# PhD studies

The PhD education is a four year program intended to teach the methods of mathematical research. An important part of the education is the writing of a thesis which is presented at a PhD defence.

On these pages you will find information about PhD studies, intended for PhD students, their supervisors and for those considering to apply.

## Admission to PhD studies

PhD student positions in mathematics, mathematical statistics and computational mathematics are usually announced in April each year.

Vacant positions at the Department of Mathematics

### Eligibility and selection for PhD studies in mathematics

#### General eligibility

A general eligibility of 240 credits is required, corresponding to 4 years full time university studies, or a university degree at an advanced (master) level or the equivalent competence.

#### Special eligibility

To be qualified you must have a university degree containing at least the following courses in mathematics:

- Algebra: groups, rings, euclidean and principal ideal rings, fields, extension fields.
- Foundation of analysis: real numbers, Bolzano-Weierstrass, derivation and integration in Rn, series of functions, implicit functions.
- Analytic functions: integral and series expansion, residue calculus, conformal mappings, harmonic functions.

The textbooks we use are

Rudin: Principles of mathematical analysis,

Beachy and Blair: Abstract algebra, and

Saff and Snider: Fundamentals of complex analysis.

#### Selection and admission

The selection of candidates is made from certificate of courses, quality of thesis, references, and interviews. Information about admission will be given latest in June.

Those who are accepted are normally financed with study grants. If you intend to finance your education in some other way, you must inform us about that.

### Eligibility and selection for PhD studies in mathematical statistics

#### General eligibility

A general eligibility of 240 credits is required, corresponding to 4 years full time university studies, or a university degree at an advanced (master) level or the equivalent competence.

#### Special eligibility

To be qualified you should have taken courses including most of the

following material:

- Probability Theory: Simultaneous and conditional distributions; conditional expectation and variance, multidimensional normal distribution, limit/convergence theorems (Law of Large Numbers; Central Limit Theorem), convergence of random variables (in distribution, probability, mean or almost surely); transforms (probability generating, moment generating, characteristic); martingales.
- Stochastic Processes: Finite state Markov processes in discrete and continuous time, in particular Poisson and birth-death processes; queueing theory; renewal processes; Brownian motion; stationary stochastic processes; methods of stochastic simulation.
- Statistical inference: Exponential families; likelihood; sufficiency; information bounds; consistency; efficiency; maximum likelihood theory; likelihood ratio tests; uniformly most powerful tests.

The books we use in courses that are prerequisites are:

Gut: An intermediate course in probability,

Ross: Introduction to probability models, and

Lindgren, B. W.: Statistical Theory.

#### Selection and admission

The selection of candidates is made from certificate of courses, quality of thesis, references, and interviews. Information about admission will be given latest in June.

Those who are accepted are normally financed with study grants. If you intend to finance your education in some other way, you must inform us about that.

### Eligibility and selection for PhD studies in computational mathematics

#### General eligibility

In order to meet the general eligibility requirements, the applicant must have completed courses equivalent to at least 240 higher education credits (4 years full time study), of which 60 credits must be in the second cycle, or have otherwise acquired equivalent knowledge in Sweden or elsewhere.

#### Special eligibility

In order to meet the special eligibility requirements, the applicant must have completed courses equivalent to at least 60 credits in mathematical subjects and at least 30 credits in either numerical analysis or computer science.

#### Selection and admission

The selection of candidates is made from certificate of courses, quality of thesis, references, and interviews. Information about admission will be given latest in June.

Those who are accepted are normally financed with study grants. If you intend to finance your education in some other way, you must inform us about that.

## Possible PhD thesis projects

This section contains information about available PhD supervisors and suggestions for PhD thesis projects. Please note that there are three separate announcements of PhD student positions, one for each of the three subjects mathematics, mathematical statistics and computational mathematics, respectively.

For further information about ongoing research at the department, please see the webpages of the research groups and the personal homepages of our researchers.

### PhD thesis projects in mathematics

This section contains information about available PhD supervisors and suggestions for PhD thesis projects in mathematics.

#### Available PhD supervisors

Algebra, Geometry and Combinatorics: Gregory Arone, Alexander Berglund, Jonas Bergström, Rikard Bøgvad, Wushi Goldring, Samuel Lundqvist, Dan Petersen, Sven Raum, Boris Shapiro.

Analysis: Pavel Kurasov, Annemarie Luger, Salvador Rodríguez-López, Sven Raum, Jonathan Rohleder, Alan Sola.

Logic: Peter LeFanu Lumsdaine.

