## JOINT MATHEMATICS COLLOQUIUM

 Square-zero matrices over commutative rings

# Luchezar Avramov

### Northeastern University

#### Tea at 4:00 p.m. in 544 Nightingale Hall

 Abstract:   The subject of the talk are $n\times n$ matrices $A$ with elements in some commutative ring $R$, satisfying $A^2=0$. We address the following open question: if $R$ is a polynomial ring in $d\ge 4$ variables over a field $k$, a matrix $A$ is upper triangular, and $ker(A)/im(A)$ is a non-zero finite dimensional $k$-vector space, does it follow that $n\ge 2^d$? I will review the origins of such a strange question (in studies of torus actions on CW complexes), its extension to noetherian commutative rings (where it ties up with major unsolved problems in the field), the partial results obtained (concerning the block structure of $A$), their applications to commutative algebra (a strong form of the New Intersection Theorem) and to algebraic geometry (structure of fibers of flat homomorphisms). The talk is based on joint work with Ragnar-Olaf Buchweitz, Srikanth Iyengar, and Claudia Miller.

 Here are some directions to Northeastern University. Lake Hall and Nightingale Hall can be best accessed from the entrance on the corner of Greenleaf Street and Leon Street. The two halls are connected, with no well-defined boundary in between. In particular, 509 Lake Hall is on the same corridor as 544 Nightingale Hall. There is free parking available for people coming to the Colloquium at Northeastern's visitor parking (Rennaisance Garage). The entrance is from Columbus Avenue. If coming by car, you should park there and take the parking talon. After the lecture, you may pick up the payment coupon from Andrei Zelevinsky.

 Web page:  Alexandru I. Suciu Comments to:  alexsuciu@neu.edu Posted:: March 12, 2008 URL: http://www.math.neu.edu/bhmn/avramov08.html