18.^{ce} 
 Differential operators on classical invariant rings do not lift modulo p,
with Anurag K. Singh,
submitted, 38 pp.
Abstract:
Levasseur and Stafford described the rings of differential operators on various classical invariant rings of characteristic zero; in each of the cases they considered, the differential operators form a simple ring. Towards an attack on the simplicity of rings of differential operators on invariant rings of linearly reductive groups over the complex numbers, Smith and Van den Bergh asked if differential operators on the corresponding rings of positive prime characteristic lift to characteristic zero differential operators. We prove that, in general, this is not the case for determinantal hypersurfaces, as well as for Pfaffian and symmetric determinantal hypersurfaces. We also prove that, with few exceptions, these hypersurfaces do not admit a mod p^{2 } lift of the Frobenius endomorphism.

17.^{ce} 
 Extensions of Primes, Flatness, and Intersection Flatness,
with Melvin Hochster,
submitted, 17 pp.
Abstract:
We study when R → S has the property that prime ideals of R extend to prime ideals
or the unit ideal of S, and the situation where this property continues to hold after adjoining the same indeterminates
to both rings. We prove that if R is reduced, every maximal ideal of R contains only finitely many
minimal primes of R, and prime
ideals of R[X_{1} , ... ,X_{n}] extend to prime ideals of
S[X_{1} , ... ,X_{n}] for all n, then S is flat over R. We give a counterexample
to flatness over a reduced quasilocal ring R with infinitely many minimal primes by
constructing a nonflat Rmodule M such that
M = PM for every minimal prime P of R. We study the notion of intersection flatness
and use it to prove that in certain graded cases it suffices
to examine just one closed fiber to prove the stable prime extension property.

16.^{c} 
 Lower Bounds on HilbertKunz Multiplicities and Maximal Fsignatures,
with Yusuke Nakajima, Ilya Smirnov, Keiichi Watanabe, and Kenichi Yoshida,
submitted, 20 pp.
Abstract:
HilbertKunz multiplicity and Fsignature are numerical invariants of commutative rings in
positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to
one and the converse holds under mild assumptions. A natural question is for what singular rings these
invariants are closest to one. For Hilbert–Kunz multiplicity this question was first considered by the last two
authors and attracted significant attention. In this paper, we study this question, i.e., an upper bound, for
Fsignature and revisit lower bounds on HilbertKunz multiplicity.

15.^{be} 
 Faithfulness of Top Local Cohomology Modules in Domains,
with Melvin Hochster,
to appear in Mathematical Research Letters, 7 pp.
Abstract:
We study the conditions under which the highest nonvanishing local cohomology module of a domain R with support in an ideal I is faithful over R, i.e., which guarantee that H^c_I(R) is faithful, where c is the cohomological dimension
of I. In particular, we prove that this is true for the case of positive prime characteristic
when c is the number of generators of I.

14.^{ce} 
 BernsteinSato Functional Equations, Vfiltrations, and Multiplier Ideals of Direct Summands,
with Josep Àlvarez Montaner, Daniel J. Hernández, Luis NúñezBetancourt, Pedro Teixeira, and Emily E. Witt,
submitted. 40 pp.
Abstract:
This paper investigates the existence and properties of a BernsteinSato functional equation in nonregular settings. In particular, we construct Dmodules in which such formal equations can be studied. The existence of the BernsteinSato polynomial for a direct summand of a polynomial over a field is proved in this context. It is observed that this polynomial can have zero as a root, or even positive roots. Moreover, a theory of Vfiltrations is introduced for nonregular rings, and the existence of these objects is established for what we call differentially extensible summands. This family of rings includes toric, determinantal, and other invariant rings. This new theory is applied to the study of multiplier ideals of singular varieties. Finally, we extend known relations among the objects of interest in the smooth case to the setting of singular direct summands of polynomial rings.

13.^{be} 
 A Transformation Rule for Natural Multiplicities,
with Ilya Smirnov,
to appear in International Mathematics Research Notices. IMRN, 12 pp.
Abstract:
For multiplicities arising from a family of ideals we provide a general approach to transformation rules for a ring extension étale in codimension one. Our result can be applied to bound the size of the local étale fundamental group of a singularity in terms of Fsignature, recovering a recent result of CarvajalRojas, Schwede, and Tucker, and differential signature, providing the first characteristicfree effective bound.

