Julian Lyczak

General

I am a postdoctoral researcher in Number Theory at the University of Bath. Previously I was in the Browning Group at the Institute for science and Technology Austria. Before I was a Ph.D. candidate at Leiden University under the supervision of dr. Martin Bright.
My research focusses on questions in number theory which have an additional analytical or geometrical flavour. I am for example interested in local to global principles such as the Hasse principle and several forms of approximation to both rational and integral points, obstructions to these principles (the Brauer–Manin obstruction, to name one). The use of analytic number theory is most prominent in my work on the Manin conjecture and the Loughran–Smeets conjecture. I also have an interest in effective techniques for points on curves and surfaces.
I am fascinated by the endless analytical and geometric techniques one can use to study some obvious and seemingly simple questions in number theory.

Contact:
Julian Lyczak
jl4212 (at) bath.ac.uk

Writings

Scientific articles:

Local solubility for a family of quadric surfaces, joint with Tim Browning and Roman Sarapin. (To appear soon).
Arithmetic of Halphen surfaces, joint with Tim Browning. (To appear soon).
Quartic del Pezzo surfaces with a Brauer group of order 4, joint with Roman Sarapin. [pdf][ArXiv]
The Batyrev-Tschinkel conjecture for a non-normal cubic surface and its symmetric square, joint with Nils Gubela. [pdf][ArXiv]
Order 5 Brauer–Manin obstructions to the integral Hasse principle on log K3 surfaces.
Accepted for publication in Annales de l'Institut Fourier. [pdf][ArXiv]
Arithmetic of affine del Pezzo surfaces.
Ph.D. thesis, Leiden University, 2019. [pdf][Webpage]
A uniform bound on the Brauer groups of certain log K3 surfaces, joint with Martin Bright.
Published in Michigan Mathematical Journal, volume 68(2), 2019, pp. 377–384. [pdf][ArXiv]

Theses:

Arithmetic of affine del Pezzo surfaces
Ph.D. thesis, written under the supervision of Martin Bright
Cobordism theories: An algebraic journey from topology to geometry
Master thesis, written under the supervision of Carel Faber
Knowledgeable Frobenius algebras and extended topological quantum field theories
Bachelor thesis, written under the supervision of Ieke Moerdijk

Here you can find a list of errata of the following notes:

Notes on spectral sequences.

Other people's scientific articles with a (minor) contribution from me:

Monodromy of the generalized hypergeometric equation in the Frobenius basis, by Leslie Molag. Published in Indagationes Mathematicae in 2015, pages 495-518.
On the equation \(x+y=1\) in finitely generated groups in positive characteristic, by Peter Koymans and Carlo Pagano.

Miscellaneous:

"Bonte was"
Article on a problem from the Baltic Way 2015, Pythagoras, February 2016
"Nines, nines, more nines"
is a translation of the article "Negens, negens en nog meer negens". This article was published in the book
"Half a Century of Pythagoras Magazine"
MAA Press, 2015.
This book and its articles are a translation of the "De Pythagoras Code" into English.
"De veelzijdigheid van de zeshoek"
Article on a problem from the Dutch National Mathematical Olympiad 2013, Pythagoras, June 2013
jointly with Daniël Kroes
"Op en neer met de roltrap"
Article on a problem from the Dutch National Mathematical Olympiad 2011, Pythagoras, February 2011
"Een extreem handig principe"
Article on the Extremal Principle, Pythagoras, April 2013
"Ramsey-kardinalen"
Article on Ramsey cardinals, Vakidioot, Number 2, 2011-2012
"De Monstergroep"
Article on the monster group, Vakidioot, Number 4, 2010-2011
"De Grieken tegen de Japanners"
Article on angle trisection by paper folding, Vakidioot, Number 1, 2010-2011
"Projectieve meetkunde"
Article on projective geometry, Vakidioot, Number 3, 2009-2010
jointly with Sjoerd Boersma
"IMO 2006 - Opgave 4"
Article on a problem from the International Mathematical Olympiad 2006, Euclides, October 2010
"Negens, negens en nog meer negens"
Article on a problem from the Dutch National Mathematical Olympiad 2009, Pythagoras, February 2009
This article was also selected, among all articles published by Pythagoras in the last 50 years, for publication in the book
"De Pythagoras Code"
Uitgeverij Bert Bakker, Amsterdam, 2011
"Materiaal voor de finaletraining van de Nederlandse Wiskunde Olympiade"
Material to prepare high school students for mathematical competitions, 2008
jointly with dr. Birgit van Dalen and dr. Quintijn Puite

The "Vakidioot" is the monthly magazine of A-Eskwadraat, the study association for mathematics in Utrecht. "Euclides" is a magazine for high school mathematics teachers. "Pythagoras" is a two-monthly magazine for high school students.

Code

Magma code for the example treated in Section 3 and 4.
Magma code for the example treated in Section 5.

