Peter Bruin
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Peter Bruin has a total of 13 co-authors. Here are the top co-authors by number of shared publications:
- Filip Najman
Faculty of Science, Department of Mathematics, University of Zagreb, Zagreb, Croatia
7 shared publications - Antonella Perucca
Department of Mathematics, University of Luxembourg, Esch-sur-Alzette, Luxembourg
2 shared publications Andrea Ferraguti
1 shared publication- Andrea Ferraguti
DICATAM, University of Brescia, via Branze 43, I-25123 Brescia, Italy
1 shared publication - Dominique Lawson
Student Leiden University
1 shared publication Filip Najman
Faculty of Science Department of Mathematics University of Zagreb Croatia
1 shared publication
Latest Publications
- Open Access
- 28 August 2025
- Published by TU Delft OPEN Publishing
Mathematics for Quantum Physics
Mathematics for Quantum Physics provides a compact introduction to the most important mathematical tools used in quantum mechanics. The text is aimed at students who already possess basic knowledge of calculus and complex numbers. It is divided into three parts: analysis, linear algebra and probability. The focus is on examples and applications, and each section comes with a collection of exercises.
- 28 August 2025
- Published by TU Delft OPEN Publishing
Open AccessMathematics for Quantum Physics
Mathematics for Quantum Physics provides a compact introduction to the most important mathematical tools used in quantum mechanics. The text is aimed at students who already possess basic knowledge of calculus and complex numbers. It is divided into three parts: analysis, linear algebra and probability. The focus is on examples and applications, and each section comes with a collection of exercises.
- Open Access
Hidden shift problem for complex functions
We study quantum algorithms for the hidden shift problem of complex scalar- and vector-valued functions on finite abelian groups. Given oracle access to a shifted function and the Fourier transform of the unshifted function, the goal is to find the hidden shift. We analyze the success probability of our algorithms when using a constant number of queries. For bent functions, they succeed with probability 1, while for arbitrary functions the success probability depends on the `bentness' of the function.
Open AccessHidden shift problem for complex functions
We study quantum algorithms for the hidden shift problem of complex scalar- and vector-valued functions on finite abelian groups. Given oracle access to a shifted function and the Fourier transform of the unshifted function, the goal is to find the hidden shift. We analyze the success probability of our algorithms when using a constant number of queries. For bent functions, they succeed with probability 1, while for arbitrary functions the success probability depends on the `bentness' of the function.
- Open Access
The Directed Van Kampen Theorem in Lean
Directed topology is an area of mathematics with applications in concurrency. It extends the concept of a topological space by adding a notion of directedness, which restricts how paths can evolve through a space and enables thereby a faithful representation of computation with their direction. In this paper, we present a Lean formalisation of directed spaces and a Van Kampen theorem for them. This theorem allows the calculation of the homotopy type of a space by combining local knowledge the homotopy type of subspaces. With this theorem, the reasoning about spaces can be reduced to subspaces and, by representing concurrent systems as directed spaces, we can reduce the deduction of properties of a composed system to that of subsystems. The formalisation in Lean can serve to support computer-assisted reasoning about the behaviour of concurrent systems.
Open AccessThe Directed Van Kampen Theorem in Lean
Directed topology is an area of mathematics with applications in concurrency. It extends the concept of a topological space by adding a notion of directedness, which restricts how paths can evolve through a space and enables thereby a faithful representation of computation with their direction. In this paper, we present a Lean formalisation of directed spaces and a Van Kampen theorem for them. This theorem allows the calculation of the homotopy type of a space by combining local knowledge the homotopy type of subspaces. With this theorem, the reasoning about spaces can be reduced to subspaces and, by representing concurrent systems as directed spaces, we can reduce the deduction of properties of a composed system to that of subsystems. The formalisation in Lean can serve to support computer-assisted reasoning about the behaviour of concurrent systems.
- Open Access
- 9 October 2023
- Published by TU Delft OPEN Publishing
Mathematics for Quantum Physics Authors
Mathematics for Quantum Physics provides a compact introduction to the most important mathematical tools used in quantum mechanics. The text is aimed at students who already possess basic knowledge of calculus and complex numbers. It is divided into three parts: analysis, linear algebra and probability. The focus is on examples and applications, and each section comes with a collection of exercises.
- 9 October 2023
- Published by TU Delft OPEN Publishing
Open AccessMathematics for Quantum Physics Authors
Mathematics for Quantum Physics provides a compact introduction to the most important mathematical tools used in quantum mechanics. The text is aimed at students who already possess basic knowledge of calculus and complex numbers. It is divided into three parts: analysis, linear algebra and probability. The focus is on examples and applications, and each section comes with a collection of exercises.
