My research interests are in the overlap of algebraic geometry, singularity theory and commutative algebra. The main focus of my work is to study how algebraic varieties and morphisms change as they vary in families, with applications to the geometry and topology of complex analytic singularities.

I introduced the local volume of a relatively very ample invertible sheaf as a fiberwise invariant to detect changes in the geometry and topology of the fibers of families of singularities. One of my main results is the Local Volume Formula (LVF) which determines the change of the local volume as one moves from a fiber to a nearby fiber. As an application of the LVF, I obtained numerical control for Whitney--Thom (differential) equisingularity for families of complex analytic sets and functions with isolated singularities.

Another application of the LVF gives rise to the notion of generalized smoothability. I introduced the class of singularities that admit deformations to deficient conormal (dc) singularities. This class contains all smoothable singularities, Cohen-Macaylay codimension 2 singularities, Gorenstein codimension 3 singularities, and determinantal singularities. Currently I am working on determining how large this class is and trying to generalize Milnor's point of view to studying the topology of isolated singularities that are not necessarily smoothable but that admit deformations to dc singularities.

Another ongoing research project is to understand the relation between equiresolution and strong forms of topological equisingularity. In a collaboration with Mourtada and Teissier I am developing an approach to prove that Zariski equisingularity implies Lipschitz equisingularity for families of hypersurfaces. 

My other research interests are in valuation theory and its connections to singularities. I proved a valuation theorem for Noetherian rings which states that under mild hypothesis the relative integral closure of a Noetherian domain is determined by finitely many discrete valuation rings. 

Papers and preprints:

Rigidity in dimension 2 (in preparation).

A Plücker formula for curves (with B. Teissier) (in preparation). 

The Milnor fiber of a smoothable curve (with Pablo Portilla) (in preparation). 

The complete intersection discrepancy of a curve I: Numerical Invariants. With an Appendix by Marc Chardin.

 (with A.Bengus-Lasnier). arXiv

A multiplicity formula for the Milnor number of smoothable curves (with A. Bengus-Lasnier and T. Gaffney). arXiv.

A valuation theorem for Noetherian rings, Michigan Math. Journal, vol. 73, issue 4, 2023, 843-851. arXiv

Local volumes, equisingularity and generalized smoothability (submitted). arXiv

The impact and progression of the COVID-19 pandemic in Bulgaria in its first two years, Vaccines  2022, 10 (11): 1901. In the special issue "SARS-CoV-2 Variants Research and Ending the COVID-19 Pandemic". (joint with G. Marinov and M. Mladenov).

SARS-CoV-2 reinfections during the first three major COVID-19 waves in Bulgaria  PLoS ONE 17(9): e0274509. (joint with G. Marinov, M. Mladenov, I. Alexiev).

The evolving notion of multiplicity as an invariant in singularity theory, Invited Address at the 51st Spring Conference of the Union of Bulgarian Mathematicians, April 5-9, 2022, 101--111.

Algebraic theory of continuity for meromorphic functions (submitted).

The demographic and geographic impact of the COVID pandemic in Bulgaria and Eastern Europe in 2020 (with G. Marinov and M. Mladenov), Scientific Reports 12, Article number: 6333 (2022).

Associated primes and integral closure of Noetherian rings, Journal of Pure and Applied Algebra, 225 Issue 5, 2021. arXiv

The A_f condition and relative conormal spaces for functions with non-vanishing derivative, Int. J. Math. vol. 30, no. 10, (2019), 12 pages. arXiv  (joint with T. Gaffney) 

Associated points and integral closure of modules, Journal of Algebra, 508 2018, pp. 301-338. arXiv

Pairs of modules and determinantal isolated singularities. arXiv (joint with T. Gaffney)

On the solvability of p-adic diagonal equations, MIT UJM 8 2007, pp. 111-124. pdf

Some sufficient conditions for the solvability of p-adic polynomial equations, Proceedings of the Thirty Fourth Spring Conference of the Union of Bulgarian Mathematicians, 2005, pp. 102--107. pdf (joint with Vesselin Dimitrov)