Commutative Algebra Seminar
Spring 2016, Friday 3:10–4:00, LCB 222
Date  Speaker  Title — click for abstract 
January 29  Graeme Milton University of Utah 
Superfunctions and the algebra of subspace collections
A natural connection between rational functions of several real or complex variables, and subspace collections is explored. A new class of function, superfunctions, are introduced which are the counterpart to functions at the level of subspace collections. Operations on subspace collections are found to correspond to various operations on rational functions, such as addition, multiplication and substitution. It is established that every rational matrix valued function which is homogeneous of degree 1 can be generated from an appropriate, but not necessarily unique, subspace collection: the mapping from subspace collections to rational functions is onto, but not one to one. For some applications superfunctions may be more important than functions, as they incorporate more information about the physical problem, yet can be manipulated in much the same way as functions. Orthogonal subspace collections occur in many physical problems, but we'll show an example of the use of nonorthogonal ones, which when substituted in orthogonal ones greatly accelerate the convergence of Fast Fourier transform methods.

February 5  Amnon Neeman
Australian National University 
Grothendieck duality via Hochschild homology
Hochschild cohomology was introduced in a 1945 paper by Hochschild, and Grothendieck duality dates back to the early 1960s. The fact that the two have some relation with each other is very new  it came up in papers by Avramov and Iyengar [2008], Avramov, Iyengar, and Lipman [2010] and Avramov, Iyengar, Lipman and Nayak [2011]. We will review this history, and the surprising formulas that come out.
We will then discuss more recent progress. The remarkable feature of all this is the role played by Hochschild homology. One example, which we will discuss in some detail, comes about as follows. The new techniques permit us to write formulas giving trace and residue maps in Grothendieck duality in terms of expressions that are very Hochschildhomological  Alonso, Jeremias and Lipman gave such a formula, but couldn't prove that it agrees with the usual formula dating back to Verdier in the 1960s. The proof that these two agree, due to Lipman and the speaker, turns out to hinge on considering the action of ordinary Hochschild homology on the various objects in the formula.

February 12  Julia Pevtsova University of Washington 
Varieties for elementary subalgebras of modular Lie algebras
Motivated by questions in modular representation theory, Carlson, Friedlander and the speaker instigated the study of projective
varieties of abelian pnilpotent subalgebras of a fixed dimension r for a pLie algebra g. These varieties are closely related to the much studied class of varieties of rtuples of commuting pnilpotent matrices which remain highly mysterious when r>2.
In this talk, I shall present some of the representationtheoretic motivation behind the study of these varieties and describe their geometry in a very special case when it is well understood: namely, when r is the maximal dimension of an abelian pnilpotent subalgebra of a semisimple Lie algebra g.
This is joint work with J. Stark.

March 4  Ian Shipman
University of Utah 
From representation varieties to CohenMacaulay modules
An important tool in the study of algebras is the
representation variety, which parameterizes representations of a fixed
dimension. These varieties are useful for organizing the representation
theory of wild algebras and also raise interesting geometric questions
themselves. It turns out that it is possible to define wellbehaved
representation varieties for commutative graded rings. These varieties
parameterize maximal CohenMacaulay (MCM) modules. In my talk, I will
explain some recent results both on representation varieties for a natural
class of finite dimensional (noncommutative) algebras and on MCM modules
over graded rings.

March 11  Kazuma Shimomoto Nihon University, Tokyo 
Fsingularities and non CohenMacaulay rings
This is a report on joint work with P.H. Quy (Hanoi).
I will talk about recent progress on the connection between Fsingularities and non CM rings, using the Frobenius action on local cohomology modules. I will also discuss
open problems.

March 11  Lars Winther Christensen Texas Tech., Lubbock 
Generic local rings interpolate between Gorenstein and Golod
Let Q be a power series ring, for example $\mathbb{Q}[\![x,y,z]\!]$, and let I be an ideal in Q. If the drop in depth from Q to the
quotient R=Q/I is at most 3, then R can be classified based on a multiplicative structure on the finite free resolution of R as a
Qmodule. The identification of the possible multiplicative structures was done 25 years ago, but the question of which structures
can actually be realized has only recently seen progress.
In joint work with Veliche and Weyman we construct families of rings $\mathbb{Q}[\![x,y,z]\!]/I$ that realize multiplicative structures which had been conjectured not to occur. In fact, rings with this structure seem to be everywhere, and
what emerges is a picture that describes generic local rings on a onedimensional scale whose end points are Gorenstein rings and Golod rings.

March 25  Brooke Ullery University of Utah 
Constructing ideals with high CastelnuovoMumford regularity
The CastelnuovoMumford regularity of a module is a homological invariant that roughly
measures complexity. Though straightforward to define, it is difficult to find ideals in
polynomial rings with high CastelnuovoMumford regularity. I will demonstrate a method
that takes as input wellunderstood modules and outputs ideals which cut out schemes
supported on linear spaces with high CastelnuovoMumford regularity and other desirable
homological properties.

April 1  Henning Krause
University of Bielefeld 
AuslanderReiten duality for Grothendieck abelian categories
A duality for representations of artin algebras was introduced in the 1970s by Auslander and Reiten. One can think of this as an analogue of Serre duality for projective varieties. The talk will present a version of AuslanderReiten duality for Grothendieck abelian categories. This makes it possible to compare both dualities.

April 6th  Thomas Polstra University of Missouri 
Global HilbertKunz Multiplicity
This talk will be a report on some results from a joint work in progress with Alessandro De
Stefani and Yongwei Yao. HilbertKunz multiplicity is a classical numerical invariant of study
attached to local rings of prime characteristic. In this talk we will discuss how to naturally extend
this numerical invariant to all rings which are not local and Ffinite in a meaningful way.

April 22  Rankeya Datta University of Michigan 
Valuations and Frobenius
We examine some usual notions of singularities in characteristic p > 0 such as Ffiniteness, Fsplitting, Fpurity etc., for valuation rings. The novelty of this approach stems from the fact that most of the characteristic p
notions of singularities have been extensively studied in the Noetherian setting, whereas valuation rings are usually highly nonNoetherian.
There is a surprising connection between the wellstudied class of Abhyankar valuations (these valuations may be viewed as higher dimensional analogues of valuations associated to prime divisors on a variety) and the class of Ffinite valuations. The talk is based on joint work with Karen Smith, and our work attempts to answer some questions asked by Karl Schwede and Zsolt Patakfalvi.

April 29  Dylan Zwick 
Symmetric bicolored trees and symmetric tropical ranks
In this talk I will define and investigate symmetric bicolored trees, the symmetric
versions of the bicolored trees studied by Markwig and Yu. In particular, I'll show
that a special subset of these symmetric bicolored trees, those with an "unbranched
stem", form a puredimensional simplicial complex, and that this complex is
shellable. I'll then describe a relationship between these symmetric bicolered trees
with unbranched stems, or "happy trees", and the notion of the symmetric tropical
rank of a matrix.

May 6  Claudiu Raicu University of Notre Dame 
The syzygies of some thickenings of determinantal varieties
The space of m x n matrices admits a natural action of the group GL_m x GL_n via row
and column operations on the matrix entries. The invariant closed subsets are the
determinantal varieties defined by the (reduced) ideals of minors of the generic
matrix. The minimal free resolutions of these ideals are wellunderstood by work
of Lascoux and others. There are however many more invariant ideals which are
nonreduced, and they were classified by De Concini, Eisenbud, and Procesi in the 80s.
I will explain how to determine the syzygies of a large class of these ideals by
employing a surprising connection with the representation theory of general linear
Lie superalgebras. This is joint work with Jerzy Weyman.

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