- Open Access
- 16 March 2023
- Published by Elsevier in Expositiones Mathematicae
- Vol. 41 (3) , 514-530
Extensions and torsors for finite group schemes
- 16 March 2023
- Published by Elsevier in Expositiones Mathematicae
- Vol. 41 (3) , 514-530
Open AccessExtensions and torsors for finite group schemes
- Open Access
Counting rational points on weighted projective spaces over number fields
Deng (arXiv:math/9812082) gave an asymptotic formula for the number of rational points on a weighted projective space over a number field with respect to a certain height function. We prove a generalization of Deng's result involving a morphism between weighted projective spaces, allowing us to count rational points whose image under this morphism has bounded height. This method provides a more general and simpler proof for a result of the first-named author and Najman on counting elliptic curves with prescribed level structures over number fields. We further include some examples of applications to modular curves.
Open AccessCounting rational points on weighted projective spaces over number fields
Deng (arXiv:math/9812082) gave an asymptotic formula for the number of rational points on a weighted projective space over a number field with respect to a certain height function. We prove a generalization of Deng's result involving a morphism between weighted projective spaces, allowing us to count rational points whose image under this morphism has bounded height. This method provides a more general and simpler proof for a result of the first-named author and Najman on counting elliptic curves with prescribed level structures over number fields. We further include some examples of applications to modular curves.
- 1 January 2023
- Published by Institute of Mathematics, Polish Academy of Sciences in Banach Center Publications
- Vol. 126, 137-159
Genus distribution of random $q$-ary lattices
The genus is an efficiently computable arithmetic invariant for lattices up to isomorphism. Given the recent proposals of basing cryptography on the lattice isomorphism problem, it is of cryptographic interest to classify relevant families of lattices acc
- 1 January 2023
- Published by Institute of Mathematics, Polish Academy of Sciences in Banach Center Publications
- Vol. 126, 137-159
Genus distribution of random $q$-ary lattices
The genus is an efficiently computable arithmetic invariant for lattices up to isomorphism. Given the recent proposals of basing cryptography on the lattice isomorphism problem, it is of cryptographic interest to classify relevant families of lattices acc
- Open AccessHyperelliptic modular curves $X_0(n)$ and isogenies of elliptic curves over quadratic fieldsLet $n$ be an integer such that the modular curve $X_0(n)$ is hyperelliptic of genus $\ge2$ and such that the Jacobian of $X_0(n)$ has rank $0$ over $\mathbb Q$. We determine all points of $X_0(n)$ defined over quadratic fields, and we give a moduli interpretation of these points. As a consequence, we show that up to $\overline{\mathbb Q}$-isomorphism, all but finitely many elliptic curves with $n$-isogenies over quadratic fields are in fact $\mathbb Q$-curves, and we list all exceptions. We also show that, again with finitely many exceptions up to $\overline{\mathbb Q}$-isomorphism, every $\mathbb Q$-curve $E$ over a quadratic field $K$ admitting an $n$-isogeny is $d$-isogenous, for some $d\mid n$, to the twist of its Galois conjugate by some quadratic extension $L$ of $K$; we determine $d$ and $L$ explicitly.Open AccessHyperelliptic modular curves $X_0(n)$ and isogenies of elliptic curves over quadratic fieldsLet $n$ be an integer such that the modular curve $X_0(n)$ is hyperelliptic of genus $\ge2$ and such that the Jacobian of $X_0(n)$ has rank $0$ over $\mathbb Q$. We determine all points of $X_0(n)$ defined over quadratic fields, and we give a moduli interpretation of these points. As a consequence, we show that up to $\overline{\mathbb Q}$-isomorphism, all but finitely many elliptic curves with $n$-isogenies over quadratic fields are in fact $\mathbb Q$-curves, and we list all exceptions. We also show that, again with finitely many exceptions up to $\overline{\mathbb Q}$-isomorphism, every $\mathbb Q$-curve $E$ over a quadratic field $K$ admitting an $n$-isogeny is $d$-isogenous, for some $d\mid n$, to the twist of its Galois conjugate by some quadratic extension $L$ of $K$; we determine $d$ and $L$ explicitly.
- Open Access
- 11 March 2022
- Published by Wiley in Journal of the London Mathematical Society
- Vol. 105 (4) , 2415-2435
Counting elliptic curves with prescribed level structures over number fields
- 11 March 2022
- Published by Wiley in Journal of the London Mathematical Society
- Vol. 105 (4) , 2415-2435
Open AccessCounting elliptic curves with prescribed level structures over number fields
- Open Access
Fields of definition of elliptic curves with prescribed torsion
We prove that all elliptic curves over quadratic fields with a subgroup isomorphic to $C_{16}$, as well as all elliptic curves over cubic fields with a subgroup isomorphic to $C_2\times C_{14}$, are base changes of elliptic curves defined over $\Q$. We obtain these results by studying geometric properties of modular curves and maps between modular curves, and then obtaining a modular description of these curves and maps.Open AccessFields of definition of elliptic curves with prescribed torsion
We prove that all elliptic curves over quadratic fields with a subgroup isomorphic to $C_{16}$, as well as all elliptic curves over cubic fields with a subgroup isomorphic to $C_2\times C_{14}$, are base changes of elliptic curves defined over $\Q$. We obtain these results by studying geometric properties of modular curves and maps between modular curves, and then obtaining a modular description of these curves and maps.