Title:  Homogeneous Gorenstein Ideals and Boij Söderberg Decompositions

Abstract:  This talk consists of two parts. Part one revolves around a construction for homogeneous Gorenstein ideals and properties of these ideals. Part two focuses on the behavior of the Boij-Söderberg decomposition of lex ideals. Gorenstein ideals are known for their nice duality properties. For codimension two and three, the structures of Gorenstein ideals have been established by Hilbert-Burch and Buchsbaum-Eisenbud, respectively. However, although some important results have been found about Gorenstein ideals of higher codimension,  there is no structure theorem proven for higher codimension cases. Kustin and Miller showed how to construct a Gorenstein ideals in local Gorenstein rings starting from smaller such ideals. We discuss a modification of their construction in the case of graded rings. In a Noetherian ring, for a given two homogeneous Gorenstein ideals, we construct another homogeneous Gorenstein ideal and so we describe the resulting ideal in terms of the initial homogeneous Gorenstein ideals. Gorenstein liaison theory plays a central role in this construction. For the second part, we talk about Boij-Söderberg theory which is a very recent theory. It arose from two conjectures given by Boij and Söderberg and their proof by Eisenbud and Schreyer.

It establishes a unique decomposition for Betti diagram of graded modules over polynomial rings. We focus on Betti diagrams of lex ideals which are the ideals having the largest Betti numbers among the ideals with the same Hilbert function. We describe Boij-Söderberg decomposition of a lex ideal in terms of Boij-Söderberg decomposition of some related lex ideals.

Title:  The complex of not 2-connected graphs

Abstract:  With every graph property that is monotone, we can associate for every positive integer n an abstract simplicial complex. The vertices of this complex are the edges of the complete graph on n nodes, and the faces are the sets of edges having this graph property. We will present and outline the proof of a result by Babson, Bjorner, Linusson, Shareshian and Welker of the homotopy type of this simplicial complex for not 2-connected graphs.

Title:  On the dimension of a certain measure arising from a quasilinear elliptic Partial Differential Equation

Abstract:  In the first part of this dissertation we will discuss the Hausdorff dimension of a measure, $\mu$, associated to a positive weak solution to a certain quasilinear elliptic PDE in simply connected domain in the plane. Our work generalizes the work of Lewis and coauthors when the PDE is p-Laplace equation and also for $p=2$ the well known theorem of Makarov regarding the Hausdorff dimension of harmonic measure relative to a point in the domain.

In the second part of the talk we will present a recent result in the study of the Hausdorff dimension of $\mu$ in space associated to a positive weak solution to the same PDE in an open subset and vanishing on a portion of the boundary of that open set. We show that this measure is concentrated on a set of $\sigma-$finite  $n-1$ dimensional Hausdorff measure for $p>n$ and the same result holds for $p=n$ with an assumption on the boundary. We also construct an example of a domain in space for which corresponding measure has Hausdorff dimension strictly less than $n-1$ for $p\geq n$.

Title:  A Characterization of Serre Classes of Reflexive Modules Over a Complete Local Noetherian Ring

Abstract:  Serre classes of modules over a ring $R$ are important because they describe relationships between certain classes of modules and sets of ideals of $R$. In this talk we characterize the Serre classes of three different types of modules. First we characterize all Serre classes of noetherian modules over a commutative noetherian ring. By relating noetherian modules to artinian modules via Matlis duality, we characterize the Serre classes of artinian modules. When $R$ is complete local and noetherian, define $E$ as the injective envelope of the residue field of $R$. Then denote $M^\nu=Hom_R(M,E)$ as the dual of $M$. A module $M$ is reflexive if the natural evaluation map from $M$ to $M^{\nu\nu}$ is an isomorphism.  The main result provides a characterization of the Serre classes of reflexive modules over such a ring. This characterization depends on an ability to ``construct'' reflexive modules from noetherian modules and artinian modules. We find that Serre classes of reflexive modules over a complete local noetherian ring are in one-to-one correspondence with pairs of collections of prime ideals which are closed under specialization.

