Helena A. Verrill September 9, 2000

      +45 39 61 77 16 • verrill@math.ku.dk • http://www.math.ku.dk/~verrill

PERSONAL RECORD

Full name:  Helena Anne Verrill
Date of birth:  June 7 1969
Nationality:  British
Departmental address: Department of Mathematics, Copenhagen University,

Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark Telephone: +45 35 32 07 25 Fax: +45 35 32 07 04 Email: verrill@math.ku.dk

Home address:  Fragariavej 7, Hellerup 2900, Denmark

Telephone: +45 39 61 77 16

Higher education:

Cambridge University, Cambridge, England:

B.A.  June 1990  Mathematics, first class.
Part III  June 1991  Mathematics, M. Sc. equivalent, distinction.
Ph.D.  November 1994 Mathematics

Thesis supervisor: Dr. N. I. Shepherd-Barron. Thesis title: Root lattices and pencils of varieties

Previous employment:

12/94–04/96 Kyoto University, Japan.

Royal Society/JSPS postdoctortal exchange fellowship

05/96–05/98 Queen’s University, Canada, postdoctortal position
05/98–05/99 Max-Planck-Institut für Mathematik, Bonn, Germany.

postdoctortal scholarship

Current employment:

05/99–05/01 University of Copenhagen, Denmark.

postdoctoral position, research adjunct

TEACHING

Queen’s University, Kingston, Canada

Instructor of the following courses:

Fall 1996  Calculus for first year engineers.
Spring 1997 Linear algebra for arts and science students.
Fall 1997  Hypergeometric functions and configuration spaces.
Winter 1997 Introduction to calculus.

Teaching Assistant of the following courses:

Winter 1997 Calculus for first year engineers.
Winter 1997 Linear algebra for first year engineers.

Cambridge University, England

1991–1993.  Tutorials: complex variables, group theory, representation theory.

Warwick University, England

Autumn 1994 Tutorials

     +45 39 61 77 16 • verrill@math.ku.dk • http://www.math.ku.dk/~verrill

RESEARCH

Research area: Arithmetic geometry, number theory, algebraic geometry,

modular forms, Galois representations, monodromy. Publications

1. Root lattices and pencils of varieties, Kyoto Journal of Mathematics, Volume 36, Number 2, July 1996, 421–446.

2. The Arithmetic of a certain Calabi-Yau threefold, Proceedings of the CNTA Meeting 1997, Carleton University, Ottawa, Canada, CRM Proceedings and Lecture Notes, Volume 19, 1999, 333–340.

3. The L-series of some Calabi-Yau threefolds, Journal of Number Theory, April 2000.

4. Thompson series, and the mirror maps of pencils of K3 surfaces, with Noriko Yui, Proceedings of the NATO ASI/CRM Summer School “The Arithmetic and Geometry of Algebraic Cycles” Banff, Alberta Canada 1998.

5. Picard-Fuchs equations of some families of elliptic curves. Proceedings of the CRM workshop on Moonshine, Montreal June 1999, to appear.

Preprints

1. On modular mod ℓ Galois representations with exceptional images. (with Ian Kiming) Submitted.

2. Some congruences related to modular forms, Max-Planck-Institut für Mathematik preprint 1999 (26).

3. The L-series of rigid Calabi-Yau threefolds from fibre products of elliptic curves, in preparation.

4. Computing the period lattice attached to a modular form, with W. A. Stein, in preparation.

5. The mirror map of certain K3 surfaces and Thompson series, with N. Yui. A research monograph, in preparation, to appear in the AMS/IP Series on Advanced Mathematics and Physics Eds. S.-T. Yau.

Services Refereed a paper for the Journal of Finite Fields. Some invited talks

99.06.1 Some examples of Picard-Fuchs equations,

CRM workshop on Moonshine, University of Montreal. 99.07.09 Some congruences related to modular forms,

Explicit methods in number theory, Oberwolfach. Recent conferences attended

July—20 July 2000 : Algorithmic Number Theory Symposium, Leiden, The Netherlands 31 July—18 August 2000: School on Automorphic forms on GL(N), ICTP, Italy

Recent research visitors

May 18-24 2000, John McKay, University of Concordia, Canada, for work

related to differential equations for modular forms and monstrous moonshine July 18-24 2000, William Stein, Harvard University, Massachusetts, USA

for work on an algorithm for computing periods of modular forms Research visits in the near future

October 15-31 2000 visiting Klaus Huleck, University of Hannover, Germany

for work related to computation of L-series of rigid Calabi-Yau threefolds November 15 - December 15 2000, MSRI Berekely, California, USA

for part of the semester on algorithmic number theory.

