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.

- N. Friedenberg, K. Mincheva, "Tropical Adic Spaces" - in preparation.

link, .in preparation. - D. Joó, K. Mincheva, "Applications of congruence methods to tropical geometry" - in preparation.

link, .in preparation. - J. Jun, K. Mincheva, and J. Tolliver, "Vector Bundles on Tropical Toric Schemes"

link, (2020).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, , 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"

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.

- 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.

- 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)
- 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)

- 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.

- I attended the Collaborate@ICERM, Tropical degree two del Pezzo surfaces and their 56 lines (August, 16-20 2021).
- I attended the VIRTUAL Workshop Women in Algebraic Geometry (July, 27-31 2020).
- I attended the AIM Workshop on Foundations of tropical schemes (April 10 - 14, 2017).
- I attended the Apprenticeship Weeks at the Fields Institute (August 21 - September 3, 2016).

- I led the tropical package group at the online Macaulay2 Workshop (June 1 - 5, 2020).
- I attended the Macaulay2 Workshop (June 4 - 8, 2018).

- I attended the Joint Math Meetings (January, 15-18 2020).
- I attended the Oberwolfach Wokshop, "Non-Archimedean Geometry and Applications" (February 23 - March 2, 2019).
- I attended the AGNES (September 21 - 23, 2018).
- I attended the ICM 2018 in Rio de Janeiro (August 1 - 10, 2018).
- I attended the Berkovich spaces - 30 years (July 9 - 13, 2018).
- I attended the AGNES (October 13 - 15, 2017).
- I attended the AGNES (April 21 - 23, 2017).
- I attended the The Georgia Algebraic Geometry Symposium (GAGS) (March 3-5, 2017).
- I attended the Workshop on Perspectives and Emerging Topics in Algebra and Konvexity (PEAK) (February 3-9, 2017).
- I attended the Workshop on Combinatorial Moduli Spaces and Intersection Theory (December 5 - 9, 2016).
- I attended the Introductory Workshop on Combinatorial Algebraic Geometry (August 15 - 19, 2016).
- I attended the AGNES (October 2 - 4, 2015).
- I attended the Oberwolfach Wokshop on Non-commutative Geometry (June 14 - 20, 2015).
- I attended the Aspects of Algebraic Geometry - Program for Women and Mathematics, IAS (May 11 - 22, 2015).
- I attended the Student Tropical Algebraic Geometry Symposium (April 17 - 19, 2015).
- I attended the AGNES (March 20 - 22, 2015).
- I attended the AGNES (October 31 - November 2, 2014).
- I attended the Workshop on Algebraic Foundations for Tropical Geometry (May 11 - 14, 2014).
- I attended the International Conference "Mathematics Days in Sofia" (July 7 - 10, 2014).
- I attended the MSRI 2012 Summer School in Non-commutative algebraic geometry (June 18 - 29, 2012).