Web Address: We have created a mirror of this site on Joshua Mullet's web page so that he can update the speaker lists while I'll be out of town the middle of July. During that time, please consult this second site for the most up-to-date list of speakers and talks.
Call for Speakers: Anyone willing to talk about a subject related to algebraic geometry (e.g., Grobner bases, abelian varieties, intersection theory) is heartily encouraged to contact either Joshua Mullet (mullet@math.uiuc.edu) or David Murphy (dcmurphy@math.uiuc.edu). We will schedule a time for your talk and post it on this web page. This is a good way to learn about topics in your research area and to see what others in our department are doing. Please email us your name, your talk's subject/title, and possibly a reference to a web page (e.g., if you are presenting a paper, you could include the mathscinet review address). Also let us know if there is a date that you would prefer to speak.
Time and Place: The seminar will meet twice each week, every Tuesday and Thursday at 2:00. We will meet in room 143 Altgeld Hall.
| Date | Speaker | Topic / Title | Abstract |
| June 5 | Joshua Mullet | Algebraic and Analytic Geometry I | Given a complex algebraic variety X, we associate a complex analytic space Xh equipped with the usual topology. In this talk, we begin to compare the geometry of X with that of Xh. For more information, see H. Cartan's review of Serre's paper GAGA. |
| June 7 | David Murphy | A Functorial Approach to Algebraic Geometry | We define schemes as a full subcategory of the category of functors from rings to sets and compare this definition with the geometric space definition we already know. The lecture is based on material from chapter 1 of Demazure's book, Groupes Algebriques. |
| June 12 | Joshua Mullet | Algebraic and Analytic Geometry II | In this talk, we compare algebraic sheaves on X with analytic sheaves on Xh and establish isomorphisms between their cohomology groups when X is projective. See H. Cartan's review of Serre's paper GAGA. |
| June 14 | David Smith | Grobner Bases and Polynomial Rings | In this first talk, we describe Buchberger's Algorithm in the case of polynomial rings. The three essentials - a partial ordering, a set of multipliers, and reduction - are defined and it is shown that Buchberger's Algorithm terminates. |
| June 19 | David Smith | Grobner Bases and Algebraic Number Rings | Based on results from his master's thesis, Buchberger's Algorithm is shown to work in orders in algebraic number fields. |
| June 21 | David Murphy | Hilbert Schemes | Following the paper Elementary Introduction to Representable Functors and Hilbert Schemes, we prove the existence of Hilbert schemes, which are used to parameterize flat families of closed subschemes of a given fixed scheme. |
| June 26 | Professor Duursma | Algebraic Decoding Using Special Divisors | Let X/k be an algebraic curve. We describe an effective algorithm that approximates vectors over k by differentials on X. We then discuss an improvement that uses special divisors. For curves with many special divisors, such as hyperelliptic curves, the improved version approximates all vectors that have a unique nearest differential. |
| June 28 | Bin Wang | Vector Bundles and Projective Modules | Richard Swan's paper from 1962 extended the work of J. P. Serre in extending the correspondence between algebraic vector bundles and finitely geneterated projective modules to the topological setting. |
| July 3 | John Jossey | The Mordell-Weil Theorem I | The Mordell-Weil Theorem asserts that the group of rational points on an abelian variety is finitely generated. This talk will begin the proof of this important theorem. |
| July 5 | John Jossey | The Mordell-Weil Theorem II | This talk will conclude the proof of the Weak Mordell-Weil Theorem. These talks are based on Silverman's books on Elliptic Curves. |
| July 10 | David Murphy | Luna's Etale Slice Theorem | The action of a reductive algebraic group G on an affine algebraic variety X is studied. In particular, if the orbit G . x is closed in X, then there is a subvariety V of X, called the etale slice of X at x, such that there is an induced etale G-morphism f : G xGx V -> X, the image U of f is open and affine, and G xGx V is isomorphic to U xU/G V/Gx. For more information, see the Mathematical Review of Luna's 1973 paper, Slices etales. |
| July 12 | John Jossey | The Mordell-Weil Theorem III | With the Weak Mordell-Weil Theorem now proven, this talk treat the full Mordell-Weil Theorem in the case K = Q: the group E(Q) of Q-rational points on the elliptic curve E is finitely generated. |
| July 17 | Soroosh Yazdani | Modular Curves | The modular curve X0(N) is defined to be the compactification of H // G0(N), where H is the upper half plane and G0(N) = { (aij) in GL2 : (det aij)2 = 1 and N divides a12 }. In this talk, we will study various aspects of X0(N). |
| July 19 | John Jossey | The Mordell-Weil Theorem IV | Weights on projective space are introduced and the proof of the Mordell-Weil Theorem is continued. |
| July 24 | Joshua Mullet | Introduction to E'tale Morphisms | The notion of an e'tale morphism may be thought of as the algebraic version of the Implicit Function Theorem, which is false in the algebraic setting. We consider the definition and some basic properties of e'tale morphisms. |
| July 26 | John Jossey | The Mordell-Weil Theorem V | The proof of the Mordell-Weil Theorem is completed. |
| July 31 | Pierrick Gaudry | An Extension of Kedlaya's Algorithm to Superelliptic Curves | Kedlaya's algorithm provides an efficient way to compute cardinalities of hyperelliptic curves over finite fields of small characteristic. We show that this extends easily to curves of the form yr = f(x), and explain how to implement it. |
There are no more meetings scheduled for this summer. We would like to thank everyone who spoke and/or participated in this seminar.