Antoni Rangachev

I am a Dickson Instructor at the University of Chicago Department of Mathematics . I obtained my Ph.D. degree at NEU in May, 2017 under the supervision of Terence Gaffney and Steven Kleiman.

I received my bachelors degree from MIT under the supervision of Steven Kleiman. Since 2008 I have been research fellow at the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences ( IMI-BAS ). I graduated from Sofia High School of Mathematics.


Research

My research interests are in the overlap of birational geometry, deformation theory, singularity theory and commutative algebra. The main focus of my work is to study how algebraic varieties change in families by detecting the changes with the help of intersection theory - specifically, with numerical invariants such as volumes of line bundles.

I am interested in finding numerical control of various equisingularity conditions. As an application of my work on local volumes, I showed how to control numerically Whitney-Thom (differential) equisingularity for families of isolated singularities.

In the deformation theory of singularities, I proposed the notion of generalized smoothabily by introducing the class of singularities that admit deformations to deficient conormal singularities. This class contains all smoothable singularities, Cohen-Macaylay codimension 2 singularities, Gorenstein codimension 3 singularities, and determinantal singularities.

My other research interests include integral closure methods, valuation theory, asymptotics of associated primes, and their connections to singularities.


Teaching

At University of Chicago I have been teaching MATH 25400 (Algebra I) and Math 25500 (Algebra II).

This June I am giving a series of lectures on resolution of singularities at the nationwide REU program run by our department.

Next year I am teaching MATH 24400 (Introduction to Algebraic Geometry).


Papers

[9] (joint with H. Mourtada and Bernard Teissier) Lipschitz geometry of quasi-ordinary singularities (in preparation).

[8] Local volumes, equisingularity and generalized smoothability; preliminary draft available upon request.

[7] Relative integral closure of Noetherian rings (in preparation).

[6] Associated primes and integral closure of Noetherian rings (submitted) arXiv

[5] (joint with T. Gaffney) The A_f condition and relative conormal spaces for functions with non-vanishing derivative, accepted in Int. J. Math. arXiv

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

Local volumes, integral closures, and equisingularity, PhD Thesis

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

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

[1] (joint with Vesselin Dimitrov) 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


Selected Invited Talks

Thom's transversality and generalized smoothability, IMPA, Rio de Janeiro, July, 2019.

Local volumes, equisingularity, and generalized smoothability, Northwestern University Algebraic Geometry Seminar, May, 2019.

Local volumes, equisingularity, and generalized smoothability, UIC Algebraic Geometry Seminar, May, 2019.

Asymptotics of intersection numbers and deficient conormal singularities, 15th International workshop on real and complex singularities, Sao Carlos, Brazil, 23rd July, 2018, Satellite Conference of the ICM.

Deformations of isolated singularities and local volumes, Singularities Seminar, IMJ-PRG, Paris, Nov. 2017.

Deformations of isolated singularities and local volumes, Route 81 Conference, Cornell University, Ithaca, Oct. 2017.

Restricted local volumes and deformation theory, University of Chicago, Algebraic Geometry Seminar, October 25th, 2016.

Restricted local volumes and deformation theory, World Wide Center of Mathematics, October 21st, 2016, youtube video available here

Local volumes and deformations, Interactions in Geometry - international conference in honor of Johan Davidov and Oleg Mushkarov, Bulgarian Academy of Sciences, Sofia, August, 2016.

Local volumes and equisingularity theory, Geometry, Algebra, Combinatorics and Singularities Seminar, Northeastern University, February 2016.

Local volumes and equisingularity theory, Algebraic Geometry Seminar, Princeton University, October 2015.

Deformation theory of multiplicities and applications, Instituto Nacional de Matematica Pura e Aplicada (IMPA), Rio de Janeiro, Brazil, June, 2015.

The epsilon multiplicity and polar varieties, 13th International workshop on real and complex singularities, Sao Carlos, Brazil, August, 2014.

The epsilon multiplicity and polar varieties, Maurice Auslander Distinguished Lectures and International Conference, Woods Hole, USA,May, 2014.

John Milnor and his contribution to differential topology and singularity theory, National colloquium of mathematics, Bulgarian Academy of Sciences, November 2011.

Equisingularity Theory - History and New Developments National colloquium of mathematics, Bulgarian Academy of Sciences, May 2008.

Equisingularity theory of complex analytic families, Steklov Mathematical Institute, Department of Algebra Seminar, Russian Academy of Sciences, December 2008.

Local solvability of diagonal forms in many variables, New Trends in Mathematics and Informatics 60 years Institute of Mathematics and Informatics, BAS, July 2007.

Density problems and Artin's conjecture on the primitive roots, Special Seminar, Institute of Mathematics and Informatics, BAS, May 2004.


Visiting Positions

IMJ-PRG, France, Paris, Visiting Research Fellow, September 2018, September 2019

IMPA, Brazil, Rio de Janeiro, Visiting Research Fellow, May-July 2015, July 2019

University of Vienna, Visiting Research Fellow, March-July 2009

Massachusetts Institute of Technology, Visiting Research Fellow, Jan.-Feb. 2009

Steklov Mathematics Institute, Russian Academy of Sciences, Moscow, Visiting Research Fellow, December 2008


Grants, Awards, Fellowships, and Projects

Principal Investigator, University of Chicago FACCTS grant ''Conormal and arc spaces in the deformation theory of singularities'' Members: Bernard Teissier (IMJ-PRG) and Hussein Mourtada (IMJ-PRG), 2018-2020.

Principal Investigator, Office of the Provost Global Faculty Grant ''University of Chicago - IMPA collaboration in algebraic geometry'' Members: Carolina Araujo (IMPA) and Eduardo Esteves (IMPA), 2019-2020.

American Mathematical Society ICM Travel Grant, 2018.

The 2017 Recipient of the Dean of College of Science Award for Excellence in Research, for "outstanding research in the mathematics of singularities," Northeastern University, Boston

Member of the project “Complex analysis, complex geometry and mathematical physics” between the Bulgarian Academy of Sciences and the Russian Academy of Sciences, 2008/2009 and 2016--

Member of the project “Complex analysis, differential geometry and topology”, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 2015--

College of Science Travel Grant, NEU, 2015/2016

Research Summer Fellowship, Department of Mathematics, NEU, 2015/2016

Ling Ma fellowship, Department of Mathematics, NEU, Summer 2014

Clay Mathematics Institute Fellowship, Clay Summer School on Resolution of Singularities, 2012

Undergraduate Research Opportunities Fellowships, MIT, 2008-2011


Service

Organizer of the University of Chicago Algebraic Geometry Seminar, 2018-

President of the Mathematics Graduate Student Association, Northeastern University, 2014-2016, re-elected in 2015.

Graduate Student Senator, representing the NEU Math Department at the Graduate Student Government, 2016-2017

Member of the Graduate Student Council at the Office of the Dean of the College of Science at NEU, 2014-2016.


References

Terence Gaffney, Northeastern University

Steven Kleiman, Massachusetts Institute of Technology

Bernard Teissier, Institut Mathematique de Jussieu-Paris Rive Gauche

Madhav Nori, University of Chicago

David Massey, Northeastern University

Bernd Ulrich, Purdue University