##
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

#### 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.

#### 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.