Algebra Seminar

2:00–3:00 Tuesdays

Physics 233

The old hand-rolled version of this page has been retired in favor of the Google calendar on the department web site. We keep this here as a historical archive of past seminars.

Olgur Celikbas, University of Kansas

Title:
*On the rigidity of Tor
*

Abstract:

I will survey some of the classical results on the rigidity of Tor for modules and discuss related questions over local rings.

Mark Kleiner, SU

Title:
*Induced modules over triangular matrix algebras
*

Mark Kleiner, SU

Title:
*Triangular matrix rings in the representation theory of algebras,
part 2
*

Mark Kleiner, SU

Title:
*Triangular matrix rings in the representation theory of algebras
*

Kosmas Diveris, SU

Title:
*On conditions concerning the vanishing of cohomology, part 2
*

Abstract:

It is known that the Auslander-Reiten conjecture holds for rings having what is called the AC property. We define a (possibly) weaker condition which all AC rings of finite injective dimension satisfy. We show that the Auslander-Reiten conjecture follows from this condition.

Kosmas Diveris, SU

Title:
*On conditions concerning the vanishing of cohomology
*

Declan Quinn, SU

Title:
*Fundamental but not well-known results on matrix rings
*

Abstract:

This would cover the Amitsur-Levitzki and Pare-Schelter theorems and introduce central polynomials if there is time.

Tom Bleier, SU

Title:
*Rational Divisor Classes on $\bar{\mathcal{M}}_g$, (Part II)
*

Abstract: PDF

Tom Bleier, SU

Title:
*Rational Divisor Classes on $\bar{\mathcal{M}}_g$
*

Abstract: PDF

Declan Quinn, SU

Title:
*Inner Gradings
*

Abstract:

The classical Skolem-Noether theorem gives conditions for an automorphism or a derivation of a central simple algebra to be inner. We will consider gradings of such algebras, what it means for them to be inner and an analogous result.

Steven P Diaz, SU

Title:
*Schubert Calculus
*

Abstract:

We will define what a Schubert variety is and explain how they are used to do computations in the Chow ring of a Grassmannian.

Claudia Miller, SU

Title:
*On a generalization of Kunz's theorem by Avramov et al., part 2
*

Abstract:

I will talk on a recently posted paper by Avramov, Hochster, Iyengar, and Yao on a generalization of Kunz's theorem from the 60s characterizing regular rings via the Frobenius map.

Claudia Miller, SU

Title:
*On a generalization of Kunz's theorem by Avramov et al.
*

Steven P. Diaz, SU

Title:
*How many lines meet four general lines in three space?
*

Abstract:

We will attempt to describe how Grassmannians which parameterize subspaces of a fixed dimension of a fixed vector space can be used to solve problems in enumerative geometry concentrating on the problem mentioned in the title. Given four general lines in projective three space, how many lines are there that meet all four lines?

Hamid Rahmati, SU

Title:
*Brauer–Thrall for totally reflexive modules
*

Abstract: PDF, or:

Let $(R, \mathfrak m , k)$ be a commutative local ring. A finite $R$-module $M$ is called totally reflexive if it is reflexive and $\rm{Ext}^i_R(M,R) = \rm{Ext}^i_R(M^*,R) = 0$ for all $i > 0$. Assume that $R$ is not Gorenstein and there are elements $w, x \in \mathfrak m$ such that $\rm {Ann}_R(x) = (w)$ and $\rm {Ann}_R(w) = (x)$. For every $n \in \mathbb N$ there exists an indecomposable totally reflexive module that is minimally generated by $n$ elements. Moreover, if $k$ is infinite then for every $n \in \mathbb N$ there are $|k|$ pairwise non-isomorphic indecomposable totally reflexive modules that are minimally generated by $n$ elements.

Marju Purin, SU

Title:
*Complexity of Trivial Extensions, Part 2
*

Abstract: PDF

Marju Purin, SU

Title:
*Complexity of Trivial Extensions, Part 1
*

Simon Smith, SU

Title:
*Permutation-ends of primitive groups
*

Abstract:

An end of an infinite graph can be thought of as a point of infinity. Now many of you will have heard, at least in passing, of ends of finitely generated groups. Beloved by combinatorial group theorists the world over, they are the ends of the Cayley graph of the group. Much has been written about them, and their relation to the structure of finitely generated groups. They are, as you say in America, a Big Deal.

Infinite primitive groups – the “atoms” of permutation group theory – are rarely finitely generated, even when one places quite strict finiteness conditions on them. Ends, one might think, are not useful when studying infinite permutation groups. But this is not the case! By imposing relatively mild finiteness conditions on these groups, it transpires that there is a very natural graph associated with each group, the ends of which I propose to call permutation-ends.

