The goal of our workshop was to explore the various initiatives currently underway in Europe aimed at developing exceptional mathematical talent at the secondary level. The outcome we hoped for was the creation of an immersion-based mathematics programme in Europe that would extend the work of this year’s CMI-PROMYS program while complementing ongoing work of others.

With the creation of PROMYS Europe, the desired outcome of the workshop has been achieved. We look forward to working with talented young mathematicians in Europe and to continuing to collaborate with the other programmes and institutions which serve them. PROMYS is grateful to the participants in the Oxford Workshop on Developing Exceptional Talent in Mathematics and to the Clay Mathematics Institute (CMI) which made possible both the workshop and PROMYS Europe.

**Organizers:**

David Conlon (University of Oxford)

Joshua Greene (COMAC Capital LLP)*

Jürg Kramer (Humboldt University of Berlin)

Dierk Schleicher (Jacobs University)

Glenn Stevens (Boston University)

Nicholas Woodhouse (CMI)

**Invited Participants:**Dan Abramson (King's College London Mathematics School)

Martin Andler (Animath)

Byoung Tae Bae (University of Cambridge)*

Marjory Baruch (Syracuse University)

Christine Binns (Mathematics Mastery)

Cecilia Busuioc (Royal Holloway, University of London)*

Michael Davies (Westminster School)

David DeRemer (ECARES / ULB)*

Helen Drury (ARK Schools)

Andrew Furnas (University of Leeds)*

Eugenio Hernandez (ESTALMAT)

Kevin Hughes (Edinburgh University)*

Po-Shen Loh (Carnegie Mellon University)

Frances Kirwan (University of Oxford)

David Miller (Google)*

Vicky Neale (University of Cambridge)

Corina Panda (Leiden University)*

Owen Patashnick (University of Bristol)*

Alice Rogers (King's College London)

Mercedes Sanchez (ESTALMAT)

Claudia Scheimbauer (ETH Zurich)*

Geoff Smith (University of Bath)

Jackie Thompson (University of Oxford)*

Günter M. Ziegler (Freie Universität of Berlin)

* = PROMYS alum

**Friday, 16 August:**

12:00 - Arrivals and Registration at Somerville College, Woodstock Road

(you may leave your luggage here)

13:00-14:00 Lunch with Students from the Oxford Masterclasses at Wadham College

14:00-17:00 Visit Masterclasses, Meet with PROMYS Alumni in New Seminar Room A at Wadham College

17:00-18:00 Lecture for Students and Visitors to the Masterclasses:

Günter M. Ziegler (Freie Universität Berlin, Bio, Abstract):

*Cannons at Sparrows: Cutting Polygons, and What That Could Lead To … * Room L3 in Mathematical Institute at 24-29 St. Giles'

18:00-18:30 Wine Reception in the Common Room at the Mathematical Institute

Participants make their own dinner arrangements

**Saturday, 17 August:** All meetings in room L3 in Mathematical Institute at 24-29 St. Giles'

9:00-9:30 Tea and Coffee. Opening Remarks: Nick Woodhouse (Bio)

9:30-10:30 Overview of the 2013 CMI-PROMYS Alliance Initiative

Panelists: Glenn Stevens (PROMYS, Bio,) and

David Conlon (Oxford Masterclasses, Bio) Abstract:

*Overview of the 2013 CMI-PROMYS International Alliance Initiative *

10:45-11:45 Advanced Maths Beyond the Classroom

Panelists: Dierk Schleicher (ISSMYS, Bio, Abstract):

The 'Modern Mathematics' International Summer School for Students

Eugenio Hernandez (ESTALMAT, Bio, Abstract): * ESTALMAT: An Enrichment Program to Develop Mathematical Talent in Spain*

Martin Andler (Animath. Bio, Abstract):

12:00-13:45 Lunch

14:00-15:00 Olympiad Training and Mentoring

Panelists: Geoff Smith (UKMT, Bio, Abstract): *Mathematics Competitions and Enrichment*

