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 J. Tolliver, "Representation theory over semifields" - in preparation.
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(2024).
in preparation.
N. Friedenberg, K. Mincheva, "Tropical Adic Spaces II: Uniform semirings" - in preparation.
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in preparation.
D. Joó, K. Mincheva, "Varieties of prime (tropical) ideals" - in preparation.
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(2024).
In this note we study the relationship between ideals and congruences of the tropical polynomial semiring.
We show that the variety of a non-zero prime ideal of the tropical polynomial semiring consists of at most one point.
We also prove a result relating the dimension of an affine tropical variety and the dimension of its "coordinate ring".
J. Jun, K. Mincheva, and J. Tolliver, "Equivariant vector bundles on tropical toric schemes"
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(2024).
We introduce a notion of equivariant vector bundles on schemes over semirings. We do this by considering the functor of points
of a locally free sheaf. We prove that every toric vector bundle on a tropical toric scheme $X$ equivariantly splits as a sum
of toric line bundles, and study the equivariant Picard group $\text{Pic}_G(X)$. Finally, we prove a tropical version of Klyachko's
classification theorem for tropical toric vector bundles.
N. Friedenberg, K. Mincheva, "Geometric interpretation of valuated term (pre)orders"
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(2023).
Valuated term orders are studied for the purposes of Gröbner theory over
fields with valuation. The points of a usual tropical variety correspond to
certain valuated terms preorders. Generalizing both of these, the set of all
"well-behaved" valuated term preorders is canonically in bijection with the
points of a space introduced in our previous work on tropical adic geometry. In
this paper we interpret these points geometrically by explicitly characterizing
them in terms of classical polyhedral geometry. This characterization gives a
bijection with equivalence classes of flags of polyhedra as well as a bijection
with a class of prime filters on a lattice of polyhedral sets. The first of
these also classifies valuated term orders. The second bijection is of the same
flavor as the bijections from [van der Put and Schneider, 1995] in
non-archimedean analytic geometry and indicates that the results of that paper
may have analogues in tropical adic geometry.
N. Friedenberg, K. Mincheva, "Tropical Adic Spaces I: The continuous spectrum of a topological semiring"
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to appear in Res. Math. Sci. (2024).
Towards building tropical analogues of adic spaces, we study certain spaces of prime congruences as a
topological semiring replacement for the space of continuous valuations on a topological ring. This requires
building the theory of topological idempotent semirings, and we consider semirings of convergent power
series as a primary example. We consider the semiring of convergent power series as a topological space
by defining a metric on it. We check that, in tropical toric cases, the proposed objects carry meaningful
geometric information. In particular, we show that the dimension behaves as expected. We give an explicit
characterization of the points in terms of classical polyhedral geometry in a follow up paper.
J. Jun, K. Mincheva, and L. Rowen "$\mathcal{T}$-Semiring Pairs"
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, Kybernetika, Special issue dedicated to Professor Martin Gavalec (2022).
We develop a general axiomatic theory of semiring pairs, which simultaneously generalizes several algebraic structures, in order to bypass
negation as much as feasible. We investigate several classical theorems and notions in this setting including fractions, integral extensions,
and Hilbert's Nullstellensatz. Finally, we study a notion of growth in this context.
J. Jun, K. Mincheva, and J. Tolliver, "Vector Bundles on Tropical Toric Schemes"
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, to appear in Journal of Algebra (2023).
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"
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, Manuscripta Mathematica (2022).
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"
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, 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"
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, 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"
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, 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"
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, 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"
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.
Theses
K. Mincheva, "Prime congruences and tropical geometry" - PhD thesis
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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
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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
Simons Foundation, Travel Support for Mathematicians (2024-2029), PI
SSE Award for Diversity and Outreach (2024)
COR internal travel grant (2023), PI
AAU Ideas Grant (2023-2024), co-PI - changing teaching evaluations
NSF Conference grant (2021-2025), PI - organize Math For All in NOLA Conference
BoR Targeted Enchancement Grant (2021-2022), co-PI
NSF Conference grant (2019), co-PI - organize JAMI 2019 Conference
AWM-NSF Mentoring grant, 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
Conferences organized
Macaulay2 workshop - TBA (2025)
Math for All (April 5 - 6, 2024)
SIAM LA-TX mini-symposium: Combinatorial algebraic geometry and rigidity theory (November 3 - 5, 2023)
In spring 2015, I participated in the Research Remix - an ongoing program of the Digital media center at Johns Hopkins that brings together
visual art and academic research. It aims at reinterpreting a research poster in a visual and artistic way. This art piece was created by Reid Sczerba based
on my joint research with Dániel Joó. You can see photo of the art piece here.