12.^{ae} 
 Quantifying Singularities With Differential Operators,
with Holger Brenner and Luis NúñezBetancourt,
Advances in Mathematics. 358 (2019), 106843, 89 pp.
Abstract:
The Fsignature of a local ring of prime characteristic is a numerical invariant that detects many interesting properties. For example, this invariant detects (non)singularity and strong Fregularity. However, it is very difficult to compute. Motivated by different aspects of the Fsignature, we define a numerical invariant for rings of characteristic zero or p>0 that exhibits many of the useful properties of the Fsignature. We also compute many examples of this invariant, including cases where the Fsignature is not known. We also obtain a number of results on symbolic powers and BernsteinSato polynomials.

11.^{ae} 
 Algebraic Signatures of Convex and Nonconvex Codes,
with Carina Curto, Elizabeth Gross, Katherine Morrison, Zvi Rosen, Anne Shiu, and Nora Youngs,
Journal of Pure and Applied Algebra, 223 (2019), 3919–3940.
Abstract:
A convex code is a binary code generated by the pattern of intersections of a collection of open convex sets in some Euclidean space. Convex codes are relevant to neuroscience as they arise from the activity of neurons that have convex receptive fields. In this paper, we use algebraic methods to determine if a code is convex. Specifically, we use the neural ideal of a code, which is a generalization of the StanleyReisner ideal. Using the neural ideal together with its standard generating set, the canonical form, we provide algebraic signatures of certain families of codes that are nonconvex. We connect these signatures to the precise conditions on the arrangement of sets that prevent the codes from being convex. Finally, we also provide algebraic signatures for some families of codes that are convex, including the class of intersectioncomplete codes. These results allow us to detect convexity and nonconvexity in a variety of situations, and point to some interesting open questions.

10.^{be} 
 Derived Functors of Differential Operators,
to appear in International Mathematics Research Notices. IMRN, 12 pp. DOI: 10.1093/imrn/rny284
Abstract:
In their work on differential operators in positive characteristic, Smith and Van den Bergh define and study the derived functors of differential operators; they arise naturally as obstructions to differential operators reducing to positive characteristic. In this note, we provide formulas for the ring of differential operators as well as these derived functors of differential operators. We apply these descriptions to show that differential operators behave well under reduction to positive characteristic under certain hypotheses. We show that these functors also detect a number of interesting properties of singularities.

9.^{ae} 
 A ZariskiNagata Theorem for Smooth ℤAlgebras,
with Alessandro De Stefani and Eloísa Grifo,
Journal für die reine und angewandte Mathematik, 761 (2020), 123–140.
Abstract:In a polynomial ring over a perfect field, the symbolic powers of a prime ideal can be described via differential operators: a classical result by Zariski and Nagata says that the nth symbolic power of a given prime ideal consists of the elements that vanish up to order n on the corresponding variety. However, this description fails in mixed characteristic. In this paper, we use pderivations, a notion due to Buium and Joyal, to define a new kind of differential powers in mixed characteristic, and prove that this new object does coincide with the symbolic powers of prime ideals. This seems to be the first application of pderivations to Commutative Algebra.

8.^{ade} 
 Polarization of Neural Ideals,
with Sema Güntürkün and Jeffrey Sun,
Journal of Algebra and Its Applications, 19 (2020), 2050146, 15 pp.
Abstract:
The "neural code" is the way the brain characterizes, stores, and processes information. Unraveling the neural code is a key goal of mathematical neuroscience. Topology, coding theory, and, recently, commutative algebra are some the mathematical areas that are involved in analyzing these codes. Neural rings and ideals are algebraic objects that create a bridge between mathematical neuroscience and commutative algebra. A neural ideal is an ideal in a polynomial ring that encodes the combinatorial firing data of a neural code. Using some algebraic techniques one hopes to understand more about the structure of a neural code via neural rings and ideals. In this paper, we introduce an operation, called "polarization," that allows us to relate neural ideals with squarefree monomial ideals, which are very well studied and known for their nice behavior in commutative algebra.

7.^{ae} 
 Local Okounkov Bodies and Limits in Prime Characteristic,
with Daniel J. Hernández,
Mathematische Annalen, 372 (2018), no. 1, 139–178.
Abstract:
This article is concerned with the asymptotic behavior of certain sequences of ideals in rings of prime characteristic. These sequences, which we call pfamilies of ideals, are ubiquitous in prime characteristic commutative algebra (e.g., they occur naturally in the theories of tight closure, HilbertKunz multiplicity, and Fsignature). We associate to each pfamily of ideals an object in Euclidean space that is analogous to the NewtonOkounkov body of a graded family of ideals, which we call a pbody. Generalizing the methods used to establish volume formulas for the HilbertKunz multiplicity and Fsignature of semigroup rings, we relate the volume of a pbody to a certain asymptotic invariant determined by the corresponding pfamily of ideals. We apply these methods to obtain new existence results for limits in positive characteristic, an analogue of the BrunnMinkowski theorem for HilbertKunz multiplicity, and a uniformity result concerning the positivity of a pfamily.