Assuming nothing more than basic undergraduate group theory, I will introduce you all to some of the research I’ve been involved in over the summer. In particular, I’ll show you how a little information about a permutation-end yields a lot of exciting information about the group.

Mark Yerrington, SU

Title:
*The lambda dimension of a commutative ring
*

Abstract: PDF

Petter Andreas Bergh,
Norwegian University of Science and Technology

Title:
*The dimension of a triangulated category
*

Abstract:

This is joint work with Srikanth Iyengar, Henning Krause and Steffen Oppermann. Given a triangulated category, its dimension measures how many steps are needed in order to generate the category from one of its objects. This concept was introduced quite recently by Rouquier, who used it to show that there exist finite dimensional algebras of arbitrarily large representation dimension. In this talk, we provide a lower bound for the dimension, and apply this to derived categories of local complete intersections and finite dimensional algebras.

Graham Leuschke, SU

Title:
*Tame and Wild Cohen-Macaulay Type, part 2
*

Abstract:

In the representation theory of finite-dimensional algebras over a field, Drozd's trichotomy theorem says that an algebra has either tame module type or wild module type. Loosely, these two possibilities correspond to: hoping for a classification theorem, or throwing up our hands in dismay. We'd very much like a similar trichotomy result in other representation-theoretic contexts, specifically for maximal Cohen--Macaulay modules over a Cohen--Macaulay local ring. I'll give a little background on the problem, including definitions of tame and wild CM type, and talk about recent work with Andrew Crabbe (Syracuse) which shows that hypersurfaces of multiplicity four or more, in three or more variables, have wild CM type.

Graham Leuschke, SU

Title:
*Tame and Wild Cohen-Macaulay Type
*

Nicole Snashall, University of Leicester

Title:
*Koszul algebras and Hochschild cohomology
*

Steven Diaz, SU

Title: Grassmannians for Mathphobics
*
*

Abstract:

A Grassmannian parameterizes subspaces of a fixed dimension of a fixed finite dimensional vector space. I will try to explain what a Grassmannian is and show that it is an example of a fine moduli space (recall the talk of Thomas Bleier). I will not succeed. Anyone with even a good undergraduate course in linear algebra should be able to understand at least some of this talk.

Susan Cooper, University of Nebraska-Lincoln

Title: Reduced VS Non-Reduced Point Sets - A Score Sheet for Invariants
*
*

Abstract:

In this talk we will compare algebraic invariants of reduced and "fat point" sets. By trimming down fat points, we will obtain upper and lower bounds for the Hilbert function, graded Betti numbers, and initial degree of any fat point set in projective 2-space. Empowered with the Hilbert function bounds, we can verify a special case of a conjectured inequality of Harbourne and Huneke involving the initial degrees of a fat point set and its support.

Dag Madsen, SU

Title: T-Koszul algebras
*
*

Abstract:

Let $\Lambda =\oplus_{n \geq 0} \Lamba_n$ be a graded algebra over a field $k$. We assume $dim_k \Lambda_i < \infty$ for all $i \geq 0$, but we do not assume that $\Lambda_0$ is semi-simple. Suppose $gldim \Lambda_0 < \infty$. Let $T$ be a graded $\L$-module concentrated in degree zero.

In this talk I propose the following new definition of T-Koszul algebras: $\Lambda$ is a $T$-Koszul algebra if both (1) and (2) hold.We prove that many quasi-hereditary Koszul algebras have a $T$-Koszul algebra structure coming form the standard (Verma) modules.

- $T$ is a tilting $\Lambda_0$-module.
- T is graded self-orthogonal as a $\Lambda$-module.

Kosmas Diveris, SU

Title: The Auslander-Reiten conjecture for Gorenstein rings.
*
*

Abstract:

Tokuji Araya has shown that if the Auslander-Reiten conjecture holds in codimension one for a commutative Gorenstein ring R, then it holds for R. We will present Araya's proof from the paper "The Auslander-Reiten conjecture for Gorenstein rings".

Andrew Crabbe, SU

Title:
*Ranks of indecomposable maximal Cohen-Macaulay modules, part 2
*

Abstract:

Let R be a one-dimensional reduced Noetherian local ring of infinite Cohen-Macaulay type, and let P_1,. . . , P_s be the minimal prime ideals of R. From work of Roger Wiegand and others, it is known that for every positive integer r there is an indecomposable maximal Cohen-Macaulay R-module M of constant rank r, i.e. M is free of the same rank r when localized at each P_i. In this talk we explore the following question: For which non-trivial s-tuples (r_1,...,r_s) is there an indecomposable maximal Cohen-Macaulay R-module M such that M is free of rank r_i when localized at P_i?

Andrew Crabbe, SU

Title:
*Ranks of indecomposable maximal Cohen-Macaulay modules
*

Claudia Miller, SU

Title: *Smoke's result on reducing modules to cyclic ones within a certain
Grothendieck group, part 2
*

Abstract:

We discuss a paper of W. Smoke on how to reduce modules to cyclic ones modulo certain modules with nice intersection properties.

