RESEARCH STATEMENT. My main area of research is in commutative algebra, the branch of mathematics that investigates commutative

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1 RESEARCH STATEMENT BETHANY KUBIK My main area of research is in commutative algebra, the branch of mathematics that investigates commutative rings. In addition to this I am also working on projects in other areas. Specifically, my areas of research are as follows: Homological Algebra (see Sections 1, 2,and 3) Algebraic Number Theory (see Section 4) Graph Theory (see Section 5) Network Science (see Section 6) I describe in Section 1 my research on quasidualizing modules. In Sections 2 and 3 I describe my research on path ideals of weighted graphs and path ideals of cayley graphs. Section 4 is a description of my research with generalized unique factorization domains. In Section 5 I describe my research on crowns. Section 6 discusses my research in network science. 1. Quasidualizing Modules My investigations consider modules over a commutative noetherian local ring R. Much of the literature in this area focuses on R-modules that are finitely generated. Since R is noetherian, these are precisely the noetherian R-modules, in other words the modules that satisfy the ascending chain condition on submodules. In my research I am working to apply techniques used to study noetherian modules to other classes of modules. The main focus of my work has been on artinian modules, those modules that satisfy the descending chain condition on submodules. Other classes of modules that I have studied are Matlis reflexive modules, and mini-max modules. Note that in general these classes are distinct. For the remainder of this note let N and N be noetherian R-modules, and let A and A be artinian R-modules. Given R-modules M and M, I am interested in understanding how properties of M and M translate to properties of their homomorphism set Hom R (M, M ) and to their tensor product M R M. For instance, it is well known that the modules Hom R (N, N ) and N R N are noetherian, and that the modules Hom R (N, A) and N R A are artinian. It is natural to ask about the properties of Hom R (A, ) and A R. In joint work with Micah Leamer and Sean Sather-Wagstaff we prove the following:

2 Theorem 1. Each of the modules Hom R (A, N) and A R A is artinian and noetherian over R, and the module Hom R (A, A ) is noetherian over the completion R. This result is surprising because, for instance, it was not clear that Hom R (A, N) and A R A would satisfy either the artinian or noetherian property, let alone both properties. An application of these ideas is contained in the next result which provides new conditions under which certain evaluation homomorphisms are isomorphisms. This was the original motivation behind the work above. Theorem 2. Let I be an injective R-module. Then the Hom-evaluation map given by θ ANI (a ψ)(φ) = ψ(φ(a)) is an isomorphism. A R Hom R (N, I) θ ANI Hom R (Hom R (A, N), I) The reason for looking at this evaluation homomorphism stems from my investigation into a particular class of dualities. The launching point for this investigation is the concept of a semidualizing module: an R-module C is semidualizing if it is noetherian and satisfies the conditions Hom R (C, C) = R and Ext i R(C, C) = 0 for each integer i > 0. For example, the module C = R is semidualizing. My research takes the tools and techniques used to understand semidualizing modules and applies them to a different class of modules, the quasidualizing modules: an R-module T is quasidualizing if it is artinian and satisfies the conditions Hom R (T, T ) = R and Ext i R(T, T ) = 0 for each integer i > 0; again R is the completion of R. For example, the injective hull E of the residue field of R is quasidualizing. Given the similarity in the definitions it is natural to ask when a quasidualizing module is semidualizing and vice versa. In general these two types of R-modules are very different. I would like to understand the relations between these two types. Again, we take our cues from the example E, and consider Matlis duality which is defined to be ( ) = Hom R (, E). It is well known that, when R is complete, this operator turns noetherian modules into artinian modules, and vice versa. This lays the foundation for my next theorem: Theorem 3. Assume that R is complete. Then the class of quasidualizing modules is in bijection with the class of semidualizing modules under Matlis duality and T is quasidualizing if and only if T is semidualizing. 2. Path Ideals of Weighted Graphs This research is in collaboration with Sean Sather-Wagstaff. In this project we use the interactions between algebra and graph theory. Let G be a finite simple graph with vertex set V = {v 1,..., v d } and let A be a non-zero commutative ring. Set S = A[X 1,..., X d ] and fix an integrer r N. Villarreal [9] [10] developed the notion of the edge ideal associated to a graph G that is generated by the edges of G. Edge ideals have been studied extensively; e.g. see [2] [4]. Villareal s notion was extended by

