My background and mathematics



When I was at the University of Illinois, I was able to talk with several professors who shaped my mathematical development. In particular, the advice, encouragement, and instruction of C. Ward Henson, Tom Nevins, Bruce Reznick, Slawomir Solecki, Jeremy Tyson, and Kim Whittlesey played a significant role in my undergraduate life. I also owe a lot to discussions with my friends Andrew Badr and Guy Bresler. At the University of Chicago, I completed my Ph. D. under the supervision of Victor Ginzburg. Sasha Beilinson also helped to shape my mathematical point of view and Benson Farb supplied much useful advice. It was a ball interacting with my fellow graduate students. From a mathematical point of view, I was particularly fortunate to go through graduate school with Mitya Boyarchenko, Tom Church, and Travis Schedler.


Mathematical work

The mathematical setting for my work is Algebraic Geometry (AG). When I say this, I am communicating which body of definitions (objects and notions I can talk about rigorously) and results (statements describing regularities in how these objects behave) I typically use to explore the mathematical universe. The origin of my subject is an effort to understand the sets of points where some given polynomials (in several variables) are zero.

My goal is for my site to be interesting and useful for some very different groups: my professional colleagues, friends, family, and any members of the public who find their way here. It may take a while, but I hope to add explainers for all of the major ideas that come up in my work. To start to see what AG is all about it is worth browsing Wikipedia on the subject.

If you want some orientation for pure math research itself, please have a look at the FAQ below. (Over the next few weeks and months, I will add explainers for the notions below and for my papers.)

Research FAQ

  1. Research in mathematics involves solving mathematical problems that have never been solved before. It is a process which typically involves two stages. In one stage, researchers identify patterns and statements that ought to be true based on working with explicit examples or intuition and heuristics. In the other stage, researchers find new ways to think about mathematical situations. These new ways of thinking might be useful in resolving unsolved problems or isolating interesting new phenomena.

    One thing that sets math research apart from other scientific fields is that math research does not involve any laboratory equipment. (Although, a powerful computer can be a useful tool in some situations.) The only requirement is a good grasp of the rules (logic) and standards (precise language).

  2. There are different motives for doing math research. Some people work for the thrill of solving hitherto unsolved problems. Others relish the opportunity to play around, looking for surprising patterns, rules, new tricks, or examples. Doing math has a lot in common with solving puzzles and can have the same appeal.

    Math can also be breathtakingly beautiful. The beauty of geometric forms is very easy to appreciate. However, the beauty can be more subtle. One of my personal favorites is Ramsey’s Theorem: given positive integer k, there is a number R(k) such that at any party with at least R(k) guests there is either (i) a group of k guests who all know one another or (ii) k guests who are mutually strangers. (It is a fun exercise to try to calculate R(3).) I find this beautiful because it reveals structure where it has no right to exist.

  3. While we do know an awful lot, every line of thinking in math leads to unresolved questions. Imagination is the only limit on the scope of mathematical investigation.

    Two great examples from number theory are Fermat’s Last Theorem and the Goldbach conjecture. We only recently learned that if $n > 2$, there are no positive integer solutions to $x^n + y^n = z^n$. We still do not know if every even integer greater than two is the sum of two prime numbers!

  4. There are so many ways to answer this! At a really high level, we talk to each other (sometimes using chalkboards, but not always), read papers, read blogs, do calculations, or just sit around and think really hard. At the most fundamental level, when we think we try to arrange our thoughts according to some core principles. We make sure that the concepts we are using have totally clear, precise, unambiguous, formal definitions (although we often ignore the most formal stuff unless we really need it). We use deductive reasoning to work out the implications of the things that we know in order to know new things.

  5. This is a hard question for any pure mathematician. Honestly, I do not know of any direct practical (technological) applications of my own research. This is OK: technology and fundamental research have a complicated relationship. One principle that seems to hold is that nobody knows in advance what knowledge will be required to reach a particular technological goal. Once enough knowledge accumulates, new technologies suddenly become possible.

    While I hope that one day some of my ideas (probably mixed with many others) make possible some useful piece of technology, I’m not holding my breath.

