# Tom Cassidy's Research

My mathematical research is in non-commutative ring theory, and in general I am interested in all things non-commutative. An operation like addition is called commutative because $$a+b=b+a$$. In contrast, subtraction is non-commutative because (in most cases) $$a-b$$ is not $$b-a$$. Other familiar non-commutative operations are the composition of functions and matrix multiplication.

In the world of algebraic structures like groups or rings, commutativity is a rarity. However, many properties known to hold for commutative rings can be extended to non-commutative rings if one employs the proper definitions. It is sometimes possible that the assumption of commutativity hides deeper facts that can be understood in the more general non-commutative setting.

Lately I have become interested in mathematical demography. Demography is the study of populations and how they change over time due to births, aging and death. Many of these processes can be modeled mathematically. During the 2009-10 academic year I worked in Germany as a Guest Researcher at the Max Planck Institute for Demographic Research.

For a more technical account of my research, take a look at the publications listed below.

# Publications

1. Global Dimension four Extensions of Artin-Schelter Regular Algebras, in the Journal of Algebra, 220 (1999), 225-254.

Artin-Schelter (AS) regular algebras are of particular interest to those working in non-commutative algebraic geometry. AS Regular algebras of global dimension three were classified in 1987 and 1990 into a quadratic and a cubic family. The AS regular algebras of global dimension four have been less tractable. In 1996, Le Bruyn, Smith and Van den Bergh classified the global dimension four AS regular algebras which are extensions of the quadratic family by a central element of degree one. Their method, however, is not applicable to the cubic family, since it relies on the fact that the quadratic AS regular algebras are Koszul algebras. I classify the global dimension four AS algebras which are extensions of the cubic family by a central element of arbitrary degree. Moreover, I extend my result and the result of Le Bruyn, Smith and Van den Bergh to include extensions by normal elements of arbitrary degree.

2. Central Extensions of Stephenson's Algebras, in Communications in Algebra, 31 (2003), 1615-1632.

In 1996 Stephenson completed the classification of AS regular algebras of dimension three by including algebras generated in degrees other than one. In Central Extensions of Stephenson's Algebras (Communications in Algebra 2003) I classify the global dimension four AS regular which are central extensions of the AS regular algebras studied by Stephenson.

3. Homogenized Down-Up Algebras, in Communications in Algebra, 31 (2003), 1765-1775.

Down-Up algebras where introduced by Benkart and Roby in 1998, and have subsequently been studied by Carvalho, Jordan, Kirkman, Kuzmanovich, Musson, Passman, Zhao and others. This paper answers questions about homogenizations of down-up algebras which were posed by Benkart and Roby in their 1998 paper.

4. Basic Properties of Generalized Down-Up Algebras, (with B. Shelton), in the Journal of Algebra, 279 (2004), 402-421.

This paper generalizes the definition of down-up algebras to include algebras studied by Smith, Rueda, Le Bruyn and others. We prove that this larger family of algebras shares many properties of down-up algebras. We determine the global (homological) dimension for all the algebras in this larger family, and we relate the representation theory of the Noetherian algebras to dynamical systems.

5. Generalized Laurent Polynomial Rings as Quantum Projective 3-Spaces, (with P. Goetz and B. Shelton), in the Journal of Algebra, 303 (2006), 358-372.

We introduce the notion of a generalized Laurent polynomial ring over a ground ring R. This class includes the generalized Weyl algebras. We show that these rings inherit many properties from the R. This construction is then used to create two new families of quadratic global dimension four Artin-Schelter regular algebras. We show that in most cases the second family has a finite point scheme and a defining automorphism of finite order. Nonetheless, a generic algebra in this family is not finite over its center.

6. PBW-deformation theory and regular central extensions, (with B. Shelton), in Crelle's Journal, 610 (2007), 1-12.

A deformation U, of a graded K-algebra A is said to be of PBW type if grU is A. It has been shown for Koszul and N-Koszul algebras that the deformation is PBW if and only if the relations of U satisfy a Jacobi type condition. In particular, for these algebras the determination of the PBW property is a finite and explicitly determined linear algebra problem. We extend these results to an arbitrary graded K-algebra, using the notion of central extensions of algebras and a homological constant attached to A which we call the complexity of A.