Vicky Neale (CMEP, NRICH, EGMO, Bio, Abstract): *CMEP, NRICH and EGMO*

15:15-16:15 Teachers and Schools

Panelists: Helen Drury (ARK, Bio)

Michael Davies (Westminster School, Bio, Abstract): * The Top Few and the Top Fifth*

Dan Abramson* *(King’s College Maths School, Bio, Abstract): * King's College London Mathematics School: the Challenges* Discussant: Joshua Greene (COMAC, London, Bio)

16:30-17:30 Special Lecture: Alice Rogers (King's College, London, Bio, Abstract): * Mathematical Talent and the Regular Classroom Diet*

19:30 - Pre-Dinner Drinks in the Somerville Gardens and Dinner in the Reading Room

at Somerville College, Woodstock Road

**Sunday, 18 August: Next Steps -** Meetings in room L3 in Mathematical Institute at 24-29 St. Giles'

9:00 - 9:30 Tea and Coffee

9:30 - 10:30 Special Lecture: Jürg Kramer (DMV, Bio, Abstract): *Interaction between Research Mathematics, Mathematics Teacher Training, *

10:45 -12:30 Where Do We Go From Here?

Open Discussion for All Participants

Dan Abramson is the first Head of the new King’s College London Mathematics School. The new Government-funded specialist mathematics school is due to open in 2014 and will involve a groundbreaking partnership of university and school. The new school will accept mathematically gifted students and will be for sixth form only (the last two years of high school).

Dan has been Head of Mathematics at Highgate School in North London for the past six years. He established and led the school’s pioneering outreach program for teaching gifted students in local state schools. Dan earned a first class degree in mathematics from Cambridge University as well as a Certificate of Advanced Study in Mathematics. He has a notable record of excellence both in teaching and in innovative curriculum design.

Martin Andler is Professor of Mathematics at the University of Versailles Saint-Quentin (UVSQ) where he has been chairperson of the mathematics department and a board member and is presently the chairperson of the scientific committee of the school of science. Martin is Chairman of Animath, an organization in France which promotes mathematics for young people of which he was one of the founders in 1998. His main research interests are representation theory of Lie groups, the history of 20th century mathematics, science communication, science education and science in the media.

Martin received his doctorate from the Université Paris 7 and then pursued teaching and research at CNRS, Université Paris 7, Ecole Normale Supérieure, Rutgers University, the Institute for Advanced Study and MIT. He has held a range of positions including Editor of the Gazette des Mathématiciens, and Vice President of the Société Mathématique de France. He is Vice President of Euroscience.

David Conlon is a University Lecturer in Discrete Mathematics at Oxford University, a Tutorial Fellow in Wadham College and a Royal Society University Research Fellow. Before this, he held a Royal Society University Research Fellowship in DPMMS and was a College Research Associate at St John's College, Cambridge. From 2007 to 2010, David was a Junior Research Fellow at St John's College. Every summer between 2004 and 2008, David taught a three-week course on theoretical physics at the Centre for Talented Youth in Ireland.

David earned his PhD from Cambridge University under the supervision of Professor W.T. Gowers. David’s research interests are extremal and probabilistic combinatorics, particularly extremal graph theory, Ramsey theory, random structures, quasirandomness and additive combinatorics. He won the European Prize in Combinatorics in 2011.

David is a member of the Organizing Committee for the CMI-PROMYS International Alliance and is the Director of the new Oxford Masterclasses in Combinatorics, August 10 – 17, 2013.

