Colloquia Patavina: Noetherian rings from a non-Noetherian perspective

Martedi' 1 Marzo 2011 - prof. Bruce Olberding


Martedi' 1 Marzo 2011 alle ore 16:00 in aula 1A150 della Torre Archimede il Professor Bruce Olberding (New Mexico State University) terra' una conferenza della serie Colloquia Patavina.

La Commissione Colloquia
P. Ciatti, M.A. Garuti, M. Pavon, F. Rossi

Noetherian rings from a non-Noetherian perspective

Bruce Olberding (New Mexico State University)

After Emmy Noether introduced the axiom of the ascending chain condition in 1921, commutative algebra developed very swiftly. From 1921 to the present, much of the impetus for this rapid growth has been to provide algebraic explanations and proofs of geometric facts; in turn, through the work of Grothendieck and many others, geometry proved useful in understanding algebraic ideas. So successful and powerful were all these efforts that today commutative algebra is sometimes considered a chapter in algebraic geometry. However, the ascending chain condition proves to be more robust than is indicated by this version of the story, and there persist classes of Noetherian rings resistant (so far) to the geometrical point of view. We discuss some of these classes, with emphasis on analytically ramified local Noetherian rings and how techniques from non-Noetherian commutative ring theory are useful in understanding such rings.

-Breve curriculum
B.S. in Mathematics with Honors with Distinction, at Baylor University, Waco, Texas, 1990.
Ph.D. in Mathematics, at Wesleyan University, Middletown, Connecticut, 1996, Advisor: Prof. J. D. Reid. Dissertation title: Torsion-Free Modules over Prüfer Domains.
Visiting Assistant Professor at Department of Mathematics, Wesleyan University, Middletown, Connecticut. (August 1996 – June 1997).
Assistant Professor at Department of Mathematics, The University of Louisiana at Monroe, Monroe, Louisiana (August 1997 – May 2002).
Assistant Professor (August 2002 – August 2004 ), Associate Professor (August 2004 – August 2009 ) and Professor (August 2009 – present) at Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico.
Editor of Journal of Commutative Algebra.
His research interests include Commutative Algebra, Module Theory and Algebraic Geometry.
He is author of more than 40 papers on commutative algebra and module theory; here is a selection of recent publications.
1. On Matlis domains and Prüfer sections of Noetherian domains, in Commutative algebra and its applications, 321--332, Walter de Gruyter, Berlin, 2009.
2. Factorization into prime and invertible ideals II, Journal of the London Mathematical Society, 80 (2009), no. 1 , 155—170.
3. Integrally closed overrings of two-dimensional Noetherian domains representable by Noetherian spaces of valuation rings, Journal of Pure and Applied Algebra 212 (2008) 1791-1821.
4. Irredundant intersections of valuation overrings of two-dimensional Noetherian domains, Journal of Algebra 318 (2007) 834-855.
5. Injective and colon properties of ideals of integral domains, Forum Mathematicum 19 (2007), no. 6, 1047-1074.
6. Holomorphy rings of function fields, in Multiplicative Ideal Theory in Commutative Algebra, 331-348, Springer-Verlag, 2006.
7. The minimal number of generators of an invertible ideal, with Moshe Roitman, in Multiplicative Ideal Theory in Commutative Algebra, 349-368, Springer-Verlag, 2006.
8. Commutative ideal theory without finiteness conditions: irreducibility in the quotient field, with Laszlo Fuchs and William Heinzer, in Abelian groups, rings, modules, and homological algebra, 121-145, Lecture Notes in Pure and Applied Mathematics, 249, Chapman & Hall/CRC, Boca Raton, FL, 2006.
9. Commutative ideal theory without finiteness conditions: completely irreducible ideals with Laszlo Fuchs and William Heinzer, Transactions of the American Mathematical Society, 358 (2006), no. 7, 3113-3131.
10. Unique irredundant intersections of completely irreducible ideals, with William Heinzer, Journal of Algebra 287 (2005), no. 2, 432-448.

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