Claudia Miller, SU

Title: *Smoke's result on reducing modules to cyclic ones within a certain
Grothendieck group
*

Dag Madsen, SU

Title: *Orbit closures in varieties parametrizing the objects in the
derived category.
*

Abstract:

A famous theorem by Zwara and Riedtmann gives an algebraic characterization of orbit closure (degeneration) in module varieties. In joint work with Jensen and Su we have proved an analogous theorem for varieties of A1-modules. Our result should be seen as the derived category version of the Zwara-Riedtmann Theorem, and contains that theorem as a special case. In this talk I will describe the varieties and orbit closures involved and show how they gives us new insight into the derived category of a nite dimensional algebra. I will also explain why new techniques might be needed for obtaining a characterization of orbit closures in varieties of algebras.

Thomas Bleier, SU

Title: *Introduction to the Moduli Space of Curves (Part II)
*

Abstract:

In this talk I will state an existence theorem for the coarse moduli space for curves of a fixed genus greater than 1. We will then discuss hyperelliptic curves and calculate the dimension of the hyperelliptic locus in the moduli space. Lastly, we will look at a modular compactification of the moduli space.

Thomas Bleier, SU

Title: *Introduction to the Moduli Space of Curves (Part I)
*

Abstract:

In this talk we will look at what it means to have a family of curves over a base scheme, and describe a functor from schemes to sets that sends a scheme to the set of all such families of curves, of a given genus, over that scheme. We will also look at what it means for a such a functor to be representable and consequences of being representable. The talk will conclude by looking at the case of elliptic curves.

Dag Oskar Madsen, SU

Title: *Orbit closures in varieties parametrizing the objects in
the derived category, Part 2.
*

Abstract:

A famous theorem by Zwara and Riedtmann gives an algebraic characterization of orbit closure (degeneration) in module varieties. In joint work with Jensen and Su we have proved an analogous theorem for varieties of A1-modules. Our result should be seen as the derived category version of the Zwara-Riedtmann Theorem, and contains that theorem as a special case. In this talk I will describe the varieties and orbit closures involved and show how they gives us new insight into the derived category of a nite dimensional algebra. I will also explain why new techniques might be needed for obtaining a characterization of orbit closures in varieties of algebras.

Steven P. Diaz, SU

Title: *What is a Weierstrass point?
*

No seminar this week: happy Thanksgiving!

Claudia Miller, SU

Title: *Limit Hilbert-Kunz Multiplicities
*

Suanne Lu, Le Moyne

Title: *Equivariant K-Theory of Toric Varieties
*

Dag Madsen, SU

Title: *Orbit closures in varieties parametrizing the objects in
the derived category.
*

Abstract:

A famous theorem by Zwara and Riedtmann gives an algebraic characterization of orbit closure (degeneration) in module varieties. In joint work with Jensen and Su we have proved an analogous theorem for varieties of A1-modules. Our result should be seen as the derived category version of the Zwara-Riedtmann Theorem, and contains that theorem as a special case. In this talk I will describe the varieties and orbit closures involved and show how they gives us new insight into the derived category of a nite dimensional algebra. I will also explain why new techniques might be needed for obtaining a characterization of orbit closures in varieties of algebras.

Suanne Lu, Le Moyne

Title: *Equivariant K-Theory of Affine Toric Varieties, part 2
*

Suanne Lu, Le Moyne

Title: *Equivariant K-Theory of Affine Toric Varieties
*

Dan Zacharia, SU

Title: *Subadditive functions, part III
*

Dan Zacharia, SU

Title: *Subadditive functions, part 2 (the rescheduling)
*

R. Inanc Baykur, Brandeis University

Title: *Smooth four-manifolds, surfaces and exotica
*

In this talk, we will focus on a 4-dimensional phenomenon: Existence of infinitely many smooth 4-manifolds, which are all homeomorphic to a standard manifold but pairwise non-diffeomorphic; and in a parallel vein, existence of infinitely many embeddings of surfaces in a fixed standard 4-manifold, which are ambiently homeomorphic but non-diffeomorphic. We will discuss how to construct such examples.

Dan Zacharia, SU

Title: *Subadditive functions, part 2
*

Dan Zacharia, SU

Title: *Subadditive functions
*

Steven P. Diaz, SU

Title: *Complex Analysis
*

Dieter Happel, Technical University of Chemnitz

Title: *The representation dimension for iterated tilted algebras
*

Roger Wiegand,
University of Nebraska-Lincoln

Title: *Commutative versions of the Brauer–Thrall
Conjectures*

Abstract:

The Brauer-Thrall conjectures concern the number of non-isomorphic indecomposable finitely generated modules over an Artinian ring. The first conjecture (now a theorem) says that if the ring has infinite representation type (that is, there are infinitely many non-isomorphic indecomposables), then there must be indecomposables of arbitrarily large finite length. The second says there are infinitely many $n$ for which there are infinitely many indecomposables of length $n$.

