# Christopher Phan's math page

## Education

• Fixed-term Assistant Professor, Department of Mathematics and Statistics, Winona State University, Winona, Minnesota, August 2012-May 2018.
• Visiting Assistant Professor, Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania, August 2010-May 2012.
• Temporary Lecturer, Department of Mathematics, University of Glasgow, Glasgow, Scotland, September 2009-August 2010.
• Adjunct Instructor, General Education Department, Cooking & Hospitality Institute of Chicago, Chicago, Illinois, August-September 2009.
• Graduate Teaching Fellow, Department of Mathematics, University of Oregon, Eugene, Oregon, September 2003-June 2009.

# Teaching demos

These are some materials that I created for use in the classroom.

# Ring Theory Research

I was a noncommutative ring theorist whose primary research interest was homological results involving graded algebras. For example, I studied the Koszul and $\mathcal{K}_2$ properties.

## Research articles

• Quotients of Koszul algebras and 2-$d$-determined algebras (with T. Cassidy), Communications in Algebra, 42 (2014), 3742–3752. Preprint available at arXiv:1210.3847 [math.RA]. MR3200055

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.

• The Yoneda algebra of a graded Ore extension, Communications in Algebra, 40 (2012) 834–844. Preprint available at arXiv:1002.2318 [math.RA]. MR2899911

Let $A$ be a connected-graded algebra with trivial module $k$, and let $B$ be a graded Ore extension of $A$. We relate the structure of the Yoneda algebra $\mathrm{E}(A):= \mathrm{Ext}_A(k,k)$ to $\mathrm{E}(B)$. Cassidy and Shelton have shown that when $A$ satisfies their $\mathcal{K}_2$ property, $B$ will also be $\mathcal{K}_2$. We prove the converse of this result.

• Localization algebras and deformations of Koszul algebras (with T. Braden, A. Licata, N. Proudfoot, and B. Webster), Selecta Mathematica, 17 (2011) 533– 572. Preprint available at arXiv:0905.1335 [math.RA]. MR2827176

We show that the center of a flat graded deformation of a standard Koszul algebra $A$ behaves in many ways like the torus-equivariant cohomology ring of an algebraic variety with finite fixed point set. In particular, the center of $A$ acts by characters on the deformed standard modules, providing a “localization map”. We construct a universal graded deformation of $A$ and show that the spectrum of its center is supported on a certain arrangement of hyperplanes which is orthogonal to the arrangement coming from the algebra Koszul dual to $A$. This is an algebraic version of a duality discovered by Goresky and MacPherson between the equivariant cohomology rings of partial flag varieties and Springer fibers; we recover and generalize their result by showing that the center of the universal deformation for the ring governing a block of parabolic category $\mathcal{O}$ for is isomorphic to the equivariant cohomology of a Spaltenstein variety. We also identify the center of the deformed version of the “category $\mathcal{O}$” of a hyperplane arrangement (defined by the authors in a previous paper) with the equivariant cohomology of a hypertoric variety.

• Noncommutative Koszul algebras from combinatorial topology (with T. Cassidy and B. Shelton), Journal für die reine und angewandte Mathematik (Crelle’s Journal), 646 (2010) 45–63. Preprint available at arXiv:0811:3450 [math.RA]. MR2719555

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 the main result of [Serconek and Wilson, J. Algebra 278: 473–493, 2004] and [Piontkovski, J. Alg. Comput. 15, 643–648, 2005].

• The Yoneda algebra of a $\mathcal{K}_2$ algebra need not be another $\mathcal{K}_2$ algebra (with T. Cassidy and B. Shelton), Communications in Algebra, 38 (2010) 46–48. Preprint available at arXiv:0810.4656 [math.RA]. MR2597480

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 $\mathcal{K}_2$.

• Generalized Koszul properties for augmented algebras, Journal of Algebra, 321 (2009) 1522–1537. Preprint available at arXiv:0711.3480 [math.RA]. MR2494406

Under certain conditions, a filtration on an augmented algebra $A$ admits a related filtration on the Yoneda algebra $\mathrm{E}(A) := \mathrm{Ext}_A(K,K)$. We show that there exists a bigraded algebra monomorphism

$$\mathrm{gr}\;\mathrm{E}(A) \hookrightarrow \mathrm{E}_{\mathrm{Gr}}(\mathrm{gr}\;A),$$

where $\mathrm{E}_{\mathrm{Gr}}(\mathrm{gr}\;A)$ is the graded Yoneda algebra of $\mathrm{gr}\;A$. This monomorphism can be applied in the case where A is connected graded to determine that $A$ has the $\mathcal{K}_2$ property recently introduced by Cassidy and Shelton.

## Dissertation

• Koszul and generalized Koszul properties for noncommutative graded algebras, University of Oregon, Department of Mathematics, 2009.

We investigate some homological properties of graded algebras. If $A$ is an $R$-algebra, then $\mathrm{E}(A):=\mathrm{Ext}_A(R,R)$ is an $R$-algebra under the cup product and is called the Yoneda algebra. (In most cases, we assume $R$ is a field.) A well-known and widely-studied condition on $\mathrm{E}(A)$ is the Koszul property. We study a class of deformations of Koszul algebras that arises from the study of equivariant cohomology and algebraic groups and show that under certain circumstances these deformations are Poincaré–Birkhoff–Witt deformations.

Some of our results involve the $\mathcal{K}_2$ property, recently introduced by Cassidy and Shelton, which is a generalization of the Koszul property. While a Koszul algebra must be quadratic, a $\mathcal{K}_2$ algebra may have its ideal of relations generated in different degrees. We study the structure of the Yoneda algebra corresponding to a monomial $\mathcal{K}_2$ algebra and provide an example of a monomial $\mathcal{K}_2$ algebra whose Yoneda algebra is not also $\mathcal{K}_2$. This example illustrates the difficulty of finding a $\mathcal{K}_2$ analogue of the classical theory of Koszul duality.

It is well-known that Poincaré–Birkhoff–Witt algebras are Koszul. We find a $\mathcal{K}_2$ analogue of this theory. If $V$ is a finite-dimensional vector space with an ordered basis, and $A:=\mathbb{T}(V)/I$ is a connected-graded algebra, we can place a filtration $F$ on $A$ as well as $\mathrm{E}(A)$. We show there is a bigraded algebra embedding

$$\Lambda: \mathrm{gr}\;\mathrm{E}(A) \hookrightarrow \mathrm{E}_{\mathrm{Gr}}(\mathrm{gr}\;A).$$

If $I$ has a Gröbner basis meeting certain conditions and $\mathrm{gr}_F\;A$ is $\mathcal{K}_2$, then $\Lambda$ can be used to show that $A$ is also $\mathcal{K}_2$.

This dissertation contains both previously published and co-authored materials.