Jerome William Hoffman

Last Updated : September 28, 2010
Rank   :   Professor of Mathematics
Degree :   Ph.D.- Harvard University, (advisor Heisuke Hironaka)
           A.B.-  Princeton University

Office :   374 Lockett Hall
Address:   Department of Mathematics, LSU, Baton Rouge, LA 70803-4918
Phone  :   225-578-1575
Fax    :   225-578-4276
E-mail :   hoffman@math.lsu.edu

Vita

Teaching

1.) Over the years I have taught a variety of advanced graduate courses, including: Algebraic Geometry,  Commutative Algebra, Elliptic Curves, Modular Forms, Algebraic Topology, Number Theory, Lie Groups and Lie Algebras.  For the notes of  a course,  taken by students on local and global fields, culminating inTate's thesis, see Spring 2000 : Math 7290

2.) I have also been teaching in the REU (Research Experience for Undergraduates) program for the past 5 summers.  See the LSU REU  link.  I have mentored students in many projects, including: zeta functions and modular forms for genus 2 curves, classical invariant theory, zeta functions of graphs. Last  summer we explored dessins d'enfants. See Vita for titles of projects.

3.) Here are notes taken by students from an Algebraic Geometry course in Fall 2005.   Math7280

4.) Here is the link to the WebWorks login page for Math 1553, Spring 2008.

5.) Here is the link for Math 4031, Fall 2008.


Research

Interests:  Algebraic Geometry.  Over  the past  few  years the main focus has been

1.)   Topology and geometry of Siegel modular threefolds. These are quotients of the Siegel upper half space
        of degree 2  and they parametrize families of abelian varieties of dimension 2. This has been a collaborative
        project with Steven Weintraub

2.)  Regularity of toric varieties. In collaboration with H. Wang, we  have introduced a generalization of Castelnuovo-
       Mumford regualrity for multiprojective spaces. This has application to algorithms for solving systems of polynomial equations.
       Further generalizations  have been introduced by  D. Maclagan and  G. Smith and are currently under investigation.
 
3.)   Pseudozeros of polynomial equations. This is a collaboration with Hong Zhang and James Madden.  This also has to
       do with efficient algorithms for solving polynomial systems.

4.)   Zeta functions of graphs and buildings.  Yasutaka Ihara introduced the zeta function of graphs and found links to the zeta
       functions of modular curves. These graphs arise as quotients of the Bruhat-Tits building for GL(2) of a local p-adic field.
       Problem: what are the analogs of these for other algebraic groups, and their connection to moduli spaces (Shimura varieties,
       Drinfeld moduli spaces)?

5)    Axiomatic homotopy theory and connections to K-theory.

Publications


Lectures:

1.) Elliptic curves and modular forms.

2.) Zeta functions of graphs, modular forms, and dynamical systems.

3.) Topology of Siegel modular threefolds.

4.) Modular forms on noncongruence subgroups and Atkin-Swinnerton-Dyer relations.

5.) Picard groups of Siegel modular threefolds and theta lifting.

6.) Zeta functions of buildings and Shimura varieties.

7.) L-functions and l-adic representations for noncongruence subgroups.

8.) Infinitesimal structure of Chow groups.

9.) Zeta functions of buildings and algebraic geometry.


10.) Algebraic curves of GL(2)-type.


11.) Arithmetic properties of Picard-Fuchs differential equations.