Research

This old website is no longer updated. My new website is https://www1.karlin.mff.cuni.cz/~kala/web/

Curriculum Vitae

My habilitation thesis (from 01/2020) gives an overview of my recent research. A (semi-)popular talk, slides.

MathSciNet

Grants and awards

Principal investigator of UFOCLAN: Universal quadratic forms and class numbers grant

- from the Czech Science Foundation GAČR, 2021 - 2025
- highly selective JUNIOR STAR project (success rate 9%, total budget 1 000 000 USD)
- postdocs: Giacomo Cherubini (from 01/2021), Alessandro Fazzari (09-12/2021), Daniel Gil Muñoz (from 08/2021), Eric Nathan Stucky (from 08/2021), Błażej Żmija (10/2021-09/2022), more to be hired in 2022

Principal investigator of Primus grant Universal quadratic forms: geometry and analysis

- from Charles University, 2020 - 2022
- postdocs: Pavlo Yatsyna (from 01/2020), Tomáš Vávra (05-08/2020), Giacomo Cherubini (09-12/2020)

Principal investigator of Junior grant Quadratic forms and numeration systems over number fields

- from the Czech Science Foundation GAČR, 2017 - 2019
- postdocs: Tomáš Hejda, Tomáš Vávra (both 01/2017 - 12/2019), Ezra Waxman (08/2018-07/2019)
- outcomes of the grant were evaluated by the grant agency as excellent (top 20%)

Neuron Impuls award for the project Semifields in number theory and geometry for 2018 - 2019

Bolzano Award for 2018 for the work Universal quadratic forms and families of number fields

University Center of Excellence: junior team member, PI Jan Krajíček

Service (selected)

Vice-Chair of the Evaluation Panel P202 "Mathematics and Informatics" of Czech Science Foundation GACR (04/2021-03/2023)

Member of the Academic Senate of the Faculty of Math. and Physics, Charles University (02/2020-01/2023)

Editorial board member for Commentationes Mathematicae Universitatis Carolinae

Editorial board member for Pokroky matematiky, fyziky a astronomie (Advances in mathematics, physics, and astronomy), Czech journal for popularization of math and physics

Articles

Most of my recent papers are on arXiv (although the published versions are typically slightly different). Please email me if you are interested in copies or preprints of my theses and other papers.

28. Number fields without universal quadratic forms of small rank exist in most degrees, 6 pp., submitted

27. (with M. Tinková) Universal quadratic forms, small norms and traces in families of number fields, 20 pp., submitted

26. (with T. Hejda) Ternary quadratic forms representing a given arithmetic progression, 10 pp., submitted

25. (with P. Yatsyna) Lifting problem for universal quadratic forms, Adv. Math. 377 (2021), 107497, 24 pp. journal

24. (with P. Yatsyna) Sums of squares in S-integers, New York J. Math. 26 (2020), 1145 - 1154. journal

23. (with T. Hejda) Additive structure of totally positive quadratic integers, Manuscripta Math. 163 (2020), 263 - 278. journal

22. (with J. Svoboda) Universal quadratic forms over multiquadratic fields, Ramanujan J. 48 (2019), 151 - 157. journal

21. (with T. Vávra) Periodic representations in algebraic bases, Monatsh. Math. 188 (2019), 109 - 119. journal

20. (with M. Korbelář) Idempotence of finitely generated commutative semifields, Forum Math. 30 (2018), 1461 - 1474. journal

19. (with A. Dahl) Distribution of class numbers in continued fraction families of real quadratic fields, Proc. Edinb. Math. Soc. 61 (2018), 1193 - 1212. journal

18. (with V. Blomer) On the rank of universal quadratic forms over real quadratic fields, Doc. Math. 23 (2018), 15 - 34. journal

17. (with P. Glivický) Fermat’s Last Theorem and Catalan Conjecture in weak exponential arithmetics, Math. Log. Quart. 63 (2017), 162 - 174. journal

16. Semifields and a theorem of Abhyankar, Comment. Math. Univ. Carolin. 58 (2017), 267 - 273. journal

15. Lattice-ordered abelian groups finitely generated as semirings, J. Commut. Algebra 9 (2017), 387 - 412. journal

14. Norms of indecomposable integers in real quadratic fields, J. Number Theory 166 (2016), 193 - 207. journal; Additional info

13. Universal quadratic forms and elements of small norm in real quadratic fields, Bull. Aust. Math. Soc. 94 (2016), 7 - 14. journal

12. (with V. Blomer) Number fields without universal n-ary quadratic forms, Math. Proc. Cambridge Philos. Soc. 159 (2015), 239 - 252. journal

11. (with J. Berg, A. Castillo, R. Grizzard, R. Moy, C. Wang) Congruences for Ramanujan's f and omega functions via generalized Borcherds products, Ramanujan J. 35 (2014), 327 - 338. journal

10. (with J. Ježek, T. Kepka) Finitely generated algebraic structures with various divisibility conditions, Forum Math. 24 (2012), 379 - 397

9. (with M. Korbelar) Congruence simple subsemirings of Q^+, Semigroup Forum 81 (2010), 286 - 296

8. (with A. Drapal, C. Hamalainen) Latin bitrades, dissections of equilateral triangles and abelian groups, J. Comb. Des. 18 (2010), 1 - 24

7. (with T. Kepka, P. Nemec) Norms on semirings I, Acta Univ. Carolin. Math. Phys. 51 (2010), 29 - 48

6. (with T. Kepka) Commutative parasemifields finitely generated as semirings, Acta Univ. Carolin. Math. Phys. 51 (2010), 49 - 56

5. (with T. Kepka, M. Korbelar) Notes on commutative parasemifields, Comment. Math. Univ. Carolin. 50 (2009), 521 - 533

4. (with A. D. Keedwell) Addendum to "The existence of Buchsteiner and conjugacy-closed quasigroups", Europ. J. Combin. 30 (2009), 1386

3. (with T. Kepka, M. Korbelar, J. D. Phillips) Various subsemirings of the field Q of the rational numbers, Acta Univ. Carolin. Math. Phys. 50 (2009), 29 - 59

2. (with T. Kepka) A note on finitely generated ideal-simple commutative semirings, Comment. Math. Univ. Carolin. 49 (2008), 1 - 9

1. (with V. Flaska, A. Jancarik, T. Kepka) Trees in commutative nil-semigroups of index two, Acta Univ. Carolin Math. Phys. 48 (2007), 81 - 101

Presentations

Typically I give blackboard talks, so here are just a few of my beamer presentations.

Warning: the slides contain intentional simplification and (certainly some!) unintentional errors - use with caution!

Lifting problem for universal quadratic forms - from XXXI^st Journées Arithmétiques

Classifications of finitely generated semifields and lattice-ordered groups - from 56th SSAOS

Universal quadratic forms over number fields - from XXX^th Journées Arithmétiques

Additively indecomposable integers in number fields - from ALaNT 4

Theses

Density of Self-Dual Automorphic Representations of GL_n(A_Q), 2014, PhD thesis, advisor F. Shahidi

Algebraic Substructures in C^m, 2013, PhD thesis, advisor T. Kepka

Simple semirings, 2009, Master's thesis (Mgr.), advisor T. Kepka

Simple semirings, 2007, undergraduate thesis (Bc.), advisor T. Kepka

Prvociselne vzorce (Prime-generating functions), 2004, high school graduation thesis, supervisor R. Kucera