Title:  Homogenization of Parabolic Operators

Abstract:  Operators that oscillate rapidly on a domain present challenges to estimating a solution. In homogenization, we hope to replace such an operator with one that is uniform in the domain.  In this talk, we will explore the problem of homogenizing a periodic parabolic operator with Dirichlet boundary conditions. We find a form for the operator using a formal expansion and then use Tartar's method of oscillating test functions to prove the result.

Title:  Spin Cobordism and Wedge Quasitoric Manifolds

Abstract:  Quasitoric manifolds are smooth 2n-manifolds admitting a "nice" action of the compact n-torus so that the quotient of this action yields a (combinatorially) simple polytope.  They are a generalization of smooth projective toric variaties and much is known about these manifolds in terms of complex cobordism theory.  In fact they were used by Buchstaber and Panov to show that every cobordism complex class contains a (connected) quasitoric manifold.



Far less is known about spin quasitoric manifolds and spin cobordism which requires the calculation of KO-characteristic classes.  We consider a procedure developed to investigate topological data for spin quasitoric manfolds which utilizes a wedge polytope operation on the quotient polytope.  We'll discuss a list of results concerning these "wedge" quasitoric manifolds, including such topics as Bott manifolds, the connected sum, the Todd genus and lastly, specific criteria in terms of combinatorial data allowing for the calculation of KO-characteristic classes of spin quasitoric manifolds.

Title:  A poset view of the major index

Abstract:  We introduce the Major MacMahon map from non-commutative polynomials in the variables a and b to polynomials in q, and show how this map commutes with the pyramid and bipyramid operators. When the Major MacMahon map is applied to the ab-index of a simplicial poset, it yields the q-analogue of n! times the h-polynomial of the poset.

Applying the map to the Boolean algebra gives the distribution of the major index on the symmetric group, a seminal result due to MacMahon.

Similarly, when applied to the cross-polytope we obtain the distribution of one of the major indexes on the signed permutations, due to Reiner.

This is joint work with Margaret Readdy

During WWII, Chaïm Perelman (1912-1984) helped found and lead the Comité de Défense des Juifs (Jewish Defense Committee), which saved the lives of 5,000 Belgian Jews. In 1947, Perelman founded the New Rhetoric Project as a response to the Holocaust and the failure of reason to prevent violence and war. In 1958, Perelman and his colleague Lucie Olbrechts-Tyteca published The New Rhetoric: A Treatise on Argumentation, which established the anchors of a new rhetoric, founded on informal logic and featuring the technique of dissociation. Professor Frank will place the New Rhetoric Project in its historical context, argue it is a Jewish rhetoric and the most important rhetoric of the 20th century, and that dissociation is a brilliant yet undeveloped notion.

The Department of Biology Undergraduate Research Showcase is scheduled for Wednesday April 30^th from 3pm-5pm on the second floor of the Thomas Hunt Morgan Building.

All Biology majors who have participated in BIO 395 research are encouraged to present their research findings.

Students wishing to earn departmental honors in biology must give a public presentation of their research.  Participation in this event would satisfy that requirement.

To have you poster printed, please fill out the form found here and send to Seth Taylor at seth.taylor@uky.edu along with the file you wish to be printed.  You must have your file submitted by noon on Wednesday, April 23^rd to be printed.

The application to present can be found at: …

If you have any questions, please contact Jacqueline Lee at j.lee@uky.edu.

The UK Department of Biology invites you and your family to attend a Reception Honoring Graduating Biology Majors

Saturday, May 10, 2014

2:00pm – 4:00pm

T.H. Morgan Building, University of Kentucky

* A ceremony recognizing graduates who have earned Honors in Biology will begin at 3:00 pm in 116 THM

Please RSVP by May 2 at https://bio.as.uky.edu/biology-graduation-rsvp