Mathematics
Mathematics
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I am a doctor in mathematics since 27 February 2025.
My math-genealogy record.
My PhD Thesis: Power Tower Theory:
Foliations and Galois Theory
in Positive Characteristic
My PhD Thesis: Power Tower Theory:
Foliations and Galois Theory
in Positive Characteristic
I am a doctor in mathematics.
The PhD defence was on 27 February 2025 at 13:00
at Agnietenkapel in Amsterdam.
My Publications:
My Publications:
On Algebraic Integrability of Vector Fields with Rational Function Coefficients that Separate Variables (2026) Arxiv
On Algebraic Integrability of Vector Fields with Rational Function Coefficients that Separate Variables (2026) Arxiv
Comment: This is my first paper that was written at KSE in Ukraine. We are building the first international math department in UA with ICMU.
This is a tangent to my work on moduli of purely inseparable subfields, which I am currently procrastinating on (Sep 2026); here is a poster I presented at GAEL33. In short: it is possible to explicitly compute any such subfield in terms of generators from derivations defining it (and back), and this way reveals structures.
Abstract: We work over a field of characteristic zero - primarily over an algebraic closure of the field of rational numbers.
We prove a necessary and sufficient condition for a vector field whose coefficients are rational functions separating variables to be algebraically integrable, that is, the subring of rational functions killed by this vector field, its ring of first integrals, is of maximal possible dimension.
We do it arithmetically by reducing the vector field modulo almost all primes.
In particular, we verify the generalized Grothendieck--Katz $p$-curvature conjecture for foliations defined by these vector fields.
Finally, we use the outcome of that verification to provide explicit formulas for coefficients of all algebraically integrable vector fields separating variables, and their first integrals.
(shorter) Abstract: The paper is about power towers. A power tower is an object analogous to a foliation, but in positive characteristic. They generalise fibrations. They include purely inseparable morphisms. We develop a Galois-type correspondence for power towers. This gives a useful framework for working with purely inseparable morphisms. In particular, we prove a formula for a pullback of a canonical divisor with respect to any such morphism.
Comment: This preprint contains the results from my PhD thesis. I defended it in February 2025.
Abstract: We explicitly compute canonical liftings modulo p^2 in a sense of Achinger--Zdanowicz of Dwork hypersurfaces. The computation involves studying a compatibility between Hodge filtrations and a crystalline Frobenius. In particular, remarkably, we explicitly compute a partial data of the crystalline Frobenius modulo p^2.
Comment: This paper covers the results from my Master thesis. I defended it in September 2020.
Abstract: For a prime number q not equal 2 and r > 0
we study, whether there exists an isometry of order q^r
acting on a free Z^p^k-module equipped with a scalar product. We investigate, whether there exists such an isometry with no non-zero fixed points. Both questions are completely answered in this paper if p is not 2 and not q. As an application we refine Naik's criterion for periodicity of links in S^3. The periodicity criterion we obtain is effectively computable and gives concrete restrictions for periodicity of low-crossing knots.
Comment: My work that I did in Borodzik's grant (2017-2018) is a part of this paper.
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