Date 
Speaker 
Title/Abstract 
9/7 
Lance Miller (UArk) 
The de RhamWitt complex and the htopology
In this series of talks, we will discuss differential forms on singular schemes. Usually, one must impose some condition to make discussions fruitful, for example to consider normal schemes. We will first survey work of Huber and Joerder, and following work of Huber, Kebekus, and Kelly on how to unify many different perspectives on differential forms using Voevodsky's htopology. We then apply these ideas to Illusie's de RhamWitt complex which is joint work with V. Ertl.

9/14 
Lance Miller (UArk) 
The de RhamWitt complex and the htopology
Continuation of last week's talk.

9/21 
Paolo Mantero (UArk) 
Properties of Reeslike algebras
Reeslike algebras have been used by McCullough and Peeva to construct counterexamples to the more general statement of the EisenbudGoto regularity conjecture. In this talk, we will explore some of their algebraic and geometric properties and illustrate some structural results. This talk is based on joint work with J. McCullough and L. E. Miller.

10/5 
Youngsu Kim (UArk) 
Generic links of determinantal varieties.
We study singularities of the generic link of a determinantal variety. Let A denote an affine space over the complex numbers, and let X and Y be reduced equidimensional subschemes of A. We say that X and Y are linked via V if there exists a complete intersection V in A such that I_X = I_V : I_Y and I_Y= I_V : I_X.
Two linked subschemes have many properties in common, and it is believed that the generic link of a variety improves singularities of the variety. Let X be a variety and Y the generic link of X. Wenbo Niu showed that the log canonical threshold, lct for short, “improves” under taking the generic link, i.e., lct Y is greater than lct X. It is not known if equality holds in general. In this talk, we show that in the case where X is a determinantal variety, we have lct X = lct Y. This is joint work with Wenbo Niu and Lance Edward Miller.

10/10 
Alex Buium (UNM) 
Lie invariance of Frobenius lifts
We show that the padic completion of any affine elliptic curve with ordinary reduction
possesses Frobenius lifts whose "normalized" action on 1forms preserves mod p the space of invariant 1forms. We also show that, after removing the 2torsion sections,
the above situation can be "infinitesimally deformed" in the sense that
the above mod p result has a mod p^2 analogue. Finally we show that the mod p result
fails for linear algebraic groups that are not tori.

10/19 
William Taylor (TSU) 
Interpolating Between Tight and Integral Closure Using sClosures
In this talk we will construct a family of closures that interpolate between integral closure and tight closure, two important operations in commutative algebra. In many rings, there are infinitely many distinct sclosures that lie between the integral and tight closure. In the local setting, the sclosures share a strong relationship with the smultiplicity, a family of multiplicity functions, in the same way that the integral and tight closure are related to the HilbertSamuel and HilbertKunz multiplicity respectively. We will discuss recent results on the properties of the sclosures in Zngraded rings and particularly in toric rings.

11/2 
Alessandra Costantini (Purdue) 
CohenMacaulayness of Rees algebras of modules
Rees algebras of ideals and modules arise in Algebraic Geometry as homogeneous coordinate rings of blow up or as graphs of rational maps. The CohenMacaulayness of the Rees algebra of an ideal I is wellunderstood in connection with the CohenMacaulayness of the associated graded ring of I, thanks to results of Huneke, Trung and Ikeda. However, there is no module analogue for the associated graded ring, so the study of CohenMacaulayness of Rees algebras of modules requires completely different techniques. In this talk, we will provide a sufficient condition for the Rees algebra of a module to be CohenMacaulay. Our result generalize results of Johnson and Ulrich, and of Goto, Nakamura and Nishida.

11/5 
Sean SatherWagstaff (Clemson) 
The power edge ideal of a graph.
Motivated by questions in electrical engineering and computer science, we associate to a finite simple graph a new squarefree monomial ideal called the power edge ideal. It is defined in terms of PMU covers where PMU is short for "phasor measurement unit." PMUs are devices placed on buses in electrical power systems to detect, e.g., power outages. In this talk, we will explain how the ideal is defined and characterize the trees for which this ideal is CohenMacaulay, showing in particular that (for trees) the CohenMacaulay property is equivalent to the unmixed property and the complete intersection property. This is joint work with Michael Cowen, James Gossell, Alan Hahn, and W. Frank Moore.

11/7 
Takumi Murayama (UMich) 
The gamma construction and applications to commutative algebra and
algebraic geometry
Hochster and Huneke introduced the gamma construction to prove that test elements (in the sense of tight closure) exist on all rings essentially of finite type over excellent local rings of characteristic p greater than 0. Tight closure and test elements were used extensively to give proofs of the BrianconSkoda theorem and a uniform comparison result between symbolic and ordinary powers of ideals on regular rings. In general, the gamma construction provides a systematic way to reduce to the case when the Frobenius homomorphism is finite. We prove new results about the gamma construction, which we use to study Fsingularities in the nonFfinite setting. A schemetheoretic version of these results are then used to study asymptotic invariants of line bundles over arbitrary fields.

11/9 
Youngsu Kim (UArk) 
Generic links of determinantal varieties Part II.
We study singularities of the generic link of a determinantal variety. Let A denote an affine space over the complex numbers, and let X and Y be reduced equidimensional subschemes of A. We say that X and Y are linked via V if there exists a complete intersection V in A such that I_X = I_V : I_Y and I_Y= I_V : I_X.
Two linked subschemes have many properties in common, and it is believed that the generic link of a variety improves singularities of the variety. Let X be a variety and Y the generic link of X. Wenbo Niu showed that the log canonical threshold, lct for short, “improves” under taking the generic link, i.e., lct Y is greater than lct X. It is not known if equality holds in general. In this talk, we show that in the case where X is a determinantal variety, we have lct X = lct Y. This is joint work with Wenbo Niu and Lance Edward Miller.
