Kalina Mincheva

Research

summary of research

My research is in algebraic geometry, in particular tropical geometry and semiring algebra. I am interested in studying the geometry of tropical schemes and varieties in terms of the congruences on the polynomial and Laurent polynomial semiring with coefficients in the tropical semifield or other idempotent semifields.

Papers and Pre-prints

  • J. Jun, K. Mincheva, and L. Rowen, "Projective Module Systems"
    link, , submitted.
    We develop the basic theory of projective modules and splitting in the more general setting of systems. This enables us to prove analogues of classical theorems for tropical and hyperfield theory. In this context we prove a Dual Basis Lemma and develop Morita theory. We also prove a Schanuel's Lemma as a first step towards defining homological dimension.
  • J. Jun, K. Mincheva, and J. Tolliver, "Picard groups for tropical toric varieties"
    link, , Manuscripta Mathematica (2018).
    From any monoid scheme one can pass to a semiring scheme (a generalization of a tropical scheme) by scalar extension to an idempotent semifield. In this note, we investigate the relationship between the Picard groups of a monoid scheme and the corresponding semiring scheme. We prove that for a given irreducible monoid scheme (with some mild conditions) the Picard group is stable under scalar extension to and idempotent semifield. Moreover, each of these groups can be computed by considering the correct sheaf cohomology groups. We also construct the group CaCl(X) of Cartier divisors modulo (naive) principal Cartier divisors for a cancellative semiring scheme X and prove that CaCl(X) is isomorphic to Pic(X).
  • L. Bossinger, S. Lamboglia, K. Mincheva, and F. Mohammadi, "Computing toric degenerations of flag varieties"
    link, , Combinatorial Algebraic Geometry. Fields Institute Communications, vol 80. Springer (2017).
    We compute toric degenerations arising from the tropicalization of the complete flag of GL(4) and GL(5). We present a general procedure to find such degenerations even in the cases where the initial ideal arising from a cone of the tropicalization is not prime. We give explicitly the Khovanskii bases obtained from maximal cones in the tropicalization. For the complete flags of GL(4) and GL(5) we compare toric degenerations arising from string polytopes and the FFLV-polytope with those obtained from the tropicalization of the flag varieties.
  • D. Joó, K. Mincheva, "On the dimension of the polynomial and the Laurent polynomial semiring"
    link, , J. of Algebra (2018).
    We prove that the Krull dimension (defined for congruences) of the n-variable polynomial and the Laurent polynomial semiring over any idempotent semiring R of finite dimension is equal to the dimension of R plus n.
  • D. Joó, K. Mincheva, "Prime congruences of additively idempotent semirings and a Nullstellensatz for tropical polynomials"
    link, , Selecta Mathematica (2017).
    A new definition of prime congruences in additively idempotent semirings, this allows us to we define radicals and Krull dimension. A complete description of prime congruences is given in certain semirings. An improvement of a result of A. Bertram and R. Easton is proven which can be regarded as a Nullstellensatz for tropical polynomials.
  • K. Mincheva, "Prime congruences and tropical geometry" - PhD thesis
    link, .
    Contains some new non-previously published results on dimension and discusses relation to tropical varieties, tropical schemes and other papers in the literature.
  • K. Mincheva, "Automorphisms of non-Abelian p-groups" - MSc thesis
    link, .
    It has been conjectured that there is no p-group with Abelian automorphism group whose center strictly contains the derived subgroup. The main focus of this thesis is to provide a counter example to this conjecture. We also discuss the minimality (in terms of number of elements) of such a group.

Invited Talks and Posters

  • Tropical geometry and moduli spaces - ICM satellite (August 13-17, 2018) talk.
  • TU Berlin - Research seminar (June 13, 2018) talk.
  • Tropical Mathematics and its Applications Meeting, Warwick (May 30, 2018) talk.
  • University of York - Algebra seminar (May 14, 2018) talk.
  • Workshop on Tropical varieties and amoebas in higher dimension (April 16-20, 2018) talk.
  • Tropical Geometry meets Representation Theory, Cologne (March 12-16, 2018) poster.
  • Workshop on Tropical algebra and applications (January 22-26, 2018) talk.
    • Toric Degenerations Focused Research Workshop in Bristol (August 21-24, 2017) talk.
    • Utah University - Algebraic Geometry Seminar (April 18, 2017) talk.
    • AIM Workshop on Foundations of tropical schemes (April 10 - 14, 2017) talk.
    • Noncommutative Geometry: Number Theory (part of Alain Connes' 70th birthday celebration) (April 4-7, 2017) talk.
    • Yale University - Algebraic and Tropical Geometry Seminar (February 23, 2017) talk.
    • AMS Sectional Meetings, Special Session on Interactions Between Algebraic and Tropical Geometry (March, 5-6 2016) talk.
    • Joint Math Meetings, AMS Session on Algebraic Geometry (January, 6-9 2016) talk.
    • George Mason University - Combinatorics, Algebra and Geometry Seminar (CAGS) (November 6, 2015) talk.
    • University of Pennsylvania - Algebra seminar (October 12, 2015) talk.
    • Binghamton University, SUNY - Algebra seminar (October 2, 2015) talk.
    • Aspects of Algebraic Geometry - Program for Women and Mathematics, IAS (May 11-22, 2015) talk.
    • Student Tropical Algebraic Geometry Symposium 2015 (April, 17-19 2015) talk.
    • Number Theory Seminar at Johns Hopkins University (March, 24 2015) talk.
    • AGNES (October 31 - November 2, 2014) poster.

Attendances, participations, future travel

Math and Art

In the spring 2015 I participated in the Research Remix - an ongoing program of the Digital media center at Johns Hopinks that brings together visual art and academic research. It aims at reinterpreting a research poster in a visual and artistic way. This art piece created by Reid Sczerba based on my joint research with Dániel Joó. You can see photo of the art piece here.