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

  • N. Friedenberg, K. Mincheva, "Tropical Adic Spaces II: Uniform semirings" - in preparation.
    link, .
    in preparation.
  • D. Joó, K. Mincheva, "Varieties of prime (tropical) ideals" - in preparation.
    link, (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, "Representation theory over semifields and matroidal representations"
    link, (2024).
    We study and classify representations of a torsion group $G$ over an idempotent semifield with special attention on the case over the Boolean semifield $\mathbb{B}$. We also study tropical subrepresentations of the Boolean regular representation $\mathbb{B}[G]$ of a finite group $G$. These are equivalent to the matroids on ground set $G$ for which left-multiplication by each element of $G$ is a matroid automorphism. We completely classify the tropical subrepresentations of $\mathbb{B}[G]$ for rank 3. When $G$ is an abelian group, our approach can be seen as a generalization of Golomb rulers. In doing so, we also introduce an interesting class of matroids obtained from equivalence relations on finite sets.
  • J. Jun, K. Mincheva, and J. Tolliver, "Tropical representations and valuated matroids"
    link, (2024).
    We explore several facets of tropical subrepresentations of a linear representation of a group over the tropical semifield $\mathbb{T}$. A key role in the study of tropical subrepresentations is played by two types of modules over a semiring: weakly free and quasi-free modules. We also investigate subgroups of $\text{GL}_n(K)$ for $K=\mathbb{T}$, $ \mathbb{R}_{\geq 0}$, and automorphisms of weakly free modules and tropical prevarieties defined by tropical linear equations. As an application of our results, we provide an intrinsic description of tropical subrepresentation via certain quasi-free modules, and prove that a tropical subrepresentation is equivalent to a valuated matroidal representation.
  • J. Jun, K. Mincheva, and J. Tolliver, "Equivariant vector bundles on tropical toric schemes"
    link, (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"
    link, (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"
    link, 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"
    link, , 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"
    link, , 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"
    link, , 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"
    link, , 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"
    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 polynomial semirings"
    link, , 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
    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.

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

  • Math for All (April 5, 2024)
  • 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)
  • SRAC - Southern Regional Algebra Conference (March 24 - 26, 2023)
  • Math for All (February 24 - 25, 2023)
    • Math for All (February 4 - 6, 2022)
    • AMS Sectional Meetings, Special Session: Tropical Geometry, $\mathbb{F}_1$-connections and Matroids (March 13-14, 2021)
    • JAMI - Japan-U.S. Mathematics Institute conference at Johns Hopkins (October 18 - 20, 2019)
    • SIAM, Applied Algebraic Geometry - Mini-symposium: Computational tropical geometry (July 9 - 13, 2019)
    • BATMOBYLE - Bi-annual Algebraic and Tropical Meetings of Brown and Yale (November 29, 2018)
    • BATMOBYLE - Bi-annual Algebraic and Tropical Meetings of Brown and Yale (November 30, 2017)

Invited Talks and Posters

  • Algebra/Geometry Seminar at University of New Mexico (December 4, 2024) talk.
  • Saint Louis University Mathematics & Statistics Colloquium (November 15, 2024) talk.
  • AMS Fall Eastern Section Meeting (October 19 - 20, 2024) talk.
  • AMS Fall Central Section Meeting (September 14 - 15, 2024) talk.
  • Noncommutativity at the interface of topology, geometry and analysis, Bologna, Italy (June 23 - 28, 2024) talk.
  • Southern Regional Algebra Conference (SRAC) (March 22 - 24, 2024) talk.
  • Lloyd Roeling Conference (March 15 - 17, 2024) talk.
    • 2023 AWM Research Symposium (September 30 - October 2, 2023) talk.
    • SIAM AG 2023 (July 10-14, 2023) talk.
    • Southern Regional Algebra Conference (SRAC) (March 24 - 26, 2023) talk.
    • South Central Topology Conference (February 11 - 12, 2023) talk.
    • Tulane University - Algebra and Combinatorics Seminar (February 1, 2023) talk.
    • The 5th Annual Meeting of the SIAM Texas-Louisiana Section (November 4 - 6, 2022) talk.
    • Second Annual Meeting of Young Bulgarian Mathematicians (June 12 - 15, 2022) talk.
    • $\mathbb{F}_1$ World Seminar (March 30, 2022) talk.
    • SIAM minisymposium: Algorithmic Algebraic Geometry (November, 2021) talk.
    • Algebra and Logic Seminar, IMI-BAS (October, 2021) talk.
    • AUBG 30th Anniversary Multidisciplinary Academic Conference (October, 2021) talk.
    • SIAM minisymposium: Foundations of Tropical Geometry (August, 2021) talk.
    • University of Pittsburgh - Algebra, Combinatorics, and Geometry Seminar (April 15, 2021) talk.
    • Tulane University - Algebra Seminar (November 11, 2020) talk.
    • Tulane University - Algebra Seminar (November 4, 2020) talk.
    • Tropical Geometry in Frankfurt (TGiF) Seminar (June 26, 2020) talk.
    • (cancelled due to COVID-19) AMS Sectional Meetings: Moduli of curves, Hilbert schemes, and tropical geometry (March 21-22, 2020) talk.
    • Utah State University - Colloquium (February 6, 2020) talk.
    • George Washington University - Colloquium (February 4, 2020) talk.
    • Tulane University - Colloquium (January 30, 2020) talk.
    • McMaster University - Algebra Seminar (January 27, 2020) talk.
    • Auburn University - Algebra seminar (January 23, 2020) talk.
    • JAMI: Riemann-Roch in characteristic one and related topics (October 18-20, 2019) talk.
    • University of Kentucky - Algebra Seminar (October 9, 2019) talk.
    • BIRS Workshop, Mutations: Mirror Symmetry, Deformations, and Combinatorics (11 August - 16 August, 2019) talk.
    • Non-Archimedean and Tropical Geometry (July 28 - August 2, 2019) talk.
    • BATMOBYLE (May 2, 2019) talk.
    • Harvard University - Algebraic Geometry seminar (April 23, 2019) talk.
    • Georgia Tech - Algebraic Geometry seminar (April 15, 2019) talk.
    • Johns Hopkins University - Algebraic Geometry seminar (March 26, 2019) talk.
    • Brown University - Algebraic Geometry seminar (October 26, 2018) talk.
    • 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: 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.

Research-in-Pairs and Research workshops

Macaulay2 Workshops

Math and Art

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.