In this talk I will transplant these conjectures to commutative algebra and survey what is known. We will consider maximal Cohen-Macaulay modules over local Noetherian rings. "Length" is replaced by "rank" or "multiplicity". For one-dimensional local rings, maximal Cohen-Macaulay modules are just non-zero torsion-free modules (also called "lattices" in algebraic number theory).

No seminar this week. Enjoy "Syracuse Showcase" instead.

Mark Kleiner, SU

Title: *Representations of quivers and root systems of Kac-Moody algebras and Coxeter groups, part 2*

Mark Kleiner, SU

Title: *Representations of quivers and root systems of Kac-Moody algebras and Coxeter groups*

Mark Kleiner, SU

Title: *Representations of quivers and root systems of Kac-Moody algebras*

Kosmas Diveris, SU

Title: *Standard resolutions and CM approximations, part 2*

Abstract:

We will summarize several results concerning resolutions over complete intersections, then define CM approximations and the related delta invariants of modules over Gorenstein rings. We will then show that over a hypersuface there is a connection between the delta invariants of a module and its standard resolution.

Kosmas Diveris, SU

Title: *Standard resolutions and CM approximations, part 1*

Janet Striuli, Fairfield University

Title: *Growth of Bass Numbers*

Abstract:

Let R be a commutative Noetherian local ring with residue field k. Important aspects of the structure of the ring can be detected by the cohomology groups Ext^i_R(k,k) and Ext^i_R(k,R). The k-vector space dimensions of the cohomology groups Ext^i_R(k,k) are called Betti numbers, while the dimensions of Ext^i_R(k,R) are known as the Bass numbers of R. While the resulting sequence of Betti numbers has been studied very extensively, little is known about the sequence of Bass numbers. We will present similarities and differences in these sequences and some recent developments on the study of Bass numbers.

Graham Leuschke, SU

Title: *Non-commutative desingularizations of determinantal varieties 3*

Abstract:

I will define and give some background on what it should mean to be a "non-commutative resolution of singularities," with emphasis on examples. Then I'll explain joint work with Buchweitz and Van den Bergh showing existence of such desingularizations for a large class of rings defined by minors of generic matrices.

This third part will focus on connections with the representation theory of quivers.

Bogdan Petrenko, SUNY Brockport

Title: *On the probability of generating the ring of integer matrices*

Abstract:

The main goal of my talk is to explain why the asymptotic probability that m random elements generate the ring of 2-by-2 integer matrices equals zeta(m)^{-1}zeta(m-1)^{-1}, where zeta is the Riemann zeta function. This formula is part of my joint work with Rostyslav Kravchenko (Texas A&M) and Marcin Mazur (Binghamton University).

Frank Moore, Cornell University

Title: *Cohomology of Products of Local Rings*

Abstract:

The Ext algebra of a commutative noetherian local ring $R$ with residue field $k$ is a graded $k$-algebra that carries subtle information about $R$. In this talk, I will discuss the structure of the Ext algebra of local rings that arise by combining local rings $S$ and $T$ with the same residue field in various ways.

Starting with $R = S \times_k T$, the fiber product of $S$ and $T$, I will describe the Ext algebra of $R$ as well as the structure of certain modules over it. Next I will discuss joint work with Avramov in which we study the connected sum $S \# T$ of $S$ and $T$, which is defined only when $S$ and $T$ are artinian Gorenstein rings. Although we have yet to describe the Ext algebra of a connected sum completely, I will discuss some partial results of ours in this direction.

Graham Leuschke, SU

Title: *Non-commutative desingularizations of determinantal varieties ii*

Abstract:

I will define and give some background on what it should mean to be a "non-commutative resolution of singularities," with emphasis on examples. Then I'll explain joint work with Buchweitz and Van den Bergh showing existence of such desingularizations for a large class of rings defined by minors of generic matrices.

Graham Leuschke, SU

Title: *Non-commutative desingularizations of determinantal varieties*

Dag Madsen, SU

Title: *AR-sequences in subcategories*

Abstract:

This is sort of a continuation of my previous talk. I will say more about how to compute AR-sequences in subcategories.

Steven P. Diaz, SU

Title: *Fun with Riemann-Roch*

Abstract:

We will show how many of the older versions of the Riemann-Roch theorem follow from the Grothendieck-Riemann-Roch theorem. If time permits we will also show some of the ways the Riemann-Roch theorem is used to study curves.

Steven P. Diaz, SU

Title: *The Grothendieck-Riemann-Roch Theorem, pt. 3*

Abstract:

This will be a brief introduction to Chern classes, the Grothendieck group and the Grothendieck-Riemann-Roch theorem.