Publications and preprints

  1. The characteristic polynomial of an alegbra and representations. (Linear Alg. App., 2017)
    Coauthor(s): - Rajesh Kulkarni - Yusuf Mustopa -
    [PDF] [DOI] [arXiv]
    In this short note, we give a new sufficient condition for a linear map from a product of copies of a field to endomorphisms of a finite dimensional vector space over the same field to be an algebra homomorphism. We expect that this result can be applied to study representations of higher-degree Clifford algebras and finite extensions of commutative rings.
  2. Irreducible components of varieties of representations II. (Math. Zeitschr., 2017)
    Coauthor(s): - Birge Huisgen-Zimmermann -
    [PDF] [DOI] [arXiv]
    This article is part of a program to evolve the generic representation theory of basic finite dimensional algebras A over an algebraically closed field K, in other words, the goal is to determine the irreducible components of the varieties Rep(A,d) parametrizing the finite dimensional representations with dimension vector d, and to generically describe the representations encoded by the components. Here we primarily target truncated path algebras, i.e., algebras of the form A = KQ/I for a quiver Q, where I is generated by all paths of some fixed length in the path algebra KQ. The main result characterizes the irreducible components of the affine (resp. projective) parametrizing variety Rep(A,d) (resp. GRASS_d(A)) in case Q is acyclic. Our classification is in representation-theoretic terms, permitting to list the components from the quiver and Loewy length of A. Combined with existing theory, it moreover yields an array of generic features of the modules parametrized by the irreducible components, such as generic minimal projective presentations, generic skeleta ("path bases" recruited from a finite set of eligible paths), generic dimensions of endomorphism rings, generic socles, etc. The information on truncated path algebras with acyclic quiver supplements the comparatively well-developed theory available in the special case where A is hereditary, i.e., for I = 0: On one hand, we add to the classical generic results regarding the d-dimensional KQ-modules, they address only the modules of maximal Loewy length. On the other hand, the more general theory for I nonzero developed here fills in generic data on the d-dimensional KQ-modules of any fixed Loewy length.
  3. The McKay correspondence, tilting equivalences, and rationality. (Michigan Math. J., 2017)
    Coauthor(s): - Morgan Brown -
    [PDF] [arXiv]
    We consider the problem of comparing t-structures under the derived McKay correspondence and for tilting equivalences. We relate the t-structures using certain natural torsion theories. As an application, we give a criterion for rationality for surfaces with a tilting bundle. In particular we show that every smooth projective surface which admits a full, strong, exceptional collection of line bundles is rational.
  4. Ulrich sheaves and higher rank Brill-Noether theory. (J. Alg., 2016)
    Coauthor(s): - Rajesh Kulkarni - Yusuf Mustopa -
    [PDF] [DOI] [arXiv]
    An Ulrich sheaf on an embedded projective variety is a normalized arithmetically Cohen-Macaulay sheaf with the maximum possible number of independent sections. Ulrich sheaves are important in the theory of Chow forms, Boij-Soderberg theory, generalized Clifford algebras, and for an approach to Lech's conjecture in commutative algebra. In this note, we give a reduction of the construction of Ulrich sheaves on a projective variety X to the construction of an Ulrich sheaf for a finite map of curves, which is in turn equivalent to a higher-rank Brill-Noether problem for any of a certain class of curves on X. Then we show that existence of an Ulrich sheaf for a finite map of curves implies sharp numerical constraints involving the degree of the map and the ramification divisor.
  5. Autoequivalences of derived categories via geometric invariant theory. (Adv. Math., 2016)
    Coauthor(s): - Daniel Halpern-Leistner -
    [PDF] [DOI] [arXiv]
    We study autoequivalences of the derived category of coherent sheaves of a variety arising from a variation of GIT quotient. We show that these automorphisms are spherical twists, and describe how they result from mutations of semiorthogonal decompositions. Beyond the GIT setting, we show that all spherical twist autoequivalences of a dg-category can be obtained from mutation in this manner. Motivated by a prediction from mirror symmetry, we refine the recent notion of "grade restriction rules" in equivariant derived categories. We produce additional derived autoequivalences of a GIT quotient and propose an interpretation in terms of monodromy of the quantum connection. We generalize this observation by proving a criterion under which a spherical twist autoequivalence factors into a composition of other spherical twists.
  6. Vector bundles whose restriction to a linear section is Ulrich. (Math. Zeitschr., 2015)
    Coauthor(s): - Rajesh Kulkarni - Yusuf Mustopa -
    [PDF] [DOI] [arXiv]
    An Ulrich sheaf on an n-dimensional projective variety X, embedded in a projective space, is a normalized ACM sheaf which has the maximum possible number of global sections. Using a construction based on the representation theory of Roby-Clifford algebras, we prove that every normal ACM variety admits a reflexive sheaf whose restriction to a general 1-dimensional linear section is Ulrich; we call such sheaves delta-Ulrich. In the case n=2, where delta-Ulrich sheaves satisfy the property that their direct image under a general finite linear projection is a semistable instanton bundle, we show that some high Veronese embedding of X admits a delta-Ulrich sheaf with a global section.