Michael Davies has been Head of Mathematics at Westminster School in London for more than twenty years. Westminster School is a selective day and boarding school for boys aged 13 to 18 and girls aged 16 to 18, which aims to be one of the foremost centres of academic excellence in the country. The school's mathematical policy is one of enrichment rather than acceleration and aims to make mathematics lessons opportunities for the collaborative development of ideas, not simply for instruction in set methods. Many Westminster students enter national mathematical challenges and competitions, and a number have reached high levels in the Mathematical Olympiads, but the school is very keen for all students to tackle problems that they find demanding and to learn to deal independently with appropriate mathematical challenges. Most Mathematics teachers at Westminster come into the profession direct from University, without teaching experience and Michael has therefore mentored many new entrants to the teaching profession. He was also a STEP (Cambridge Mathematics entry) examiner for a number of years, so he knows how hard it is to set questions which are challenging but not impossible for bright students.

Helen Drury has taught mathematics and studied mathematics education since 2002. Whilst remaining a classroom practitioner, she carried out research with Oxford University and the Open University, and was awarded a PhD in Mathematics Education.

Helen has been Director of Mathematics for ARK Schools from September 2010, leading the development and introduction of the Mathematics Mastery curriculum and developing the network’s approach to professional development in mathematics. With teaching and leadership experience in diverse inner city comprehensives, Helen is concerned with how to bring effective research-based approaches into schools to close the attainment gap and ensure success for every child. The Education Endowment Foundation has funded Mathematics Mastery training, curriculum materials and independent evaluation for 90 schools, of which 36 began implementing the approach this academic year.

ARK Schools is a network of high-achieving, non-selective schools and one of the country’s top-performing academy groups operating 18 academies in London, Birmingham and Portsmouth, educating around 9,000 pupils. ARK's aim is to create outstanding schools that give every pupil the opportunity to go to university or pursue the career of their choice.

Joshua Greene is a partner and the head of research and analysis at COMAC Capital LLP. COMAC Capital is a leading global macro hedge fund manager based in Europe.

Joshua earned a joint AM/AB summa cum laude in mathematics from Harvard University and is a CFA Charterholder. He has a deep longstanding interest and involvement in mathematical education both in the UK, where he currently lives, and in the US. He is a Trustee of the PROMYS Foundation.

Joshua is a member of the Organizing Committee for the CMI-PROMYS International Alliance.

Eugenio Hernandez is Professor of Mathematical Analysis at the Autonomous University of Madrid. His research interests are harmonic analysis, Fourier transform, wavelets, image compression, and non-linear approximation of functions. Eugenio earned his PhD from Washington University in St. Louis in 1981.

Since 2004, Eugenio has coordinated Estalmat, which detects, teaches, guides, and encourages the talents of 12- to 15-year-old students in Spain who are exceptionally gifted in mathematics. Eugenio has published extensively, written several undergraduate textbooks, and been the recipient of numerous research grants related to the detection, support, and development of exceptional mathematical talent. of young students. Eugenio has also organized conferences, meetings and courses on the stimulation of precocious mathematical ability.

Jürg Kramer is Professor of Mathematics at the Humboldt University of Berlin. His research interests focus on problems in Arakelov geometry and on the theory of automorphic forms, in particular the theory of modular forms as well as problems at the interface of these two fields.

Jürg’s educational activities are devoted to the education and training of mathematics teachers and to the advancement of mathematically talented and mathematically interested high school students. Jürg has launched many initiatives for subject-oriented as well as more practice-oriented teacher training. In particular, the Berlin Network of Schools Specializing in Mathematics and the Sciences has been established for the advancement of mathematically talented high school students. Jürg’s current projects include being Deputy Chair of the Berlin Mathematical School (BMS), Director of the German Center for Mathematics Teacher Education (DZLM), and member of the Executive Board of the DFG Research Center MATHEON. Presently, Jürg is President of the German Mathematical Society (DMV).

Jürg is a member of the Organizing Committee for the CMI-PROMYS International Alliance.

Vicky Neale is a Senior Teaching Associate in the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge, where she is a Director of the Cambridge Mathematics Education Project (CMEP). She is also a Fellow and the Director of Studies in Mathematics at Murray Edwards College, University of Cambridge. Vicky earned her PhD at Trinity College, Cambridge, having done research in analytic number theory and additive combinatorics.