No seminar. Happy Thanksgiving!

Dag Madsen, SU

Title: *AR-sequences for modules with a Verma flag*

Abstract:

The category of modules orthogonal to a given tilting module has Auslander-Reiten sequences. In this talk I will illustrate a method for computing these sequences using the category of so(4)-modules with a Verma flag as an example.

Alex Dugas, UC Santa Barbara

Title: *Periodic Algebras*

Steven P. Diaz, SU

Title: *The Grothendieck-Riemann-Roch Theorem, Part 2*

Abstract:

This will be a brief introduction to Chern classes, the Grothendieck group and the Grothendieck-Riemann-Roch theorem.

Steven P. Diaz, SU

Title: *The Grothendieck-Riemann-Roch Theorem*

Marju Purin, SU

Title: *Complexity and Periodicity, Part 2*

Abstract:

We study modules over an artin k-algebra Λ where k is a commutative artin ring. We introduce the notions of complexity and periodicity of a module. We proceed to show that under a certain ﬁnite generation assumption bounded resolutions must become eventually periodic. This talk is based on the paper ”Complexity and Periodicity” by Petter Bergh.

Marju Purin, SU

Title: *Complexity and Periodicity*

Steven P. Diaz, SU

Title: *Determinantal Varieties*

Abstract:

The talk will cover some of the basic properties of determinantal varieties and also include a brief introduction to Grassmannians.

Dieter Happel, TU Chemnitz

Title: *On the coefficients of the Coxeter polynomial*

Andrew Crabbe, SU

Title: *Building Indecomposable Modules*

Abstract:

A common way to understand a ring is to study certain subcategories of its modules, in particular, the indecomposable objects in such categories. Over rings with dimension 0, it's reasonable to look at the entire category of modules, however, over larger rings, its beneficial to restrict to a more tractable collection. The subcategory of greatest interest has been that of all maximal Cohen-Macaulay modules. By knowing the number of indecomposable maximal Cohen-Macaulay modules or whether there is a bound on the "size" of these modules, one can gain important information about the ring (for instance, the dimension of its singular locus). But what happens if you go away from the subcategory of maximal Cohen-Macaulay modules?

One purpose of this talk is to show that over certain rings with dimension greater than 1, one can build indecomposable modules that are arbitrarily "large" (where "large" could refer to the multiplicity, or in our case, the rank on the punctured spectrum), even if the ring does not permit the construction of arbitrarily "large" indecomposable maximal Cohen-Macaulay modules. These constructions are achieved using results for the generalized Hilbert-Samuel polynomials.

Andrew Crabbe, SU

Title: *Generalized Hilbert-Samuel polynomials*

Abstract:

The purpose of this talk is to introduce a generalization of the classical Hilbert-Samuel polynomial. These generalized polynomials have been used to build "big" indecomposable modules over certain rings (which will be discussed in a 2nd talk), as well as to provide information about the dimensions of large syzygies of modules. The talk should be accessible and many of the basic ideas will be explained.

Dag Madsen, SU

Title: *Hochschild homology and global dimension*

Abstract:

The aim of the talk is to give a gentle introduction to my recent paper with Petter A. Bergh (with the same title).

Organizational Meeting

Dan I. Zacharia, SU

Title: *Auslander-Reiten triangles in derived categories, the return*

Gary Kennedy, Ohio State Mansfield

Title: *Monodromy of plane curves and quasi-ordinary surfaces*

Abstract:

A quasi-ordinary surface f(x,y,z)=0 is one for which, at each singular point, there is a theory of Puiseux expansion as for plane curves. Taking a transverse slice with x=constant, one obtains a singular plane curve. Its Milnor fiber f(x,y)=e has two sorts of monodromy: (1) the Milnor monodromy (also called its horizontal monodromy), in which x is fixed while e varies around a small circle, (2) the vertical monodromy, in which e is fixed while x varies. In joint work with my colleague Lee McEwan, we have discovered simple recursive formulas for both monodromies.

Dan I. Zacharia, SU

Title: *Auslander-Reiten triangles in derived categories*

Kosmas Diveris, SU

Title: *What makes a complex exact?*

**Special Two-fer Seminar!** (no fooling!)

Cornelia Yuen, SUNY Potsdam

Title: *A minimal reduction of Ferrers Ideals*

AND

Jason Howald, SUNY Potsdam

Title: *Motivic integration and monomial ideals*

Liana Sega, UMKC

Title: *Free resolutions over a class of
local rings*

Abstract:

I will consider a class of local Koszul rings with radical cube zero. Assuming the existence of a certain element, which we call Conca generator, we can describe the Poincare series of the residue field, and determine the structure of Ext(M,k) for a finite R-module M. The existence of the Conca generator guarantees that free resolutions over such rings have some remarkable properties: every finite module admits a syzygy which is Koszul and every module annihilated by the Conca generator is Koszul. We can describe completely the Koszul modules over Gorenstein rings with radical cube zero, and, in general, we have preliminary results towards understanding the Koszul property for modules with a fixed Hilbert series. This talk covers joint work with L. Avramov and S.Iyengar.