#### Suggestions for PhD thesis projects

In the 2021 call with deadline April 23, we especially seek to recruit PhD students for the following three projects:

Mathematics - Goldring, Generation by groups (pdf) (178 Kb)

Mathematics - Kurasov, Quasicrystals and networks (pdf) (5602 Kb)

Mathematics - Lumsdaine, Homotopy type theory as a foundation for mathematics (pdf) (74 Kb)

Further project suggestions are listed below

Mathematics - Berglund, Topics in algebraic topology (pdf) (12 Kb)

Mathematics - Bergström, Moduli spaces, varieties over finite fields, Galois representations (pdf) (63 Kb)

Mathematics - Lundqvist, Topics in (computational) commutative algebra (pdf) (161 Kb)

Mathematics - Raum, Operator algebras and group theory (pdf) (12 Kb)

Mathematics - Rohleder, Spectral properties of differential operators on domains and graphs (pdf) (27 Kb)

### PhD thesis projects in mathematical statistics

When you apply for a PhD student position in mathematical statistics, you must choose one of the projects suggested in the announcement.

Mathematical statistics - Lindensjö, Stochastic control - with applications to finance (pdf) (347 Kb)

Mathematical statistics - Lindskog, Valuation and decision making in insurance (pdf) (66 Kb)

### PhD thesis projects in computational mathematics

When you apply for a PhD student position in mathematical statistics, you must choose one of the projects suggested in the announcement.

Computational Mathematics - Arvestad and Li, project descriptions (pdf) (38 Kb)

## Study plans

The study plan describes the content of the PhD program and contains information about admission and eligibility. The individual study plan, which is created at the start of the program, contains specifications such as the choice of courses and the research plan.

Information for PhD students from the Faculty of Science

Handbook for postgraduate students from the Swedish National Agency for Higher Education

If you have any questions please contact the Directors of PhD Studies. You can find their contact details at the bottom of the page.

### Study plans - files

#### Individual study plan

Individual Study Plan (odt) (279 Kb)

ISP appendix (docx) (77 Kb)

#### General study plans for PhD students admitted after 2017-07-01

Study plan for PhD studies in mathematics, starting after 2017-07-01, Swedish (pdf) (247 Kb)

Study plan for PhD studies in mathematical statistics, starting after 2017-07-01 (pdf) (414 Kb)

Study plan for PhD studies in computational mathematics (pdf) (165 Kb)

#### General study plans for PhD students admitted before 2017-07-01

Study plan for PhD studies in mathematics, starting before 2017-07-01 (pdf) (212 Kb)

Study plan for PhD studies in mathematical statistics, starting before 2017-07-01 (pdf) (231 Kb)

## PhD students' council

The PhD students at the Department of Mathematics have a council (doktorandrådet) which looks after their interests and has a representative on the department board.

The council is headed by the following PhD students:

Chairman: Måns Karlsson

Secretary: Erik Thorsén

Treasurer: Hadrien Espic

## Preparing for thesis defence (PhD and Licentiate)

Before the defence of the doctoral dissertation some formal procedures have to be dealt with.

Stockholm University information about PhD defence

Guidelines from the Faculty of Science

These documents contain practical information specifically for the Department of Mathematics.

Checklist for PhD thesis defense (pdf) (36 Kb)

Checklist for licentiate thesis presentation (pdf) (112 Kb)

See also the following guidelines for information on how the PhD defence is conducted in Sweden.

Guidelines for opponents (pdf) (16 Kb)

## PhD courses

The planned PhD-Courses in mathematics, mathematical statistics and computational mathematics, during the academic year 2020/2021, can be seen here.

Mathematics Autumn 2020 | Mathematics Spring 2021 |
---|---|

Modular forms, Lilian Matthiesen, KTH | Combinatorial and Algebraic Statistics, Liam Solus, KTH |

Geometric function theory, Alan Sola, SU, and Fredrik Viklund, KTH | Geometric group theory, Sven Raum, SU |

Random Matrices, Kurt Johansson, KTH | Indefiniteness, Annemarie Luger, SU |

Tropical Combinatorics and Geometry (reading course), Johannes Hofscheier, Nottingham, and Katharina Jochemko, KTH | |

Mixed Shimura varieties and other advanced topics about families of mixed Hodge structures (reading course), Wushi Goldring, SU |

Mathematical statistics Autumn 2020 | Mathematical statistics Spring 2021 |
---|---|

Networks and epidemics, Tom Britton, Mia Deijfen, Pieter Trapman, SU | Soft skills for mathematicians, Tom Britton, SU |

Probability theory, Guo Jhen Wu, KTH |

Computational Mathematics Autumn 2020 | Computational Mathematics Spring 2021 |
---|---|

