Main publications
(my newer preprints can be obtained from this arXiv page. Here is a recent list of publications.)
Free resolutions, syzygies, Koszul algebras
* Linearity defect of edge ideals and Fröberg's theorem, [pdf]
(with Thanh Vu), J. Algebr. Combin. 44 (2016), 165–199.
We classify graphs whose edge ideals have linearity defect 1. They are weakly chordal graphs with induced matching number 2. This is motivated by Fröberg's theorem, which is equivalent to the statement that the edge ideal of a graph has linearity defect zero if and only if the graph in question is weakly chordal with induced matching number 1.
The accompanying Macaulay2 package LinearityDefect (Last updated on the 7th of Jan. 2021)
* Notes on the linearity defect and applications, [pdf]
Illinois J. Math. 59 (2015), 637–662.
A main result is the following change-of-rings result for the linearity defect. If R-->S is a surjection of noetherian local rings such that ld_R S=0, then for any finitely generated S-module N, there is an equality ld_R N=ld_S N. As an application, if R is a standard graded k-algebra over which any module has a finite linearity defect (such a ring is called absolutely Koszul), and x is a regular linear form, then R/(x) is also absolutely Koszul.
* Regularity bounds for complexes and their homology, [pdf]
Math. Proc. Cambridge Phil. Soc. 159 (2015), 355–377.
A study of the Castelnuovo-Mumford regularity using derived category methods.
* Absolutely Koszul algebras and the Backelin-Roos property, [pdf]
(with Aldo Conca, Srikanth B. Iyengar and Tim Römer), Acta. Math. Vietnam. 40 (2015), 353–374.
We study several related questions on an analogue of regular local rings, the so-called absolutely Koszul rings. Regular local rings, after Auslander-Buchsbaum-Serre, are local rings over which the minimal free resolution of each f.g. module terminates after finitely many steps. Absolutely Koszul rings, roughly speaking, are local rings over which the minimal free resolution of each f.g. module becomes "linear" after finitely many steps. Formally, this means that every f.g. module has finite linearity defect, a notion originating in work of Eisenbud-Fløystad-Schreyer and defined by Herzog and Iyengar. Having finite linearity defect implies a finiteness property of the free resolution of the module in question: its Poincaré series is rational.
In our paper, the following questions are consider: Are Veronese subrings of polynomial rings over a field absolutely Koszul? We have been able to confirm this for polynomial rings up to dimension 3, and some more cases. The idea is showing that such Veronese rings have the Backelin-Roos property, namely being a Golod map away from a quadratic complete intersection.
More surprisingly perhaps, there is a numerical obstruction on h-vector of rings having the Backelin-Roos property, which resembles the obstruction on Hilbert function of algebras which are Koszul. Conjecturally, this obstruction rules out most Veronese subrings. The moral is that being absolutely Koszul (or having the Backelin-Roos property) is a stringent condition and it is desirable to "classify" absolutely Koszul rings with minor restrictions, like being Veronese subrings, or being a tensor product.
* Koszul determinantal rings and 2xe matrices of linear forms, [pdf]
(with Phong Dinh Thieu and Thanh Vu), Michigan Math. J. 64 (2015), 349–381.
Classify Koszul rings defined by the 2-minors of 2xe matrices of linear forms (where e>=2 is an integer) using the Kronecker--Weierstrass theory of matrix pencils.
* Regularity over homomorphisms and a Frobenius characterization of Koszul algebras, [pdf]
(with Thanh Vu), J. Algebra 429 (2015), 103–132.
We prove that if k is a field of characteristic p>0 and R is a standard graded k-algebra, then R is a Koszul algebra if and only if the Frobenius pullback of R has finite regularity as an R-module. This is an analogue of similar Frobenius characterizations for regular rings (Kunz), complete intersections (Avramov--Iyengar--Miller), Gorenstein rings (Iyengar--Sather-Wagstaff), Cohen--Macaulay rings (Takahashi--Yoshino).
Ordinary and symbolic powers of ideals
* Symbolic powers of sums of ideals, [pdf]
(with H. Tài Hà, Ngô V. Trung and Trần N. Trung), Math. Z. 294 (2020), no. 3-4, 1499–1520.
We establish the following formula for the symbolic powers of sums of ideals living in different polynomial rings: (I+J)^(s)=I^(s)+I^(s-1)J^(1)+....+I^(1)J^(s-1)+J^(s).
This a yields formula for the Waldschmidt constant of the sum I+J. a Cohen-Macaulayness criterion for the symbolic powers of such a sum.
* Depth fuctions of symbolic powers of homogeneous ideals, [pdf]
(with Ngô V. Trung), Invent. math. 218 (2019), 779–827. Corrigendum in the same issue on pp. 829–831.
One of the questions we would like to address here is: What are the possible depth sequences of the symbolic powers of homogeneous polynomial ideals? We prove that for any asymptotically periodic positive numerical function f(t), there exists a field k, a polynomial ring over k, and a homogeneous ideal I of R such that depth R/I^(t)=f(t) for all t.
"If we reject the question of value and settle for a description (thematic, sociological, formalist) of a work (of a historical period, culture, etc.); if we equate all cultures and all cultural activities (Bach and rock, comic strips and Proust); if the criticism of art (meditation on value) can no longer find room for expression, then the "historical evolution of art" will lose its meaning, will crumble, will turn into a vast and absurd storehouse of works." (Milan Kundera)