  7. Representation schemes and rigid maximal Cohen-Macaulay modules. (Selecta Math., 2015)
    Coauthor(s): - Hai Long Dao -
    [PDF] [DOI] [arXiv]
    Let k be an algebraically closed field and A be a finitely generated, centrally finite, non- negatively graded (not necessarily commutative) k-algebra. In this note we construct a representation scheme for graded maximal Cohen-Macaulay A modules. Our main application asserts that when A is commutative with an isolated singularity, for a fixed multiplicity, there are only finitely many indecomposable rigid (i.e, with no nontrivial self-extensions) MCM modules up to shifting and isomorphism. We appeal to a result by Keller, Murfet, and Van den Bergh to prove a similar result for rings that are completion of graded rings. Finally, we discuss how finiteness results for rigid MCM modules are related to recent work by Iyama and Wemyss on maximal modifying modules over compound Du Val singularities.
  8. A geometric approach to Orlov's theorem. (Comp. Math., 2012)
    [PDF] [DOI] [arXiv]
    A famous theorem of D. Orlov describes the derived bounded category of coherent sheaves on projective hypersurfaces in terms of an algebraic construction called graded matrix factorizations. In this article, I implement a proposal of E. Segal to prove Orlov's theorem in the Calabi-Yau setting using a globalization of the category of graded matrix factorizations (graded D-branes). Let X be a projective hypersurface. Already, Segal has established an equivalence between Orlov's category of graded matrix factorizations and the category of graded D-branes on the canonical bundle K of the ambient projective space. To complete the picture, I give an equivalence between the homotopy category of graded D-branes on K and Dcoh(X). This can be achieved directly and by deforming K to the normal bundle of X, embedded in K and invoking a global version of Knörrer periodicity. We also discuss an equivalence between graded D-branes on a general smooth quasi-projective variety and on the formal neighborhood of the singular locus of the zero fiber of the potential.
  9. Generalized Weyl algebras: category O and graded Morita equivalence. (J. Alg., 2012)
    [PDF] [DOI] [arXiv]
    We study the structural and homological properties of graded Artinian modules over generalized Weyl algebras (GWAs), and this leads to a decomposition result for the category of graded Artinian modules. Then we define and examine a category of graded modules analogous to the BGG category O. We discover a condition on the data defining the GWA that ensures O has a system of projective generators. Under this condition, O has nice representation-theoretic properties. There is also a decomposition result for O. Next, we give a necessary condition for there to be a strongly graded Morita equivalence between two GWAs. We define a new algebra related to GWAs, and use it to produce some strongly graded Morita equivalences. Finally, we give a complete answer to the strongly graded Morita problem for classical GWAs.
  10. On representation schemes and Grassmannians of finite dimensional algebras and a construction of Lusztig. (Math. Research. Lett., 2012)
    [PDF] [DOI] [arXiv]
    Let I be a finite set and CI be the algebra of functions on I. For a finite dimensional C algebra A with C I contained in A we show that certain moduli spaces of finite dimsional modules are isomorphic to certain Grassmannian (quot-type) varieties. There is a special case of interest in representation theory. Lusztig defined two varieties related to a quiver and gave a bijection between their C-points (citation in article). Savage and Tingley raised the question (citation in article) of whether these varieties are isomorphic as algebraic varieties. This question has been open since Lusztig's original work. It follows from the result of this note that the two varieties are indeed isomorphic.
  11. Hyper-minimizing minimized deterministic finite state automata. (Theor. Inform. Appl., 2009)
    Coauthor(s): - Andrew Badr - Villiam Geffert -
    [DOI] [arXiv]
    We present the first (polynomial-time) algorithm for reducing a given deterministic finite state automaton (DFA) into a hyper-minimized DFA, which may have fewer states than the classically minimized DFA. The price we pay is that the language recognized by the new machine can differ from the original on a finite number of inputs. These hyper-minimized automata are optimal, in the sense that every DFA with fewer states must disagree on infinitely many inputs. With small modifications, the construction works also for finite state transducers producing outputs. Within a class of finitely differing languages, the hyper-minimized automaton is not necessarily unique. There may exist several non-isomorphic machines using the minimum number of states, each accepting a separate language finitely-different from the original one. We will show that there are large structural similarities among all these smallest automata.
  12. The distinguishing number of the iterated line graph. (2005)
    We show that for all simple graphs G other than the cycles C_3,C_4,C_5, and the claw K_1,3 there exists a K > 0 such that whenever k > K the k-th iterate of the line graph can be distinguished by at most two colors. Additionally we determine, for trees, when the distinguishing number of the line graph of T is greater than the distinguishing number of T.