My name is Nawaj KC and I am interested in commutative algebra.  Currently I am a postdoc at Utah. 

CV

 nawaj.kc@utah.edu

The little corner of math that I work in has recently seen some extraordinary (!!) progress. These developments are closely connected to my current research, and my immediate research plans will focus on understanding these proposed proofs and their consequences.

The first preprint below is motivated, in part, by the recent counterexample to the dimension inequality conjecture of Peskine--Szpiro. This conjecture had stood for over fifty years! 

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My most recent work has focused on the following two questions.


1) Modules with unbalanced support.  Preliminary draft.

Recently, I have been thinking about what I call unbalanced support : on a space with several components, a module has positive-dimensional support on one component but collapses to a point on another. What homological signals does this leave behind? In particular, what happens when a module with unbalanced support has finite projective dimension?

The New Intersection theorem forces a module with finite projective dimension and an unbalanced support to contain nonzero finite-length torsion. I call the positive-depth quotient obtained by removing this torsion the spine. What I have found is that finite projective dimension requires a delicate balance between the torsion and the spine: the geometry forces torsion to appear, but the homological condition constrains how these two pieces can fit together.

This also leads to a quantitative question: how small can this torsion be? Can its length be bounded below in terms of the multiplicity of the ring? Recent progress by OpenAI on related length conjectures concerning modules of finite projective dimension makes this question especially interesting. Here, however, the torsion is necessarily of infinite projective dimension. The challenge is to find bounds that draw on the underlying support geometry.


2) On an ideal membership problem. (Joint with M. Hochster, J. Jeffries, and A. K. Singh). Preliminary draft.  

I have also been working on a rather interesting ideal membership problem. As a corollary to the Briançon–Skoda theorem, Mel Hochster observed that for any n+1 polynomials in a polynomial ring in n variables, the nth power of their product belongs to the ideal generated by their individual (n+1)st powers.  In joint work with Mel, Jack, and Anurag, we ask: is this bound optimal?  

For general forms in large enough characteristic, we show that a much stronger statement holds: their product already belongs to the ideal generated by their squares. Any examples witnessing optimality must therefore involve special polynomials.

One of our key observations is that the nth power of the polynomials witnessing optimality must admit a syzygy of an unusually low degree. This led us directly to an example. We then construct examples witnessing optimality in every dimension over fields of positive characteristic---the source of the required syzygy is the Frobenius endomorphism! Whether the bound is optimal in characteristic zero in dimensions three and higher remains open. 

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My past work has been on differential characterizations of regularity in ramified mixed characteristic, liftings of modules along surjective local maps, and lower bounds on lengths and Loewy lengths of finite-length modules of finite projective dimension. I have also worked on constructible stratifications by the Betti tables of tangent cones at points of an excellent scheme.

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Publications: 

Preprints: 

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Recorded Talks: 

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"But here you are, Sil. The times make the man, honey, not the other way around."
Gabriella Dante, Mob wife in New Jersey.