Magma code to compute models of del Pezzo surfaces of degree 5 with a specific action of Galois on the ten lines. This document also helps with the computation of the Brauer–Manin obstruction coming from the algebraic Brauer group on the complement of an effective anticanonical divisor. The algebraic Brauer group of such a log K3 surfaces is cyclic of order 5.

Magma code to compute the subgroups of the finite automorphism group of the Picard group of del Pezzo surfaces.
Magma code to compare the first cohomology groups of subgroups of the Weyl group acting on two free \(\mathbb Z\)-modules. This is used to compare the algebraic Brauer group of a log K3 surface with an irreducible ample anticanonical bundle, to the Brauer group of the corresponding del Pezzo surface.

Talks

Autumn 2021

The Loughran–Smeets conjecture over an arbitrary base
At the Browning group working seminar at IST.
The notes where accompanied by some pictures.

Spring 2021

Locally soluble members of a family with multiple fibres
At the Number theory program, Institut Mittag-Leffler, Sweden.
Locally soluble members of a family with multiple fibres, Part I: geometry
At the Browning group working seminar at IST.
Manin's conjecture for symmetric squares of surfaces
At the Oberseminar Zahlentheorie at Georg-August-Universität Göttingen.
Manin's conjecture for non-normal cubic surfaces
At the Browning group working seminar at IST.

Autumn 2020

Manin's conjecture for symmetric squares of surfaces
At the Zahlentheoretisches Kolloquium at Technische Universität Graz.
At the Oberseminar Zahlentheorie und Arithmetische Geometrie at Leibniz Universität Hannover.

Spring 2020

The Brauer group of a complete intersection
At the Browning group working seminar at IST.
The geometry of Egyptians fractions
At the Browning group working seminar at IST.
Why do I want to learn about Coleman–Chabauty?

An informal seminar on the method of Chabauty and Coleman

Manin's conjecture for lattices
At the Browning group working seminar at IST.

Autumn 2019

On the study of integral points in families
At the Browning group working seminar at IST.

Spring 2019:

The geometry of integral points
At the Browning group working seminar at IST.
Explicit examples of Brauer–Manin obstructions to integral points
At the Browning group working seminar at IST.

Spring 2018

Brauer–Manin obstructions of order 5 on log K3 surfaces
At the conference Rational Points on Schiermonnikoog.
What it is I do
At the PhD Colloquium in Leiden.
Abstract tropical curves
At the Tropical Geometry Seminar in Leiden.

Autumn 2017

Explicit Brauer–Manin obstructions of order 5
At the DIAMANT symposium in Breukelen, The Netherlands.
Why your piano is never in tune!
A lunch lecture organized by De Leidsche Flesch in Leiden.

Spring 2017

A uniform bound on Brauer groups of certain log K3 surfaces
At XXX-émes Journées Arithmétic in Caen, France.
A uniform bound on Brauer groups of certain log K3 surfaces
In the Algebra, Geometry and Number Theory seminar.
Het Hasse-principe (in Dutch)
In the bachelor seminar algebra, geometry and number theory.
Obstructions to local-global principles on del Pezzo surfaces
In the seminar on Obstructions to local-global principles.
Spectral sequences
In the seminar on Obstructions to local-global principles.
Introduction to the etale fundamental group
in the seminar on Good reduction of abelian varieties at the University of Amsterdam.

Spring 2016

Trace fields of knots
in the Knot theory seminar at Leiden University.

Spring 2015

Nodal curves on surfaces, an application of algebraic cobordism
Presentation of master thesis in the Algebra, Geometry and Number Theory seminar

Older talks (at the University of Utrecht)

Nodal curves on surfaces, an application of algebraic cobordism
Defense of master thesis
The slides were accompanied by a black board presentation
Taylor's formula
in a seminar on Vertex Operator Algebras
Obstruction theory
in a seminar on Characteristic Classes
Relative \(K_0\) and \(K_1\)
in a seminar on Algebraic K-theory

Seminars, conferences etc.

Ongoing

Browning group working seminar, IST.
Algebraic Geometry and Number Theory Seminar, IST.

Autumn 2021

Conférence en l'honneur d'Alexei N. Skorobogatov, Institut Henri Poincaré, Paris, November 3rd–5th, 2021.
Arithmetic statistics and local-global principles, ESI, Vienna, September 20th–24th, 2021

Spring 2021

Memorial conference for Sir Peter Swinnerton-Dyer, Isaac Newton Institute, Cambridge, May 17th–20th, 2021.
Number theory program, Institut Mittag-Leffler, Sweden, January 11th–April 30th, 2021.

Autumn 2020

DIAMANT symposium, November 26th–27th, 2021.
Modern Breakthroughs in Diophantine Problems, Banff International Research Station, August 31st–September 4th, 2020.

Spring 2020