Steven P. Diaz, SU

Title: *There is more to resolutions than graded Betti numbers,
Part 2*

Abstract:

This is joint work of Diaz and Geramita. We find two finite sets of points in projective 3 space that differ geometrically, yet the graded minimal free resolutions of their ideals have the same graded Betti numbers. Inspired by this example we define a (hopefully new) discrete invariant one can associate to a graded minimal free resolution. This invariant distinguishes between these two sets of points.

Claudia Miller, SU

Title: *Rigidity of Frobenius, part deux*

Claudia Miller, SU

Title: *Rigidity of Frobenius*

Steven P. Diaz, SU

Title: *There is more to resolutions than graded Betti numbers*

Abstract:

This is joint work of Diaz and Geramita. We find two finite sets of points in projective 3 space that differ geometrically, yet the graded minimal free resolutions of their ideals have the same graded Betti numbers. Inspired by this example we define a (hopefully new) discrete invariant one can associate to a graded minimal free resolution. This invariant distinguishes between these two sets of points.

Declan Quinn, SU

Title: *Kronecker products of representations of the symmetric
group, part iv*

Declan Quinn, SU

Title: *Kronecker products of representations of the symmetric
group, part iii*

Declan Quinn, SU

Title: *Kronecker products of representations of the symmetric
group, part ii*

Declan Quinn, SU

Title: *Kronecker products of
representations of the symmetric group*

Bogdan Petrenko, SUNY Brockport

Title: *Some formulas for the smallest number of generators for finite direct sums of matrix algebras*

Abstract:

Any finite direct sum of matrix algebras over an infinite field has 2 generators. This is no longer true in general for a finite direct sum of matrix algebras over other commutative rings. We obtain an asymptotic upper bound for the smallest number of generators for a finite direct sum of matrix algebras over a finite field. We also obtain an exact formula for the smallest number a_{n}(q) of generators for a direct sum of n copies of the 2-by-2 matrix algebra over a field with q elements. We show that a__{n}(2) is also the smallest number of generators for a direct sum of n copies of the 2-by-2 integer matrix ring. We remark that generators for a finite direct sum of integer matrix rings can be used as generators for a similar direct sum where the ring of integers is replaced with any ring with 1.

The talk will be based on my joint work with Said Sidki (Journal of Algebra, 310 (2007), no. 1, 15--40) and Rostyslav Kravchenko(arXiv:math/0611674).

Mark Kleiner, SU

Title: *Finite and Infinite Coxeter Groups*

Mark Kleiner, SU

Title: *More on Coxeter groups*

Declan Quinn, SU

Title: *An introduction to group representations and the symmetric group*

Steven Diaz, SU

Title: *Schur Functors, Part 3*

Steven Diaz, SU

Title: *Schur Functors, Part 2*

Steven Diaz, SU

Title: *Schur Functors*

Jinjia Li, SU

Title: *The Game of Tor and Frobenius, part 2*

Abstract:

The Frobenius endomorphism is an important tool in studying homological questions in commutative algebra. I will first give a brief survey on topics related to homological properties of Frobenius, assuming only minimum background in commutative algebra and homological algebra. Then I will discuss some recent progress made in joint work with I. Aberbach of University of Missouri and, respectively, with C. Miller.

Jinjia Li, SU

Title: *The Game of Tor and Frobenius, part 1*

Mark Kleiner, SU

Title: *Coxeter groups and representations of algebras, part 4*

Mark Kleiner, SU

Title: *Coxeter groups and representations of algebras, part 3*

Mark Kleiner, SU

Title: *Coxeter groups and representations of algebras, part 2*

Mark Kleiner, SU

Title: *Coxeter groups and representations of algebras, part 1*

Organizational Meeting

Steven P. Diaz, SU

Title: *The Gale Transform and Multi-Graded Determinantal Schemes, Part 5*

Abstract:

This is joint work with Susan M. Cooper of California Polytechnic State University. Eisenbud and Popescu showed that certain finite determinantal subschemes of projective spaces defined by maximal minors of adjoint matrices of homogeneous linear forms are related by Veronese embeddings and a Gale transform. We extend this result to adjoint matrices of multihomogeneous multilinear forms. The subschemes now lie in products of projective spaces and the Veronese embeddings are replaced with Segre embeddings.