In addition to giving undergraduate lectures and looking after her undergraduate mathematicians at Murray Edwards, Vicky spends much of her time working on CMEP, creating resources and working with colleagues to develop the project. She also works closely with colleagues from NRICH, creating some of the site’s resources for teachers and students and helping to moderate Ask NRICH, a free online maths discussion forum.

Vicky is involved with several other Cambridge-based outreach activities, such as the Millenium Mathematics Project (MMP), the Cambridge Science Festival, and the Further Maths Support Programme, for whom Vicky has led workshops. She dreamed up and, with others, launched the Cambridge Maths Circle for children and young people aged 5 to 18. She is committed to the university's efforts to widen access to students from all backgrounds, and regularly gives lectures and leads workshops for groups of visiting students at open days and summer schools, including the Sutton Trust Maths summer school in Cambridge (for 17-year olds from disadvantaged backgrounds thinking of applying to study Maths at Cambridge).

Vicky is a mentor on the UK Mathematics Trust (UKMT) Senior Mentoring Scheme and has been involved with the British Mathematical Olympiad (BMO) and the National Mathematics Summer School in various ways including directing several summer schools for students aged 15 and 16. Vicky was on on the organising committee of the inaugural European Girls' Mathematical Olympiad (EGMO), held at Murray Edwards College in April 2012. She has also led Royal Institution masterclasses.

Vicky is one of the London Mathematical Society's popular lecturers in 2013, and was a guest on BBC Radio 4's 'In Our Time' programme in October 2012.

Vicky has a blog, Theorem of the Week, aimed at people who are not expert mathematicians and at her undergraduate students.

Alice Rogers is Professor of Mathematics and a member of the Theoretical Physics Group in the Mathematics Department at King’s College London. She obtained her Ph.D. from Imperial College in 1981 with a thesis on supermanifolds. Her first degree is in mathematics from the University of Cambridge. Alice’s publications relate to geometrical and analytic aspects of supermanifolds together with applications in theoretical physics, and she is the author of a book entitled ‘Supermanifolds: Theory and Applications’. From 2001 until 2004, Alice was Head of Department.

As a member from 2007-2011 of the Advisory Committee on Mathematics Education (ACME), Alice has been involved in national education policy. She is currently the Education Secretary of the London Mathematical Society. She also co-runs the King's Factor, which gives sixth form students taking mathematics A-level the opportunity to tackle challenging problems in mathematics which enrich and develop their mathematical thinking.

Dierk Schleicher is Professor of Mathematics at Jacobs University Bremen. He obtained his PhD at Cornell University and held visiting positions in Berkeley, Stony Brook, Paris, Toronto, and München. His main research interests are in dynamical systems and chaos, especially in holomorphic dynamics and the Mandelbrot set, and the dynamics of Newton's root-finding method.

Dierk was one of the chief organizers of the 50th International Mathematical Olympiad (IMO) in 2009 and was featured as Math Activist of the Month in July 2009 by the German Mathematical Society. He is one of the initiators of the "Modern Mathematics" international summer school for students that has so far taken place three times (2011 Breman, 2012 Lyon, 2013 Bremen).

Dierk is a member of the Organizing Committee for the CMI-PROMYS International Alliance.

Geoff Smith (email: G.C.Smith@bath.ac.uk) is a Senior Lecturer in Mathematics at the University of Bath, where he works in group theory and geometry. He attended Keble College, Oxford, the University of Warwick and the University of Manchester, where he gained a Ph.D in 1983.

Geoff has just returned from IMO 2013 in Colombia, where the UK IMO team 2013 achieved its highest finish since 1996, coming 9th out of 97 participating countries. He has led the UK IMO team for 10 of the past 12 years.

Geoff is one of the five elected members of the IMO Advisory Board, and in 2014 he is one of the two candidates contesting the IMO Chair (the senior position in the competition) for the following four years. He is also vice-chair of the United Kingdom Mathematics Trust (UKMT), the organization which administers all the nationwide mathematics competitions in secondary schools in the UK. He chairs the British Mathematical Olympiad (BMO), the hardest domestic UK mathematics competition.

