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Every scientific endeavor relies on the ability to solve equations. It should come as no surprise then that one of the most fundamental objectives in all of mathematics is the study of finding simultaneous solutions to a given system of $m$ equations $f_1=0,\dots,f_m=0$ in $n$ variables, $x_1,\dots,x_n$. When these equations are linear, we learn as undergraduates in any elementary linear algebra course how to solve such a system, but when these equations are not linear, finding all simultaneous solutions becomes extremely difficult in general, even when the equations are polynomial. The effort to classify solutions to systems of nonlinear polynomial equations is the essence of modern algebraic geometry, a subject of mathematics which manifests much of mankind's cumulative knowledge in pure mathematics. Commutative ring theory is an essential ingredient of algebraic geometry. My field of study is in commutative ring theory. David Hilbert, widely acknowledged as one of the greatest mathematicians who ever lived, noticed in 1890 that an important aspect of understanding solutions to systems of polynomial equations was to study their algebraic dependence on one another. This leads to what has come to be known in commutative ring theory as a free resolution of the ideal $I=(f_1,\dots,f_m)$ over the polynomial ring $R=k[x_1,\dots,x_n]$ . Sometimes one is only interested in algebraic dependence modulo some other equations, and this amounts to studying the free resolution of $I$ over a quotient $R/J$ of the polynomial ring $R$ by some other ideal $J$. The study of free resolutions constitutes a huge portion of current research in algebraic geometry and commutative ring theory. My work focuses mainly on understanding free resolutions, and is clustered around the areas listed below. We assume that $R$ is a polynomial ring or a quotient, and a localization of such or graded, although the statements below hold for rings more general than this. {\bf Lifting Resolutions.} This classical problem concerns lifting a free resolution over a given fixed quotient ring $R/J$ to a free resolution over the ambient ring $R$. It is fundamental yet very difficult, and advances in the lifting problem have a wide variety of important applications. Most of my work in this area has centered around constructing resolutions which are unliftable. For example, I have shown that if $J$ is a principal ideal generated by an element of degree at least two, and $R$ has depth at least five, then there always exist finite free resolutions over $R/J$ which do not lift to $R$. One outcome of this is that some classical theorems on lifting are now known to be best possible. One would like to systematically describe the class of resolutions which are known to be unliftable, with the goal of classifying, perhaps in terms of classical invariants, resolutions which do lift. It is easy and well-known that resolutions of complete intersection ideals, in other words Koszul complexes, are liftable. A natural next step is to study resolutions of Gorenstein ideals. These resolutions, like the Koszul ones, have many nice properties; in particular, they are symmetric. It is known that resolutions of Gorenstein ideals of length three are always liftable. It would appear plausible that all such resolutions of finite length are liftable. However, it has been known to deformation theorists for some time that there exist resolutions of Gorenstein ideals of length four over a double point which fail to lift. I have recently discovered several other examples over quadric cones. One would like to better understand the connection with deformation theory. These examples suggest that our understanding of Gorenstein singularities is rudimentary at best. If $R/J$ is a complete intersection ring (meaning $J$ is a complete intersection ideal), then it is also a Gorenstein ring ($J$ is a Gorenstein ideal), but not every Gorenstein ring is a complete intersection. Reading on, one will come to see that a major theme of my work is that the gap between $R/J$ being a complete intersection and $R/J$ being Gorenstein is really quite enormous. Maybe a more fruitful direction to go in for this topic is to give some relative quantitative information on the abundance of unliftable modules of finite projective dimension over $R/J$. For example, some precise statement along the lines of ``almost all modules of finite projective dimension over $R/J$ are unliftable,'' or maybe the opposite is true. {\bf Fitting Ideals.} Free resolutions are sequentially defined and often infinite. This topic deals with the problem of determining properties of the entire resolution, in particular whether or not it is finite, from looking at the first step. In theory, all of the information about a given resolution is present in the first step; the challenge lies in knowing how to extract it. My collaborators, C. Huneke and D. Katz, and I have had some success with this by investigating properties of an associated Fitting ideal. For instance, we have shown that if $J$ is generated by a regular sequence and $M$ is an $R/J$-module with a rank, then $M$ has finite projective dimension over $R/J$ if and only if the Fitting ideal of any higher syzygy module over $R$ is grade unmixed. We have another interesting theorem in codimension two which determines the support variety of an $R/J$-module by analyzing characteristics of the generators of the Fitting ideal. In higher codimension things are naturally more difficult, but it seems analyzing generators deeper in the fiber ring of the Fitting ideal may lead to some significant higher-dimensional results. \medskip Another means of studying resolutions is through the derived functors, Ext and Tor, which are again defined sequentially, and the sequences are often infinite. These functors act somewhat like a prism: they separate, or isolate, various aspects of a free resolution. They return coarser information about the resolution, but since they may be computed in various ways, they are more tractable and often easier to understand. Generally, one is interested in whether or not the derived functors Ext and Tor are zero, a condition which we refer to as vanishing. Most of my contributions thus far have been in the study of vanishing Ext and Tor. {\bf Vanishing Cohomology.