TBA | TBA |

### Previous years' PhD courses in mathematics

#### Academic Year 2019/20

- Spectral theory of quantum graphs and inverse problems, Pavel Kurasov, SU, Autumn 2019
- Infinity categories, Peter LeFanu Lumsdaine, SU, Autumn 2019
- Mathematics education at university level, Torbjörn Tambour, SU, Autumn 2019
- Reading seminar on the Kervaire invariant one problem (Reading course), Gregory Arone, SU, Autumn 2019
- Introduction to operator algebras, Sven Raum, SU, Spring 2020
- Geometric measure theory, John Andersson, KTH, Spring 2020
- Complex algebraic geometry, David Rydh, KTH, Spring 2020

#### Academic Year 2018/19

- Theory of distributions, Pavel Kurasov, SU Autumn 2018.
- Semi-Riemannian geometry (reading course), Mattias Dahl, KTH Autumn 2018
- Partial Differential Equations, John Andersson, KTH Autumn 2018
- Fourier analysis methods for PDEs, Salvador Rodriguez-Lopez och Odysseas Bakas, SU Spring 2019
- Characteristic classes, Tilman Bauer, KTH Spring 2019
- Tannakian categories, Wushi Goldring och Andreas Holmström, SU Spring 2019
- Advanced Topics in Proof Theory and the Foundations of Mathematics, Erik Palmgren, SU from Autumn 2018 to Spring 2019
- Cohen-Macaulay Complexes (mini-course), Afshin Goodarzi, KTH, Spring 2019

#### Academic Year 2017/18

- Algebraic Groups, Wushi Goldring, SU Autumn
- Introduction to Sectorial Operators, Jonathan Rohleder, SU Autumn
- Topics in Analysis, Danijela Damianovic and Henrik Shahgholian, KTH Autumn
- Constructive and Computational Mathematics, Peter LeFanu Lumsdaine, SU Spring
- Geometric Function Theory, Alan Sola (SU) and Fredrik Viklund (KTH), Spring
- Polynomial Functors in Algebra and Topology, Greg Arone, SU Spring
- Riemann-Hilbert Methods in Asymptotic Analysis, Maurice Duits, KTH Spring

#### Academic Year 2016/17

- Computational Clgebraic Geometry (Reading course), Mats Boij, (KTH) and Samuel Lundqvist (SU), Autumn
- Constructions in Dynamical Systems (Reading course), Kristian Bjerklöv and Maria Saprykina, KTH Autumn
- Differential Geometry, Mattias Dahl and Hans Ringström, KTH Autumn
- Mathematical Analysis For All!, John Andersson, KTH Autumn
- Real and Complex Analysis (Reading course), 15hp, Fredrik Viklund, KTH Full year
- Algebraic Topology, Gregory Arone (SU), Tilman Bauer and Wojciech Chacholski (KTH), Spring
- Classical Combinatorics, Linusson and Brändén, KTH Spring
- An Introduction to Pseudodifferential Operators, Rodriguez-Lopez, SU Spring
- Matematical Didactics, Tambour, SU Spring
- Mathematical Physics. (Topics in Mathematics IV), Lundholm, KTH Spring
- Realizability: Computational Interpretations of Logic, Erik Palmgren, SU Spring
- Topics in Applied Algebraic Geometry, Dickenstein and Di Rocco, KTH Spring

#### Academic Year 2015/16

- Commutative Algebra, Roy Skjelnes, Autumn
- Cluster Algebras, Michael Shapiro, Autumn
- Elliptic Partial Differential Equations and Harmonic Function Theory (Reading course), Jonatan Lenells and Henrik Shahgolian, Autumn
- Random Matrices, Maurice Duits and Kurt Johansson, Autumn
- Model Theory, Erik Palmgren, Autumn
- Algebraic and Enumerative Combinatorics, Petter Bränden, Spring
- Clifford Algebras, Douglas Lundholm and Lars Svensson, Spring
- Étale Cohomology, Jonas Bergström and David Rydh, Spring
- Spectral Theory for Quantum Graphs, Pavel Kurasov, Spring
- Several Complex Variables (Reading course), Håkan Hedenmalm, Spring
- Operads in Algebraic Topology, Alexander Berglund and Stephanie Ziegenhagen, Spring

#### Academic Year 2014/15

- Infinity-categories and Homotopy Type Theory, Ph. Hackney and P. Lumsdaine, Autumn
- Riemann Surface, Analytic and Algebraic Aspects, J. E. Björk and B. Shapiro, Autumn
- Gröbner Bases, J. Backelin, Spring
- Spectra of (non-selfadjoint) Matrices and Operators, A. Luger and Christiane Tretter (Bern), Spring

#### Academic Year 2013/14

- Commutative Algebra II, Christian Gottlieb, SU Autumn
- Type Theory, Erik Palmgren, SU Autumn
- Fourieranalys, KTH Autumn
- Analysis on Manifolds, Olga Rossi, Spring
- Spectral Theory for Quantum Graphs, Pavel Kurasov, Spring
- Cathegory theory, Spring

#### Spring 2000 to spring 2013

Courses given between the spring 2000 and the spring 2013 can be found on the Swedish version of this page.