Over recent years, he has facilitated a substantial expansion in the range of international maths competitions in which the UK participates. It now includes the Balkan Mathematical Olympiad, the Romanian Master of Mathematics competition and the European Girls' Mathematical Olympiad. The last competition was founded by UKMT in 2012, and Geoff sits on its Advisory Board (Council). Geoff has also been involved in the expansion of the UK training regime for mathematics competitions, with four national mentoring systems, annual camps in Trinity College, Cambridge; The Queen's College, Oxford; Oundle School and two joint international camps with both Australia and Hungary.

He was awarded an MBE in June 2011 for services to mathematics education, the citation mentioning his work in supporting Royal Institution Mathematics Masterclasses over 20 years.

**Glenn Stevens **

Glenn Stevens has been Director of PROMYS since he co-founded the program in 1989. He is Professor of Mathematics at Boston University where he has taught and conducted research since 1984. He earned his Ph.D. in Mathematics from Harvard University in 1981. His research specialties are Number Theory, Automorphic Forms, and Arithmetic Geometry. He has authored or edited three books and published numerous articles on these topics. Glenn has organized two major research conferences including the Conference on Modular Forms and Fermat's Last Theorem held at Boston University in 1995. Glenn is Principal Investigator of the NSF-funded Focus on Mathematics Math and Science Partnership and co-Principal Investigator of the NSF Noyce grant, Math for America Boston: Teaching Scholars Program. He is also President of Math for America Boston.

Glenn is a member of the Organizing Committee for the CMI-PROMYS International Alliance.

Nick Woodhouse is President of the Clay Mathematics Institute, Professor of Mathematics and a Senior Research Fellow of Wadham College. He was Chairman of Mathematics at Oxford University. Up until 2009, he was Treasurer of the London Mathematical Society. Nick earned his Ph.D. at King’s College, London. His research interests are twistors and the isomonodromy deformation problem, geometric quantization, and general relativity.

Nick is a member of the Organizing Committee for the CMI-PROMYS International Alliance.

Günter M. Ziegler is Professor of Mathematics at Freie Universität, Berlin. His research interests connect discrete and computational geometry (especially polytopes), algebraic and topological methods in combinatorics, discrete mathematics, and the theory of linear and integer programming. He earned his PhD from MIT.

Günter’s many honors include the Leibniz Prize from the German Research Foundation DFG, the Chauvenet Prize from the Mathematical Association of America, and the Communicator Award from DFG and Stifterverband. Günter’s writings include “Proofs from THE BOOK” which has now been published in fourteen languages. From 2006 – 2008, Günter was the President of the German Mathematical Society (DMV). He currently heads the Media Office of the DMV and the School University Network Office, which coordinates the nationwide student and teacher activities of the DMV.

**Dan Abramson: ***King's College London Mathematics School: the challenges*

Dan Abramson will explore the challenges of setting up this new school for highly motivated students aged 16-19 with a particular aptitude and enthusiasm for mathematics. The school aims to be a place for the brightest and the best as well as to improve access to high quality mathematics teaching in London. The challenges include recruiting and selecting pupils, developing a successful partnership between school and university, and defining a curriculum that is engaging, challenging and effective.

**Martin Andler: ***Animath*

Animath was created in 1998, on the initiative of the Société mathématique de France, with two roles, to « promote mathematical activities among young people, in all possible forms: science fair, competitions, clubs... in middle and high schools, while developing the pleasure to do mathematics ». It would both offer some activities of its own and serve as an umbrella organisation. Since 1998, Animath has grown, fulfilling some of its initial ambitions and adding some new items to its initial agenda.

In the presentation, I will focus on three aspects :

1° how and why we received a 3M€ grant from the government in 2011, under the name « Cap'Maths »;

2° give an overview of what Animath and sister organisations are doing;

3° discuss the goals of our action.

