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

**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. **

**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. **

**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. **