Steven P. Diaz, SU

Title: *The Gale Transform and Multi-Graded Determinantal Schemes, Part 4*

Steven P. Diaz, SU

Title: *The Gale Transform and Multi-Graded Determinantal Schemes, Part 3*

Steven P. Diaz, SU

Title: *The Gale Transform and Multi-Graded Determinantal Schemes, Part 2*

Steven P. Diaz, SU

Title: *The Gale Transform and Multi-Graded Determinantal Schemes, Part 1*

Graham Leuschke, SU

Title: *The Most Preposterous Theorem I Know, part 2*

Abstract:

I'll discuss the weirdest theorem I've ever heard of: Osofsky's result that projective dimension depends on your set theory.

Graham Leuschke, SU

Title: *The Most Preposterous Theorem I Know*

Claudia Miller, SU

Title: *Intersection Numbers, part 3*

Greg Piepmeyer, U. Nebraska

Title: *Breaking inseparable*

This talk describes a problem raised by Larry Levy and Lee Klingler on resolving which rings have extensions of Steinitz' theorem. They addressed all cases except when the residue field extension is characteristic two and inseparable. They set up, but do not solve, a matrix problem to answer the general case of Dedekind-like rings, but use other methods to handle all other cases. Klingler and I address this matrix problem.

Greg Piepmeyer, U. Nebraska

Title: *New Intersection via Adams Operations*

Dan I. Zacharia, SU

Title: *A quick introduction to Auslander-Reiten theory, part 2*

Dan I. Zacharia, SU

Title: *A quick introduction to Auslander-Reiten theory*

Claudia Miller, SU

Title: *Intersection Numbers, part 2*

Claudia Miller, SU

Title: *Intersection Numbers*

Philip Lynn, SU

**PhD Dissertation Defense:** *Deformations of Plane Curve Singularities of Constant Class*

Declan Quinn, SU

Title: *Galois actions, part 3*

Declan Quinn, SU

Title: *Galois actions, part 2*

Declan Quinn, SU

Title: *Galois actions*

Jinjia Li, SU

Title: *Homology of perfect complexes over Cohen-Macaulay local rings, part 2*

Abstract:

A perfect complex is a bounded complex of finite rank free modules. If all the homology of such a complex have finite length, one defines the Euler characteristic to be the alternating sum of the lengths of the homology. We will first discuss a property enjoyed by the homology of perfect complexes over a Cohen-Macaulay local ring A. Then, we apply this property to the situations: (1) the trivial extension of A and its canonical module, (2) the Frobenius endomorphism (in positive characteristic case), to obtain interesting results regarding the behavior of the Euler characteristics.

Jinjia Li, SU

Title: *Homology of perfect complexes over Cohen-Macaulay local rings*

Steven P. Diaz, SU

Title: *Veronese Subrings, continued*

Steven P. Diaz, SU

Title: *Veronese Subrings, continued*

Steven P. Diaz, SU

Title: *Veronese Subrings*

Faculty meeting; no seminar

Allen Pelley, SU

Title: *Representations of Quivers and Weyl groups of Kac-Moody Algebras (part 2)*

Abstract:

In this series of talks, we will see a brief introduction to representations of quivers and how they relate to Weyl groups of Kac-Moody algebras. Specifically, we will use representations of quivers to prove "for any Coxeter element of the Weyl group associated to an indecomposable symmetrizable generalized Cartan matrix, the group is infinite if and only if the powers of the element are reduced words." This strengthens known results of Howlett, Fomin-Zelevinsky. These talks should be accessible to graduate students who have completed MAT631-632.

Allen Pelley, SU

Title: *Representations of Quivers and Weyl groups of Kac-Moody Algebras (part 1)*

Mark Kleiner, SU

Title: *Preprojective modules and reduced words in the Weyl group of a quiver*

Abstract:

This is joint work with Allen Pelley. Given a finite quiver without oriented cycles, the indecomposable preprojective modules over the path algebra are classified by and computed in terms of special sequences of vertices of the quiver. To each such sequence one associate a word in the Weyl group of the quiver, and the goal is to study the resulting set of words. It turns out that these words have very nice properties. In particular, they are reduced, and they detect isomorpism classes of indecomposable preprojectives.

Graham Leuschke, SU

Title: *The McKay Correspondence III*

Abstract:

I will describe a confluence of ideas from group representation theory, commutative algebra, non-commutative algebra, and algebraic geometry, which centers around two-dimensional rings of invariants. Conjectural extensions to higher dimension are a very active area of research, and apparently come up in mathematical physics, while the two-dimensional theory is just beautiful mathematics.