**Michael Davies**:* The Top Few and the Top Fifth*

Is there a discontinuity between the mathematical education of the exceptionally talented and that of other strong students? I will suggest, on the contrary, that the same aims should apply when teaching mathematics to a substantial proportion of, if not all, young people as applies to the exceptional few; discuss what this means in practice and describe the sort of resources which need to be available if one wants to teach in this way.

**Eugenio Hernandez ** *ESTALMAT: A enrichment program to develop mathematical talent in Spain *

ESTALMAT is a program started by Professor Miguel de Guzmán (1936-2004) in 1998 in the region of Madrid with 25 students. Since then, it has extended to several regions in Spain, selecting around 250 students each year. The objective is to detect, stimulate and guide the talent of young mathematically gifted children. I will present the key points of the selection process and the structure of the program.

**Jürg Kramer****: ***Interaction between research mathematics, mathematics teacher training, and mathematics education at schools*.

In our presentation we will show how our activities for the advancement of mathematical talents reach from the primary school via the lower secondary to the upper secondary school level. These activities find their continuation into the fast track program of the Berlin Mathematical School. This concept enables us to connect our educational work with the manifold projects for research mathematics, as well as with our projects for the education and the training of mathematics teachers.

**Vicky Neale: ***CMEP, NRICH and EGMO*

I'll give a very quick introduction to three projects with which I have been involved: the Cambridge Mathematics Education Project (CMEP), the NRICH project, and the European Girls' Maths Olympiad (EGMO).

**Alice Rogers**: *Mathematical Talent and the Regular Classroom Diet*

Most schools will from time to time have an exceptionally talented pupil who they must provide for in a school which, even if selective, will have pupils of average or near average as well as some who are strong but not exceptional at mathematics. I will consider some of the issues faced by schools in providing for those of high talent alongside their general responsibilities for teaching mathematics to all their pupils.

**Dierk Schleicher: ***The 'Modern Mathematics' International Summer School for Students*

The "Modern Mathematics" International Summer School for Students intends to bring together some of the leading international mathematicians of today and (hopefully) of tomorrow. It takes place some 10 days during the summer, alternating in location between Bremen (Germany) and Lyon (France) and is open to highly talented students from around the world that are old enough to understand serious mathematics and young enough so they are not yet specialized. We try to invite some of the most inspiring active research mathematicians.

This summer school is a joint French-German initiative and funded by the Clay Mathematics Foundation, the German Volkswagen Stiftung, and the French Excellence Initiative.

The story told in this lecture starts with an innocuous little geometry problem, posed in a September 2006 blog entry by R. Nandakumar, an engineer from Calcutta, India: “Can you cut every polygon into a prescribed number of convex pieces that have equal area and equal perimeter?” This little problem is a “sparrow”, tantalizing, not as easy as one could perhaps expect, and Recreational Mathematics: of no practical use.

I will sketch, however, how this little problem connects to very serious mathematics: For the modelling of this problem we employ insights from a key area of Applied Mathematics, the Theory of Optimal Transportation, which leads to weighted Voronoi diagrams with prescribed areas. This will set up the stage for application of a major tool from Very Pure Mathematics, known as Equivariant Obstruction Theory. This is a “cannon”, and we'll have fun with shooting with it at the sparrow.

On the way to a solution, combinatorial properties of a very classical geometric object, the permutahedron, turn out to be essential. These will, at the end of the story, lead us back to India, with some time travel 100 years into the past: For the last step in our (partial) solution of the sparrows problem we need a simple divisibility property for the numbers in Pascal’s triangle, which was first observed by Balak Ram, in Madras 1909.

But even if the existence problem is solved, the little geometry problem is not: If the solution exists, how do you find one? This problem will be left to you. Instead, I will comment on the strained relationship between cannons and sparrows, and to this avail quote a poem by Hans Magnus Enzensberger.

** **