Commutative Algebra Seminar
Fall 2018, Friday 2:30–3:20, LCB 215
Date  Speaker  Title — click for abstract 
August 24  Michael Brown University of Wisconsin, Madison 
ChernWeil theory for matrix factorizations
This is a report on joint work with Mark Walker. Classical algebraic ChernWeil theory provides a formula for the Chern character of a projective module P over a commutative ring in terms of a connection on P. In this talk, I will discuss an analogous formula for the Chern character of a matrix factorization. Along the way, I will provide background on matrix factorizations, and also on classical ChernWeil theory.

August 31  Ben Briggs University of Utah 
LusternikSchnirelmann category in commutative algebra
LS category is a numerical invariant originating in differential geometry, which has since found an important place in rational homotopy theory. In the early 80s commutative algebraists and rational homotopy theorists began to uncover deep parallels between their disciplines, greatly enriching both sides. Constructions and theorems on one side were found to have "lookingglass" analogues on the other. For example, LS category can be defined for local rings or homomorphisms, and this invariant has been used to prove important results in local commutative algebra. Nonetheless, LS category has not been well studied in commutative algebra in the years since. I will talk about an attempt to develop some of the properties of this invariant. In particular, I'll present a looking glass analogue of the mapping theorem, which is invaluable for computing LS category in rational homotopy theory. I will also be very careful to explain any topology I might want to use.

September 7  Thomas Polstra University of Utah 
Prime characteristic invariants and birational maps
We explore the behavior of prime characteristic invariants, such as Fsignature and HilbertKunz multiplicity, under
birational morphisms. We will discuss both positive and negative behavior.

September 14  Jenny Kenkel University of Utah 
Local cohomology of thickenings
Let R be a standard graded polynomial ring that is finitely generated over a field, and let I be a homogenous prime ideal. Bhatt, Blickle, Lyubeznik, Singh, and Zhang examined the local
cohomology of R/I^t, as t goes to infinity, which led to the development of an asymptotic invariant by Dao and Montanõ. I will discuss their results, and give concrete examples of the
calculation of this new invariant in the case of determinantal rings.

September 21  Henning Krause University of Bielefeld 
The Morita theory for derived categories revisited
A fundamental theorem of Rickard states for a pair of coherent rings that the categories of perfect complexes are triangle equivalent if and only
if their bounded derived categories of finitely presented modules are triangle equivalent. The proofs of both directions are somewhat delicate
and not treated properly in any text book. The aim of this talk is to propose new proofs, which should be of independent interest; they involve the notion of Cauchy completion for one direction, and the notion of a distinguished tstructure for the other direction.

September 28  

October 5  Rebecca R.G. George Mason University 

October 12  

October 19  

October 26  Cris Negron M.I.T., Boston 

November 2  Jon Carlson University of Georgia, Athens 

November 9  Linquan Ma Purdue University 

November 16  Lars Winther Christensen Texas Tech., Lubbock 

November 30  Robert M. Walker University of Michigan 

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