       +45 39 61 77 16 • verrill@math.ku.dk • http://www.math.ku.dk/~verrill

RESEARCH STATEMENT My research is in the areas of arithmetic geometry and number theory. I am interested in modular forms and modular Galois representations, particularly those coming from geometry. There are many open problems in this area, such as the conjectural modularity of 2-dimensional Galois representaions coming from geometry. I have investigated this in the case of some Galois representations obtained from the Galois action on the middle cohomology of certain Calabi-Yau three-folds. In this case one obtains weight 4 modular forms as the Mellin transform of the L-series of the representation. As in the elliptic curve case, the coefficients a_{p} of the weight 4 modular forms in question can be written in terms of the numbers of points on the corresponding variety over 𝔽_{p}. Some of this work has appeared in my recent paper in the Journal of Number Theory. I am also interested in studying the intermediate Jacobian of certain rigid Calabi-Yau threefolds. This leads to the problem of computing periods of modulars forms. In a paper with William Stein, currently in preparation, we developed an algorithm for numerically computing the period lattice of an arbitrary modular form of weight k ≥ 2. This generalizes the computational work of Cremona for k = 2 and a result about modular symbols due to Manin, for k = 2, to higher weights. I hope to develop these algorithms further, so that it will be possible to computationally verify certain special cases of the Bloch-Kato conjectures. Another approach I have used in computing L-series of fibered varieties is the relationship with the variation of Hodge structure of the fibration. This is related to the study of the Picard-Fuchs differential equation of a family of varieties. The connection between the solutions of the Picard-Fuchs equation and the L-series of the ambient space leads to mod-p congruences for the coefficients of the modular forms, which I wrote about in a Max-Planck preprint. I am also interested in other questions related to modular Galois representations, and recently I have been working with Ian Kiming on the study of modular mod p Galois representations with exceptional image. In the future I intend to continue my work in investigating geometry and arithmetic of modular forms, particularly using computational methods, and developing algorithms for making computations possible which will help verify some of the many conjectures in this area.

       +45 39 61 77 16 • verrill@math.ku.dk • http://www.math.ku.dk/~verrill

TEACHING STATEMENT I have taught a variety of courses at different levels and with different class sizes. When I was a graduate student at Cambridge University I supervised undegraduates for various courses. This is a teching system with very small groups of students, usually two students per tutor. At Queen’s University I taught far larger sized classes, lecturing courses in calculus and linear algebra, as well as giving a more advanced course on hypergeometric functions. I have taught a wide range of students, including mathematics majors, engineering students, and students with very little mathematical background. I use computer programs substantially in my own work, and so it is natural for me to incorporate computers into my courses. I have used MATLAB in teaching linear algebra, and Maple in teaching calculus for engineering students. I also wrote web pages for some of the courses I gave at Queen’s, which were very popular with the students. Some of these can still be seen at: http://hverrill.net/˜ helena/math006/ In teaching, effective communication is of upmost importance. I aim to explain ideas as clearly as possible using a variety of techniques and media as appropriate, such as computer algebra packages, and web pages, as well as lecutres and tutorials. It is also important to listen to the students and understand their needs, aims and background, and how the course is relevant to them. In addition to the formal teaching of required courses, I enjoy getting ideas across in general and sharing my knowledge. I have given several colloquia, including one of the Coleman-Ellis Lectures, a special undergraduate lecture series at Queen’s. I talked about “tilings and patterns”, and introduced some simple group theory, using eye-catching tessellation designs. I’ve also given talks on mathematical and computational problems arising from origami at the computer science departments of Queen’s University, and of the University of Waterloo in Canada. I also gave a talk on mathematical origami at the 1998 Bridges “Mathematical Connections in Art, Music, and Science” Conference, at Southwestern College, Kansas. I also gave a class in mathematical origami at the Art and Math Conference at the Department of Computer Science in Berkeley California in 1998. Also I’ve written popular web pages on recreational mathematics, about which I get positive feed back from many teachers who can use this material at lower levels. The mathematical origami web pages were also recently used as part of the mathematics display of the Kansas State University open house, in April of this year, which was aimed at high school students.

+45 39 61 77 16 • verrill@math.ku.dk • http://www.math.ku.dk/~verrill

REFEREES • Ian Kiming +45 35 32 07 25 Department of Mathematics Universitetsparken 5 DK-2100 Copenhagen Ø Denmark

kiming@math.ku.dk

• Noriko Yui +1 (613) 533.2421 Department of Mathematics Jeffery Hall Queen’s University Kingston, Ontario, Canada K7L 3N6

yui@ny.mast.queensu.ca

• Don Zagier +49 (228) 402.0 Max-Planck-Institut für Mathematik P.O.Box: 7280, D-53072 Bonn Germany

don@mpim-bonn.mpg.de

• (Teaching reference) Peter Taylor +1 (613) 533.2434 Department of Mathematics Jeffery Hall Queen’s University Kingston, Ontario, Canada K7L 3N6

taylorp@post.queensu.ca