### Previous years' PhD courses in mathematical statistics

#### Academic Year 2019/20

- Data-driven statistical modelling with optimisation, Tobias Rydén, SU, Autumn 2019
- Mathematics education at university level, Torbjörn Tambour, SU, Autumn 2019
- Stochastic finance in discrete time, Filip Lindskog, SU, Autumn 2019
- Advanced probability, Henrik Hult, KTH, Autumn 2019
- Unsupervised learning, Chun-Biu Li, SU, Spring 2020

#### Academic Year 2018/19

- Optimal Stochastic Control, Kristoffer Lindensjö, SU Autumn 2018
- Probability Theory, Jimmy Olsson, KTH Autumn 2018
- Topics in Discrete Probability, Timo Hirscher, SU Spring 2019
- Computational Methods for Stochastic Differential Equations, Mattias Sandberg och Anders Szepesy, KTH Spring 2019
- Computational Methods for Stochastic Differential Equations, Mattias Sandberg och Anders Szepesy, KTH Spring 2019
- Soft Skills for Mathematicians, Tom Britton, SU
- Causal Inference, Timo Koski, KTH

#### Academic Year 2011/12

- Statistical Constultancy Methodology, Rolf Sundberg, Autumn
- Computer-intensive Statistical Methods, Tom Britton, Alexander Ploner and Niclas Noren, Spring
- Probability Theory IV, Dmitrii Silvestrov, Spring
- Statistical Models, Rolf Sundberg, Spring

#### Academic Year 2010/11

- Statistical Constultancy Methodology, Rolf Sundberg, Autumn
- Standastic Processes III, Pieter Trapman, Autumn
- Writing and Presenting Mathematical Papers, Tom Britton, Autumn
- Statistical Models, Rolf Sundberg, Spring
- Standastic Processes IV, Dmitrii Silvestrov, Spring

#### Academic Year 2009/10

- Computer Intensive Statistical Methods, Britton et al, Autumn
- Probability Theory IV, Dmitrii Silvestrov, Spring
- Statistical Models, Rolf Sundberg, Spring

#### Academic Year 2008/09

- Study Group in Random Networks, Britton, Autumn
- Standastic Processes III, Hössjer, Autumn
- Probability Theory, Gut, Uppsala University
- Study Group in Phylogenetics and Comparative Genomics: Bio, Maths, Stats and Algorithms, Britton

#### Academic Year 2007/08

- Probability Theory, Hössjer, Autumn
- Statistical Methods of Population Genetics and Gene Mapping, Palmgren and Hössjer, Spring
- Writing and Presenting Mathematics/Statistics, Britton, Spring

#### Academic Year 2006/07

- Large Deviations, Martin-Löf, Autumn
- Inference for Standastic Processes, Svensson, Spring
- Likelihood Based Inference, Pawitan, KI
- Random Graphs (Study Circle), Britton

#### Academic Year 2005/06

- Standastic Methods of Population Genetics, Hössjer, Autumn

#### Academic Year 2004/05

- Statistical Theory for Exponential Families, Sundberg
- Statistical Genetics and Bioinformatics (Seminar Series), Palmgren
- Markov Population Models, Martin-Löf
- Statistical Consulting Methodology, Sundberg
- Probabilistic Properties of DeSemesteristic Systems, Tyrcha
- Statistics for Microarrays, Sundberg

#### Academic Year 2003/04

- Standastic Epidemic Models and Their Statistical Analysis, Britton
- Probability Theory, Gut, Uppsala University
- Statistical Modeling and Inference using Likelihood, Pawitan, KI

#### Academic Year 2002/03

- Statistical Genomics, Greenwood
- Statistical Methods in Molecular Medicine and Genetic Epidemiology, Palmgren

#### Academic Year 2001/02

- Att skriva matematisk statistik, Svensson
- Statistisk inferensteori

### Previous years' PhD courses in computational mathematics

#### Academic Year 2019/20

- Data-driven statistical modelling with optimisation, Tobias Rydén, SU, Autumn 2019
- Mathematics education at university level, Torbjörn Tambour, SU, Autumn 2019

## Contact

**Director of studies for the PhD programme in mathematics**

**Director of studies for the PhD programme in mathematical statistics**

**Director of studies for the PhD programme in computational mathematics**

Last updated: April 8, 2021

Source: Department of Mathematics