7. Generalizing the notion of Koszul Algebra, (with B. Shelton), in Mathematische Zeitschrift, 260 (2008), no. 1, 93-114.

We introduce a generalization of the notion of a Koszul algebra, which includes graded algebras with relations in different degrees, and we establish some of the basic properties of these algebras. This class is closed under twists, twisted tensor products, regular central extensions and Ore extensions. We explore the monomial algebras in this class and we include some well-known examples of algebras that fall into this class.

8. Generalizations of graded Clifford algebras and of complete intersections, (with M. Vancliff), in the Journal of the London Mathmatical Society, 81 (2010) no. 2,
91-112

We introduce graded skew Clifford algebras and associate to such an algebra the notion of a quadric system in the spirit of the non-commutative algebraic geometry of Artin, Tate and Van den Bergh. We prove that a graded skew Clifford algebra is quadratic and regular if and only if its associated quadric system is normalizing and base-point free. In the process, we extend the notions of complete intersection, base-point free, quadratic form and symmetric matrix to the non-commutative setting. We then use our results to produce several families of quadratic regular skew Clifford algebras of global dimension four.

9. The Yoneda algebra of a K2 algebra need not be another $$\mathcal K_2$$ algebra, (with C. Phan and B. Shelton), in Communications in Algebra, vol. 38 (2010), 46-48.

The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. $$\mathcal K_2$$ algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a $$\mathcal K_2$$ algebra would be another $$\mathcal K_2$$ algebra. We show that this is not necessarily the case by constructing a monomial $$\mathcal K_2$$ algebra for which the corresponding Yoneda algebra is not $$\mathcal K_2$$.

10. Noncommutative Koszul algebras from combinatorial topology, (with C. Phan and B. Shelton), in Crelle's Journal, 646 (2010), 45–63.

Associated to any uniform finite layered graph Gamma there is a noncommutative graded quadratic algebra A(Gamma) given by a construction due to Gelfand, Retakh, Serconek and Wilson. It is natural to ask when these algebras are Koszul. Unfortunately, a mistake in the literature states that all such algebras are Koszul. That is not the case and the theorem was recently retracted. We analyze the Koszul property of these algebras for two large classes of graphs associated to finite regular CW complexes, X. Our methods are primarily topological. We solve the Koszul problem by introducing new cohomology groups $$H_X(n,k)$$, generalizing the usual cohomology groups $$H^n(X)$$. Along with several other results, our methods give a new and primarily topological proof of a result of Serconek and Wilson and of Piontkovski.

11. Quadratic algebras with Ext algebras generated in two degrees, in the Journal of Pure and Applied Algebra, 214 (2010), 1011-1016.

Green and Marcos call a graded k-algebra delta-Koszul if the corresponding Yoneda algebra $$Ext (k,k)$$ is finitely generated and $$Ext^{i,j} (k,k)$$ is zero unless $$j=\delta(i)$$ for some function $$\delta: \mathbb N \to \mathbb N$$. For any integer $$m\ge 2$$ we exhibit a non-commutative quadratic delta-Koszul algebra for which the Yoneda algebra is generated in degrees (1,1) and (m,m+1). These examples answer a question of Green and Marcos. These algebras are not Koszul but are m-Koszul (in the sense of Backelin).

12. How slowing senescence translates into longer life expectancy, (with J. Goldstein) in Population Studies, 66 (2012), 23-39.

Mortality decline has historically been largely a result of reductions in the level of mortality at all ages. A number of leading researchers on aging, however, suggest that the next revolution of longevity increase will be the result of slowing down the rate of aging. In this paper, we show mathematically how varying the pace of senescence influences life expectancy. We provide a formula that holds for any baseline hazard function. Our result is analogous to Keyfitz's entropy'' relationship for changing the level of mortality. Interestingly, the influence of the shape of the baseline schedule on the effect of senescence changes is the complement of that found for level changes. We also provide a generalized formulation that mixes level and slope effects. We illustrate the applicability of these models using recent mortality decline in Japan and the problem of period to cohort translation.

