Martin Lorenz

Click on the links to open the presentation files (pdf):

• "Growth of groups and algebras" (Algebra Seminar at DePaul University, Chicago, June 4, 2021) This was a survey talk on growth functions that are associated to groups and algebras with a given finite weighted generating set. The focal point was a theorem of Benson (1983) proving rationality of such series for virtually abelian groups. The talk tentatively explored the possibility of constructing a ring-theoretic proof of this result using the framework of "representable" algebras.
• "On the importance of being rational" (AMS special session "Homological Aspects of Quantum Symmetries and Related Topics", JMM, virtual, January 8, 2021) The title of the talk, a nod to to political events, is borrowed from a related survey article by Jason Bell (On the importance of being primitive, Rev. Colombiana Mat. 53, 2019). The overarching theme is the description of noncommutative prime spectra by taking advantage of quantum symmetries. The talk offered some speculations and preliminary results concerning primitivity/rationality and primeness of (infinite-dimensional) Hopf algebras. I did not actually get to this last part in my talk; the notes posted here give an indication of what I had intended to say.
• "Torus actions on noncommutative algebras" (AMS special session "Quantum Groups and Noncommutative Algebraic Geometry", Tulane University, New Orleans, October 13, 2012) This talk describes the Stratification Theorem for rational torus actions on arbitrary algebras. The paper on which the main result of this talk is based is here.
• "Prime ideals and group actions in noncommutative algebra" (Plenary talk at the BMC 2012, University of Kent, Canterbury, April 19, 2012) Survey talk on rational group actions on noncommutative prime spectra, with historical background and examples.
• "Group actions and stratifications of prime spectra" (Conference "New Trends in Noncommutative Algebra", UW-Seattle, Aug 10, 2010) This talk describes some results concerning actions of affine algebraic groups on the prime spectrum of an arbitrary associative algebra. The articles on which the talk is based are Group actions and rational ideals and Algebraic group actions on noncommutative spectra.
• "Ring theoretic methods in the representation theory of Hopf algebras" (Conference "Hopf Algebras and Noncommutative Algebra", Ben-Gurion University, Sde-Boker, May 26, 2010) The talk discusses a unified ring theoretic approach, based on the theory of Frobenius algebras, to a variety of results on Hopf algebras. These include the so-called class equation and the semisimplicity locus of the Grothendieck ring of all finite-dimensional representations of a semisimple Hopf algebra. The first half of this talk is based on an expository paper of mine that you can find here.
• "Koszul algebras and the master theorem" (Algebra Seminar, University of Pennsylvania, December 3, 2007) This is a report on joint work with Benoit Kriegk and Phung Ho Hai. The talk explains how algebraic tools from the theory of N-Koszul superalgebras can be used to derive generalizations of MacMahon's celebrated "master theorem". A version of the master theorem involving Berezinians is stated as well as some binomial identities that follow from it. The paper on which this talk is based is here.
• "How large can a finite group of matrices be?" (Blundon Lecture, Fredericton, October 13, 2007) This talk was given to an audience of undergraduate students. The goal was to give an introduction to group theory and some of its history, old and new. The problem of bounding the size of finite matrix groups over the rationals (Minkowski's Theorem) was discussed in some detail. The article on which this part of the talk was based can be found here.
• "Multiplicative invariant theory" (Lie Groups Seminar, Rutgers University, February 9, 2007) The talk gives an introduction to multiplicative invariant theory highlighting regularity and the Cohen-Macaulay property.