6.^{be} 
 Mapping Toric Varieties into Low Dimensional Spaces,
with Emilie Dufresne,
to appear in Transactions of the American Mathematical Society, 28 pp. DOI: 10.1090/tran/7026
Abstract:
A smooth ddimensional projective variety X can always be embedded into 2d+1dimensional space. In contrast, a singular variety may require an arbitrary large ambient space. If we relax our requirement and ask only that the map is injective, then any ddimensional projective variety can be mapped injectively to 2d+1dimensional projective space. A natural question then arises: what is the minimal m such that a projective variety can be mapped injectively to mdimensional projective space? In this paper we investigate this question for normal toric varieties, with our most complete results being for SegreVeronese varieties.

5.^{ae} 
 What Makes a Neural Code Convex?,
with Carina Curto, Elizabeth Gross, Katherine Morrison, Mohamed Omar, Zvi Rosen, Anne Shiu, and Nora Youngs,
SIAM Journal of Applied Algebraic Geometry, 1 (2017), no. 1, 222–238.
Abstract:
Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus, where neurons exhibit convex receptive fields. What makes a neural code convex? That is, how can we tell from the intrinsic structure of a code if there exists a corresponding arrangement of convex open sets? In this work, we provide a complete characterization of local obstructions to convexity. This motivates us to define max intersectioncomplete codes, a family guaranteed to have no local obstructions. We then show how our characterization enables one to use free resolutions of StanleyReisner ideals in order to detect violations of convexity. Taken together, these results provide a significant advance in understanding the intrinsic combinatorial properties of convex codes.

4.^{a} 

Separating Invariants and Local Cohomology,
with Emilie Dufresne,
Advances in Mathematics, 270 (2015) 565–581.
Abstract:
The study of separating invariants is a recent trend in invariant theory. For a finite group acting linearly on a vector space, a separating set is a set of invariants whose elements separate the orbits of G. In some ways, separating sets often exhibit better behavior than generating sets for the ring of invariants. We investigate the least possible cardinality of a separating set for a given Gaction. Our main result is a lower bound that generalizes the classical result of Serre that if the ring of invariants is polynomial then the group action must be generated by pseudoreflections. We find these bounds to be sharp in a wide range of examples.

3.^{a} 

Multiplicities of Classical Varieties,
with Jonathan Montaño and Matteo Varbaro,
Proceedings of the London Mathematical Society, 110 (2015), no. 4, 1033–1055.
Abstract:
The jmultiplicity plays an important role in the intersection theory of St&\uuml;ckradVogel cycles, while recent developments confirm the connections between the εmultiplicity and equisingularity theory. In this paper we establish, under some constraints, a relationship between the jmultiplicity of an ideal and the degree of its fiber cone. As a consequence, we are able to compute the jmultiplicity of all the ideals defining rational normal scrolls. By using the standard monomial theory, we can also compute the j and εmultiplicity of ideals defining determinantal varieties: The found quantities are integrals which, quite surprisingly, are central in random matrix theory.

2.^{a} 

Nonsimplicial Decompositions of Betti Diagrams of Complete Intersections,
with Courtney Gibbons, Sarah
Mayes, Claudiu Raicu, Branden Stone, and Bryan White,
Journal of Commutative Algebra, 7 (2015), no. 2, 189–206.
Abstract:
We investigate decompositions of Betti diagrams over a polynomial ring within the framework of BoijSoederberg theory. That is, given a Betti diagram, we decompose it into pure diagrams. Relaxing the requirement that the degree sequences in such pure diagrams be totally ordered, we are able to define a multiplication law for Betti diagrams that respects the decomposition and allows us to write a simple expression the decomposition of the Betti diagram of any complete intersection in terms of the degrees of its minimal generators. In the more traditional sense, the decomposition of complete intersections of codimension at most 3 are also computed as given by the totally ordered decomposition algorithm obtained from (EisenbudSchreyer, 2009). In higher codimension, obstructions arise that inspire our work on an alternative algorithm.

1.^{a} 

The jmultiplicity of Monomial Ideals,
with Jonathan Montaño,
Mathematical Research Letters, 20 (2013) no. 4, 729–744.
Abstract:
We prove a characterization of the jmultiplicity of a monomial ideal as the normalized volume of a polytopal complex. Our result is an extension of Teissier's volumetheoretic interpretation of the HilbertSamuel multiplicity for mprimary monomial ideals. We also give a description of the epsilonmultiplicity of a monomial ideal in terms of the volume of a region.