3 Paulsen and Sather-Wagstaff in [6] to the edge ideal of a weighted graph G ω, that is a graph G with weight function ω : E N. Instead of using edges to generate the ideal, our work uses paths of length r. For a given weighted graph G ω, we define I r (G ω ) = Xei1 i 1 X ei r+1 i r+1 to be the weighted r-path ideal of G ω. v i1... v ir+1 is a path in G with e i1 = ω(v i1 v i2 ), e ij = max{ω(v ij 1 v ij ), ω(v ij v ij+1 )} for 1 < j r and e ir+1 = ω(v ir v ir+1) S We investigate the properties of I r (G ω ). In particular we attain the following decomposition result. Theorem 4. Given a weighted graph G ω one has I r (G ω ) = P (W,σ) = (W,σ) (W, σ) min P (W,σ) where the first intersection is taken over all weighted r-path vertex covers of G ω, and the second intersection is taken over all minimal weighted r-path vertex covers of G ω. Moreover, the second intersection is irredundant. For the complete graph and the case where r = 2, we ascertain when the ideal I 2 (K n ω) is Cohen-Macaulay. Theorem 5. Let K n ω be a complete graph. Then I 2 (K n ω) is Cohen-Macaulay if and only if every sub-3-clique K 3 ω is of the form v i a v j b a v k where a b. Next we plan to investigate the conditions under which we can expect path ideals I r (K n ω) to be Cohen- Macaulay where r 3 3. Path Ideals of Cayley Graphs My work with path ideals of Cayley graphs is in collaboration with Branden Stone. Let G be a group and S a generating set. The Cayley graph of G with respect to S, denoted C(G; S), is the graph where each element g G is a vertex and the edge set is described by (g, gs) where g G and s S. In the next two theorems, we let o(g) denote the order of g G. If G is a graph with edge set E then the edge ideal of G is the ideal generated by the edges of G I(G) = (x i x j {v i, v j } E(G)).

4 Given a Cayley graph, we specify the dimension of k[c(g; S)]/I[C(G; S)] in the following theorem. Theorem 6. Let G be a finite group with G =< g 1,..., g r > where o(g 1 ) = n and G = N. Let C(G; S) be the Cayley graph of G. Then we have (1/2)N if 2 n dim(k[c(g; S)]/I[C(G; S)]) )N if 2 (n + 1) with equality holding if o(g i ) 2 for all 2 i r. ( n 1 2n The next theorem extends the above results. In particular, given a graph H, we only need a subgraph isomorphic to a Cayley graph to identity the dimension of k[h]/i[h]. Theorem 7. Let G be a finite group with G =< g 1,..., g r > where o(g 1 ) = n and G = N. Let C(G; S) be the Cayley graph of G. If a graph H on N vertices has a subgraph isomorphic to C(G; S), then for any i we have N 2 if 2 n dim(k[h]/i(h)) Nn N 2n if 2 (n + 1) where n = g i. Given a graph G, a subdivision of G occurs when we add a vertex onto a currently existing edge of the graph. We are investigating further the dimension of k[h]/i[h] where H is obtained from a Cayley graph through a series of subdivisions. 4. Generalized Unique Factorization Domains This work is in collaboration with Jim Coykendall and Richard Hasenauer. Let R be an integral domain. We say that R is a Generalized Unique Factorization Domain, or GUFD, if every element that factors into irreducible elements does so uniquely. Every Unique Factorization Domain(UFD) is clearly a GUFD. Also, any antimatter domain, that is, a domain where there are no irreducibles, is vacuously a GUFD (but certainly not a UFD). Let Irr(R)(R) denote the set of all irreducible elements in R. Then we have the following theorem. Theorem 8. Let R be a domain. Then the following are equivalent: (1) R is a GUFD. (2) If a 1,..., a r Irr(R) are distinct and a ej j t (in R) for all j where e j t Irr(R)(R). N, then a e1 1 aer r t (in R) or (3) For all a, x Irr(R), if x (at) (in R), then x = ua where u U(R), x t (in R), or t A(R).

