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
N. Friedenberg, K. Mincheva, "Tropical Adic Spaces" - in preparation.
D. Joó, K. Mincheva, "Applications of congruence methods to tropical geometry" - in preparation.
J. Jun, K. Mincheva, and J. Tolliver, "Vector Bundles on Tropical Toric Schemes"
We define vector bundles for tropical schemes, and explore their properties. The paper largely consists of three parts; (1) we study free modules over zero-sum free semirings, which provide the necessary algebraic background for the theory (2) we relate vector bundles on tropical schemes to topological vector bundles and vector bundles on monoid schemes, and finally (3) we show that all line bundles on a tropical scheme can be lifted to line bundles on a usual scheme.
J. Jun, K. Mincheva, and L. Rowen, "Homology of systemic modules"
, to appear in Manuscripta Mathematica (2021).
In this paper, we develop the rudiments of a tropical homology theory. We use the language
of "triples" and "systems" to simultaneously treat structures from various approaches to
tropical mathematics, including semirings, hyperfields, and super tropical algebra. We enrich
the algebraic structures with a negation map where it does not exist naturally. We obtain an
analogue to Schanuel's lemma which allows us to talk about projective dimension of modules in
this setting. We define two different versions of homology and exactness, and study their
properties. We also prove a weak Snake lemma type result.
J. Jun, K. Mincheva, and L. Rowen, "Projective systemic modules"
, Journal of Pure and Applied Algebra (2019).
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"
, 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"
, 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 polynomial semirings"
, Journal 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"
, 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
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
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.
Grants and Awards
NSF Conference grant, PI - Math For All in NOLA Conference
BoR Targeted Enchancement Grant, Co-PI
NSF Conference grant, Co-PI - organize JAMI 2019 Conference
AWM-NSF Mentoring grant to spend a month at Warwick University, mentor Diane Maclagan 2018
AMS travel grant to attend ICM 2018
Clay Mathematics Institute scholarship to participate in the Apprenticeship weeks at the Fields Institute 2016
(declined) INdAM-COFUND-2012 Fellowships in Mathematics and/or Applications for experienced researchers co-funded by Marie Curie actions
Math for All (February 4-6, 2022)
AMS Sectional Meetings, Special Session: Tropical Geometry, F1-connections and Matroids (March 13-14, 2021)
JAMI - Japan-U.S. Mathematics Institute conference at Johns Hopkins (October 18-20, 2019)
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.