13. Quotients of Koszul algebras and 2-d-determined algebras, (with C. Phan) in Communications in Algebra, 42 Issue 9 (2014), 3742-3752.

Vatne and Green & Marcos have independently studied the Koszul-like homological properties of graded algebras that have defining relations in degree 2 and exactly one other degree. We contrast these two approaches, answer two questions posed by Green & Marcos, and find conditions that imply the corresponding Yoneda algebras are generated in the lowest possible degrees.

14. A cohort model of fertility postponement, (with J. Goldstein) in Demography, 51 No. 5 (2014), 1797-1819.

We introduce a new formal model in which demographic behavior, such as fertility, is postponed by differing amounts depending only on cohort membership. The cohort-based model shows the effects of cohort shifts on period fertility measures, and provides an accompanying tempo adjustment to determine the period fertility that would have occurred without postponement. Cohort-based postponement spans multiple periods and produces `fertility momentum' with implications for future fertility rates. We illustrate several methods for model estimation, and apply the model to fertility rates in several countries. We also compare the fit of period-based and cohort-based shift models to the recent Dutch fertility surface, and show how both cohort- and period-based postponement can occur simultaneously.

15. Corrigendum: Generalizations of graded Clifford algebras and of complete intersections, (with M. Vancliff) in the Journal of the London Mathematical Society (2), 90 No. 2 (2014), 631-636.

We provide a correction for Proposition 3.5 in article 10 above. The correction is: if $$S$$ is a skew polynomial ring on finitely many generators of degree 1 that are normal elements in $$S$$, and if $$I$$ is a homogeneous ideal of $$S$$ that is generated by a normalizing sequence, then $$dim_k(S/I)$$ is finite if and only if $$S/I$$ has no point modules and no fat point modules. A similar correction is provided for Corollary 3.6 of the same article.

16. Modules with pure resolutions over Koszul algebras revisited, to appear in the Journal of Algebra and its Applications, 15 No. 2 (2016).

I construct a Koszul algebra $$A$$ and a finitely generated graded $$A$$-module $$M$$ that together form a counterexample to a recently published claim. $$M$$ is generated in degree $$0$$ and has a pure resolution, and the graded Jacobson radical of the Yoneda algebra of $$A$$ does not annihilate the Ext module of $$M$$, but nonetheless $$M$$ is not a Koszul module.

17. Amplified changes: an analysis of four dynamic fertility models, (with J. Goldstein) in Dynamic Demographic Analysis, ed. R. Schoen, Springer (2016).

Fertility change over time can be modeled in a variety of ways. Implicit in each model is a story of the behavior driving fertility, and the assumptions behind each model provide insights into the forces that influence fertility. We present Ryder's classic formulation of the translation between period and cohort measures of fertility, Lee's moving-target model connecting fertility goals with period rates, the period-shift model of Bongaarts and Feeney, and the Goldstein and Cassidy cohort-shift model. All of these models have in common a simplified view of how fertility change occurs. An important lesson of all these formulations is that small variations in timing or targets can produce large fluctuations in period fertility, telling us that period fertility is particularly sensitive to changes in underlying aspects of the fertility process.

18. Periodic free resolutions from twisted matrix factorizations, (with A. Conner, E. Kirkman, amd W. F. Moore) in the Journal of Algebra, 455 (2016), 137-163.

The notion of a matrix factorization was introduced by Eisenbud in the commutative case in his study of bounded (periodic) free resolutions over complete intersections. Since then, matrix factorizations have appeared in a number of applications. In this work, we extend the notion of (homogeneous) matrix factorizations to regular normal elements of connected graded algebras over a field. Next, we relate the category of twisted matrix factorizations to an element over a ring and certain Zhang twists. We also show that in the setting of a quotient of a ring of finite global dimension by a normal regular element, every sufficiently high syzygy module is the cokernel of some twisted matrix factorization. Furthermore, we show that in the noetherian AS-regular setting, there is an equivalence of categories between the homotopy category of twisted matrix factorizations and the singularity category of the hypersurface, following work of Orlov.