Graham Leuschke, SU

Title: *The McKay Correspondence II*

Graham Leuschke, SU

Title: *The McKay Correspondence*

Izuru Mori, SUNY Brockport

Title: Co-point Modules over Koszul Algebras

Abstract:

Let $A$ be a graded algebra finitely generated in degree 1 over a field $k$. Point modules over $A$ introduced by Artin, Tate and Van den Bergh play an important role in studying $A$ in noncommutative algebraic geometry. In this talk, we define a dual notion of point module in terms of Koszul duality, which we call a co-point module. Using co-point modules, we will construct counter-examples to the following condition due to Auslander: for every finitely generated right module $M$ over a ring $R$, there is a natural number $n_M$ such that, for any finitely generated right module $N$ over $R$, $Ext ^i_R(M, N)=0$ for all $i>>0$ implies $Ext ^i_R(M, N)=0$ for all $i>n_M$.

Dan I. Zacharia, SU

Title: *Representation Dimension IV*

Dan I. Zacharia, SU

Title: *Representation Dimension III*

Dan I. Zacharia, SU

Title: *Representation Dimension II*

Dan I. Zacharia, SU

Title: *Representation Dimension*

Markus Reitenbach, SU

Title: *Projectivization of Poset Representations*

Abstract:

I will give a brief introduction into the theory of representations of partially ordered sets and their relationship to representations of quivers respectively finite dimensional algebras. In particular, I will indicate how an algebraic technique called projectivization leads to the notion of a certain "derivative" of a given poset, and how this derivative can be used to answer questions about the representations of the original poset.

Susan Cooper, SU

Title: *Geometric Consequences of the Eisenbud-Green-Harris Conjecture*

Abstract:

The Eisenbud-Green-Harris conjecture gives rise to a charaterization of the Hilbert functions of point sets which are subsets of certain families of complete intersections. In this talk we will review the characterization of such functions and investigate geometric consequences. In particular, we will see that subsets with certain Hilbert functions are guaranteed to have the "Cayley-Bacharach Property". We will also show that certain collections of subsets must have a "maximal Hilbert function".

Frédérick Latour, SU

Title: *Central extensions of preprojective algebras*

Dave Jorgensen, University of Texas at Arlington

Title: *Gorenstein rings: not so nice*

Steve Diaz, SU

Title: *Almost Split Sequences, Preprojective Algebras, and Multiplication Maps of Maximal Rank, II*

Steve Diaz, SU

Title: *Almost Split Sequences, Preprojective Algebras, and Multiplication Maps of Maximal Rank*

Lars Winther Christensen, University of Nebraska

Title: *Stability of G-dimensions*

Abstract:

Auslander and Bridger's 1966 definition of the so-called G-dimension for finitely generated modules over a commutative, noetherian ring marked the start of the study of Gorenstein homological dimensions. In commutative algebra, these homological invariants are used to characterize Gorenstein rings in much the same way the usual homological dimensions characterize regular rings.

This talk will center around some questions on stability of Gorenstein dimensions. It will be an introduction/survey illustrated by some tangible examples to shed light on these questions.

Jinjia Li, University of Illinois

Title: *Intersection Multiplicity and Asymptotic Length of Homologies*

Mark Kleiner, SU

Title: Algebras with Smallest Resolutions of Simple Modules, II

Mark Kleiner, SU

Title: Algebras with Smallest Resolutions of Simple Modules

Sebastian Burciu, SU

Title: Representations of the Drinfel'd double, II

Sebastian Burciu, SU

Title: Representations of the Drinfel'd double

Claudia Miller, SU

Title: Extremal Algebras, II

Claudia Miller, SU

Title: Extremal Algebras

Graham Leuschke, SU

Title: Matrix factorizations and factoring matrices, II

Abstract:

Graham Leuschke, SU

Title: Matrix factorizations and factoring matrices: the generic adjoint and maximal Cohen-Macaulay modules

Abstract:

I'll talk about joint work with Ragnar-Olaf Buchweitz (Toronto) inspired by the following question of G. Bergman: "Can one factor the classical adjoint of a generic matrix?". This unassuming (and not terribly motivated) question has unexpected connections to topology, representation theory, and commutative algebra. I'll describe some of these connections, focusing on the connection with the theory of matrix factorizations (a.k.a. maximal Cohen-Macaulay modules).

Moira McDermott, Gustavus Adolphus College

Title: Hilbert-Kunz Functions, II

Moira McDermott, Gustavus Adolphus College

Title: Hilbert-Kunz Functions

Abstract:

This talk will give an introduction to Hilbert-Kunz functions, including basic facts and what is currently known about them. In addition, I will present a new result (joint with Craig Huneke and Paul Monsky) regarding the existence of a second coefficient in the Hilbert-Kunz function of an m-primary ideal in a normal local ring of positive characteristic.

Sebastian Burciu, SU

Title: Representations of Degree Three for Semisimple Hopf Algebras, II

Sebastian Burciu, SU

Title: Representations of Degree Three for Semisimple Hopf Algebras

No seminar today due to the A&S Faculty Meeting.

No seminars this week.

Frédérick Latour, SU

Title: Representations of Cherednik algebras in positive characteristic

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