} In my earlier work I improved rigidity theorems in the case where $R$ is a complete intersection by determining the smallest number of consecutive vanishing $\Tor_i^R(M,N)$ (or $\Ext_R^i(M,N)$) required to force the vanishing of all successive $\Tor_i^R(M,N)$ (or $\Ext_R^i(M,N)$). I initiated the connection between vanishing $\Tor^R(M,N)$ and the support varieties of the modules $M$ and $N$, the complete picture being finished by Avramov and Buchweitz. I have also studied, together with C. Huneke and R. Wiegand, the connection between the vanishing of $\Tor^R(M,N)$ and depth properties of the tensor product $M\otimes_R N$ over a complete intersection $R$. As the vanishing properties of $\Ext_R(M,N)$ and $\Tor^R(M,N)$ over a complete intersection became quite well-understood, I began studying vanishing more generally over Gorenstein rings. Huneke and I introduced {\it AB rings\/}, which are rings whose modules satisfy the following strong form of a conjecture of Auslander: if $\Ext^i_R(M,N)=0$ for all $i\gg 0$, then they vanish for all $i$ greater than some fixed integer $n$ depending on the ring $R$. We proved that if $R$ is an AB ring then $\Ext^i_R(M,N)=0$ for all $i\gg 0$ if and only if $\Ext^i_R(N,M)=0$ for all $i\gg 0$. It is easy to see that complete intersections are AB rings. Hence this symmetry in vanishing of Ext result recovers part of a theorem of Avramov and Buchweitz for complete intersections, this time without using the machinery of support varieties. Huneke and I also gave examples of AB rings which are not complete intersections, so that our theorem in fact generalizes the result of Avramov and Buchweitz. It was only recently that \c Sega and I gave a Gorenstein counterexample to the conjecture of Auslander mentioned above, thus showing that not all Gorenstein rings are AB rings. Subsequently, \c Sega and I constructed examples of doubly infinite exact sequences of free $R$-modules $\cdots\to F_2\to F_1\to F_0\to F_{-1}\to F_{-2}\to\cdots$ Such that the sequence $\{\rank F_i\}_{i\ge 0}$ grows exponentially, yet the sequence $\{\rank F_i\}_{i\le 0}$ is constant. One application of these examples is that we now know the conditions defining {\it Gorenstein dimension zero\/} are in general independent, meaning that $\Ext^*_R(M,R)=0$ does not necessarily imply $\Ext_R^*(\Hom_R(M,R),R)=0$. We also have such examples when $R$ is Gorenstein, and these show that complete resolutions can have totally asymmetric growth, This is something that cannot happen for {\it complete resolutions\/} over complete intersections. These examples also show that the symmetry in vanishing of Ext property above does not hold for Gorenstein rings in general. I am also interested in the following commutative local version of a conjecture of Auslander and Reiten: if $\Ext^i_R(M,M\oplus R)=0$ for all $i>0$, then $M$ is free. Other plans of mine are to broaden the class of {\it AB rings\/}, and investigate whether this class localizes. {\bf Realizing Cohomology} Another question I have worked on is (one not of vanishing but of) what possible $\Ext_R(M,N)$ can occur when $N=k$, the coefficient field of $R$. L. Avramov and I have developed a fairly complete answer when $R$ is a complete intersection, and recently we have extended the results to rings whose Koszul complex is quasiisomorphic to a Koszul algebra $E$. In this case there is a normal injection of graded $k$-algebras $\mathcal R=\Ext_E(k,k)\hookrightarrow\mathcal E=\Ext_R(k,k)$. We have shown that, if $R$ is as above, then given a graded left module $\mathcal M$ over $\mathcal R$ with a finite free resolution over $\mathcal R$, there exists a finite $R$-module $M$ such that $\Ext_R(M,k)$ is isomorphic as an $\mathcal E$-module (and therefore also as an $\mathcal R$-module) to truncation of $\mathcal E\otimes_{\mathcal R} \mathcal M$. The place of truncation (at least in the complete intersection case) depends on the Castelnuovo-Mumford regularity of $\mathcal M$ over $\mathcal R$. This theorem comes from establishing equivalences between the derived categories of categories of DG modules over certain DG algebras. One equivalence is effectively a DG version of a duality of Bernstein-Gel'fand-Gel'fand type, only we establish the duality for any Koszul dual pair and do not use typical adjoint functors. The full story of the equivalences of categories is not completely understood, however, and the investigation is ongoing. One question is why we have chosen to work with the derived category of DG modules rather than that of graded modules. My graduate student, Kristen Beck, has written a Master's thesis elaborating on our justification for doing so. An interesting sidelight of our theorem, which was in fact the motivating problem, is that we are now able to construct modules having virtually any admissible homological behavior. This is a very exciting development. In particular, we are able to construct modules having any prescribed support variety. There are alternate methods for doing this using Hochschild homology along the lines of Carlson. Another question on this topic for the future is to compare the different constructions.