties, including a version of Riemann hypothesis. I will also explain how its construction can be associated to subgroups of the automorphism group of a tree, in this way creating an analogy with Selberg zeta function. Ihara zeta function also comes up in the context of modular curves; I will try to say a few words about this connection.


January 25, 2005
Speaker:    Ahmad El-Guindy (Texas A&M)
Title:          L-functions in Number Theory, Part II

Abstract:

We continue our discussion of L-functions by taking a closer look at the ones attached to elliptic curves and modular forms. We discuss their definition, the connections they have among one another and among other number theoretic object. We also try to touch on some of the techniques used in proving important theorems about them.


February 1, 2005
Speaker:    Ahmad El-Guindy (Texas A&M)
Title:          L-functions in Number Theory, Part III

Abstract:

We attempt to give a modest overview of Gross and Zagier's ground-breaking "Heegner points and derivatives of L-series". This dictates the pleasant tasks of discussing modular curves, elliptic curves with complex multiplication, and how Heegner points relate to both. We also hope to reach the applications to the Birch and Swinnerton-Dyer conjecture, and perhaps talk about some ingredients of the proof.


February 8, 2005
Speakers:    Ahmad El-Guindy and Matt Papanikolas (Texas A&M)
What:          Informal Q&A session on elliptic curves, modular forms, and related L-functions

Abstract:

Ahmad and Matt will answer questions about elliptic curves, modular forms, and related L-functions that were coming up in the previous talks by Ahmad, as well as (perhaps) quite informally continue the discussion of these subjects.


February 15, 2005
Speaker:    Lenny Fukshansky (Texas A&M)
Title:          On Ihara Zeta Function, Part II

Abstract:
I will continue talking about zeta function of a finite graph. I will briefly recall its definition and some of its properties. I will also review an alternative construction which associates a zeta function to subgroups of the automorphism group of a tree, in this way creating an analogy with Selberg zeta function; this was the original construction of Ihara. Ihara zeta function also comes up in the context of modular curves; I will try to concentrate on this important connection.



February 22, 2005
Speaker:    Bogdan Petrenko (Texas A&M)
Title:          A Theorem of Burnside on Matrix Rings

Abstract:

I plan to discuss the proof of this beautiful theorem in detail. All the regular participants of the seminar have more than enough prerequisites to understand it. Depending on the circumstances, I may begin speaking on my recent research that would be impossible without this result of Burnside.


March 1, 2005
Speaker:    Bogdan Petrenko (Texas A&M)
Title:          On Pairs of Matrices That Generate Matrix Rings, Part I

Abstract:

I plan to discuss my recent research, which is substantially based on the Burnside's lemma discussed previous time. All the regular participants of the seminar have more than enough prerequisites to understand my talk.


March 8, 2005
Speaker:    Bogdan Petrenko (Texas A&M)
Title:          On Pairs of Matrices That Generate Matrix Rings, Part II

Abstract:

I plan to continue to discuss my recent research. All the regular participants of the seminar have more than enough prerequisites to understand my talk.


March 22, 2005
Speaker:    Bogdan Petrenko (Texas A&M)
Title:          On Pairs of Matrices That Generate Matrix Rings, Part III

Abstract:

I will continue to discuss my  recent research with Said Sidki. In particular, I plan to prove that the presentation suggested last time is indeed a presentation of the ring of n-by-n matrices with integral entries.



April 12, 2005
Speaker:    Lenny Fukshansky (Texas A&M)
Title:          Bezout's Theorem and the Theory of Heights, Part I

Abstract:

I will give a brief review of a few basic concepts in intersection theory, and will state the classical Bezout's theorem in its general form. Then I will introduce height functions and use them as an arithmetic analogue of degree (i.e. another measure of complexity) on a variety or on an intersection cycle. This will require defining Chow forms. The eventual goal, which is not necessarily going to be achieved in the first talk, is to state a version of arithmetic Bezout's theorem.


April 21, Thursday, at 1:00 pm in Milner 317    -   Notice the unusual time and place
Speaker:    Lenny Fukshansky (Texas A&M)
Title:          Bezout's Theorem and the Theory of Heights, Part II

Abstract:

I will continue talking about height functions, including heights of varieties and intersection cycles. This will require defining Chow forms. The eventual goal is to state a version of arithmetic Bezout's theorem. I will also discsuss some additional topics from the theory of heights if time permits.