Dr. Sophie Huczynska

Please note that, upon my return from leave in April 2008, I will be taking the remainder of my Fellowship on a half-time basis. I can be contacted in the department during weekday mornings.

BCC 2009: I am Local Organizer for the next British Combinatorial Conference, which will be held at the University of St Andrews from 5th-10th July 2009. The conference webpage can be found here; preliminary registration is now open.

Current Position

I am currently a Royal Society Dorothy Hodgkin Research Fellow in Pure Mathematics at the School of Mathematics and Statistics at the University of St Andrews. My research is mainly in the areas of number theory and combinatorics; I also have interests in algebra and theoretical computer science. I am a member of the Centre for Interdiscipinary Research in Computational Algebra (CIRCA).

Brief career history

I obtained my PhD in the Department of Mathematics at the University of Glasgow. I was a postgraduate student there from October 1999 to December 2002. I spent the first year of my postgraduate studies working under the supervision of the late Professor Bob Odoni. Subsequently, I moved into the area of Galois fields, and worked on this topic for my PhD thesis, supervised by Professor Steve Cohen.

From January 2003 to August 2003, I was a Research Associate in the Centre for Intelligent Systems and their Applications (CISA), part of the School of Informatics at the University of Edinburgh. My work there was in the area of automating discovery and reasoning in mathematics. I maintain an interest in automated reasoning and theorem-proving.

Since September 2003, I have been a member of the School of Mathematics and Statistics at the University of St Andrews. I was a Teaching Fellow in Pure Mathematics until October 2004, when I began my Dorothy Hodgkin Research Fellowship.

Teaching

Session 2003-04: Groups-MT4603 and Linear Algebra-MT2001. Course materials for Groups-MT4603 may be found here.

Session 2005-06: Finite Fields - MT5826. Course materials for Finite Fields may be found here.

Session 2006-07: Linear Algebra - MT2001.

Supervision

Fiona Brunk is working on a PhD in the area of combinatorics, jointly supervised by Nik Ruskuc and myself.

Current research interests

• Finite fields I am interested in many aspects of finite fields, especially the existence and properties of primitive, free elements (elements which are simultaneously additive and multiplicative generators of the field). The Primitive Normal Basis Theorem guarantees that we can find a primitive free element for any extension GF(q^n) of a finite field GF(q); a natural question is what extra conditions we can demand of this element. Answering this question is of particular interest to practitioners, since it reduces the search space when seeking a generating element in a field.
• Combinatorial designs I enjoy finding out about the construction and properties of a range of combinatorial objects and designs, in particular those constructible using finite fields.
• Combinatorics of permutations One of my main interests at present may be broadly described as ``things we can do with permutations". I'm interested in all sorts of combinatorial topics related to this: below are a couple of general headings.
  • Permutation arrays are appealing combinatorial objects, which arise from coding with permutations and recently been applied to powerline communications (the technology which enables data to be transmitted through the electric power grid).
  • Permutation patterns When permutations are considered under the pattern containment order, the resulting area of combinatorics is not only interesting in its own right, but has a natural connection with sorting machines, familiar from theoretical computer science.

Public understanding

• To read a non-specialist account of my work and its possible applications, see the article "Euler's Revolution" by Ian Stewart, which appeared in the New Scientist magazine (issue 24-30 March 2007).

Publications

• "The Primitive Normal Basis Theorem- without a computer", S.D.Cohen and S.Huczynska, Journal of the London Mathematical Society, 67 (2003), no 1, 41--56 .
• "Primitive free quartics with specified norm and trace", S.D.Cohen and S.Huczynska, Acta Arithmetica 109 (2003), no. 4, 359--385.
• "Primitive free cubics with specified norm and trace", S.Huczynska and S.D.Cohen, Trans. Amer. Math. Soc. 355 (2003), no. 8, 3099--3116.
• "Grid classes and the Fibonacci dichotomy for restricted permutations", S. Huczynska and V. Vatter, The Electronic Journal of Combinatorics 13 (2006), no.1.
• "Frequency permutation arrays", S.Huczynska and G.L.Mullen, Journal of Combinatorial Designs 14 (2006), 463--478.
• "Powerline communication and the 36 officers problem", S. Huczynska, Phil. Trans. R. Soc. A, 364 (2006).
• "Pattern classes of permutations via bijections between linearly ordered sets", S. Huczynska and N. Ruskuc, European Journal of Combinatorics 29 (2008), no. 1, 118--139.
• "Simple permutations and algebraic generating functions", R. Brignall, S. Huczynska and V. Vatter, Journal of Combinatorial Theory, Series A, 115, no.3, 423--441.
• "Decomposing simple permutations, with enumerative consequences", R. Brignall, S. Huczynska and V. Vatter, Combinatorica 28 (2008), no. 4, 385--400.
• "The extent to which subsets are additively closed", S.Huczynska, G.L.Mullen and J.Yucas, Journal of Combinatorial Theory, Series A (in press).
• "The strong primitive normal basis theorem", S.D.Cohen and S.Huczynska (preprint).

Various work-related activities

• I organized the CIRCA Undergraduate Summer School, June-August 2004. Reports and pictures are available here.
• I was on the organizing committee of the Third International Conference on Pattern Avoiding Permutations , which was held in Florida in March 2005.
• I was on the organizing committee of the Edinburgh Mathematical Society Postgraduate Meeting at The Burn, May 2005 and May 2006.
• I was on the organizing committee of the Fourth International Conference on Pattern Avoiding Permutations , held in Iceland, 12-16 June 2006.
• I will be the local organizer of the British Combinatorial Conference in 2009, which will be held in St Andrews.

Contact details:

Sophie Huczynska
School of Mathematics and Statistics
University of St Andrews
Mathematical Institute (Room 214)
North Haugh
St Andrews, Fife, KY16 9SS
SCOTLAND

Tel: +44 (0)1334 463711
E-mail: My departmental email address is: sophieh 'at' mcs.st-and.ac.uk
I am also contactable at: sophie.huczynska 'at' st-and.ac.uk.