5 We explore the relationship with polynomial rings next. An AP domain is a domain where every irrecuible element (or atom) in the ring is also a prime element. AP stands for atoms are prime. We have the following result. Theorem 9. Let R be a domain. The following conditions are equivalent. (1) R[x] is GUFD. (2) R[x] is AP domain. (3) R is AP domain. 5. Generalized Crowns This research is in collaboration with Rebecca Garcia, Pamela Harris, and Shannon Talbott. Our research focuses on a special family of posets called generalized crowns. This definition was originally introduced by Trotter; see [7] [8]. Let n, k N with n 3 and k 0. Then the generalized crown, denoted S k n is bipartite graph where the bottom and top layers each consist of n + k vertices. The edges are described as follows: Fix a vertex b i on the top layer of the graph. Then b i has edges with vertices a i+k+1, a i+k+2,..., a i 1 on the bottom layer where the subscripts are written modulo n + k. Thus each vertex on top misses" (or has no edges with) k + 1 vertices on the bottom layer and hits" (or has an edge) with n 1 vertices on the bottom layer. The following is an example of a generalized crown, in particular S 2 4. b 1 b 2 b 3 b 4 b 5 b 6 a 1 a 2 a 3 a 4 a 5 a 6 Garcia extended this definition by placing multiple copies of a crown on top of the original crown to create layered crowns; see [3]. Our research characterizes the adjacency matrix of the skeleton of the strict hypergraph of generalized crowns. In particular, the resulting matrix is symmetric. 6. Network Science In collaboration with Jocelyn Bell and Brian MacDonald, we constructed a network with individual nodes representing faculty members of the West Point Department of Mathematical Sciences to model how a disease spreads. Edges represented shared office space or shared classroom space. We computed the standard centrality measures of eigenvector, betweenness, closeness, and degree [5].

6 With a computer program written in R, we modeled an SIR (susceptible, infected, recovery) situation with the probability of contracting the infection from an infected node at 37% (known as the epidemic threshold). Each run of the program started with a single infected node and ran until no nodes were infected. Over numerous runs, we were able to conclude which faculty members would be likely to infect the most number of people. We found that the best centrality measure to predict a high infection rate was eigenvector closely followed by degree. Jocelyn Bell and I are extending this research by investigating subgroup centrality measures (see [1]) in the case of weighted graphs. The objective is to better predict team effectiveness through these measures. References 1. J. Bell, Subgroup Centrality Measures, preprint. 2. C. A. Francisco, A. Hoefel, and A. Van Tuyl, EdgeIdeals: a package for (hyper)graphs, J. Softw. Algebra Geom. 1 (2009), 1 4. MR (2012m:05243) 3. R.E. Garcia, D. Silva, Order Dimension of Layered Generalized Crowns. Ars Combinatoria, Volume CXIIIA (2014) pp S. Morey and R. H. Villarreal, Edge ideals: algebraic and combinatorial properties, Progress in commutative algebra 1, de Gruyter, Berlin, 2012, pp MR M. Newman, Networks: An Introduction, Oxford University Press, C. Paulsen and S. Sather-Wagstaff, Edge ideals of weighted graphs, J. Algebra Appl. 12 (2013), no. 5, 24 pp. MR W.T. Trotter, Combinatorics and partially ordered sets: Dimension Theory, The Johns Hopkins University Press, Baltimore, MD (1992). 8. W.T. Trotter, Jr. Dimension of the crown S k n, Discrete Mathematics 8 (1974) R. H. Villarreal, Cohen-Macaulay graphs, Manuscripta Math. 66 (1990), no. 3, MR (91b:13031) 10., Monomial algebras, Monographs and Textbooks in Pure and Applied Mathematics, vol. 238, Marcel Dekker Inc., New York, MR (2002c:13001)