Covering the years 2008-2012, this book profiles the life and work of recent winners of the Abel Prize:

· John G. Thompson and Jacques Tits, 2008

· Mikhail Gromov, 2009

· John T. Tate Jr., 2010

· John W. Milnor, 2011

· Endre Szemerédi, 2012.

The profiles feature autobiographical information as well as a description of each mathematician's work. In addition, each profile contains a complete bibliography, a curriculum vitae, as well as photos ― old and new. As an added feature, interviews with the Laureates are presented on an accompanying web site (http://extras.springer.com/).

The book also presents a history of the Abel Prize written by the historian Kim Helsvig, and includes a facsimile of a letter from Niels Henrik Abel, which is transcribed, translated into English, and placed into historical perspective by Christian Skau.

This book follows on The Abel Prize: 2003-2007, The First Five Years (Springer, 2010), which profiles the work of the first Abel Prize winners.

and F : [0, 1]n → Rq is a continuous map, then one of the fibers of F has diameter ≥1. Linear Algebra Theorem 3 If n > q, then any linear map from Rn to Rq has kernel of dimension at least n − q. This theorem is a stronger version of Linear Algebra Theorem 2. In particular, if n is much bigger than q, then the fibers of a linear map Rn → Rq must be very large. The Lebesgue covering lemma and its corollary do not provide a good geometric analogue for this stronger theorem. They say that a

development. The working group’s presentation also highlighted the Nobel Prizes as the obvious ideal for the new prize, both in organizational and economical terms: “If handled properly, an annual Abel Prize might draw great attention and in time achieve the same status as the Nobel Prizes within the other sciences”.21 The responses were welcomed and strongly supported the need to make the importance of mathematics much more visible to the general public. This, many of the respondents maintained,

The set π0 = π0 (G) of prime divisors p of |G| for which G possesses a subgroup isomorphic to Zp × Zp × Zp is introduced and then partitioned into subsets π3 = π3 (G) and π4 = π4 (G). A prime p lies in π3 if and only if some Sylow p-subgroup lies in the normalizer of some nonidentity q-subgroup for some prime q ̸= p. Otherwise p ∈ π4 . (This is not actually their definition but a consequence of their definition.) Such a partition makes sense in any finite group containing Zp × Zp × Zp , and it

non-degenerate 2-form, and a contact structure on a (2n + 1)-dimensional manifold N is a completely non-integrable hyperplane field ξ ⊂ T N . The complete non-integrability can be expressed by the Frobenius condition: if one defines ξ (locally) by a Pfaffian equation α = 0, then the 2-form dα|ξ is non-degenerate, see Sect. 6.1 for more discussion. A formal solution for the corresponding differential relation is a non-degenerate (not necessarily closed) 2-form in the symplectic case, and a pair

-directed if it projects as an immersion to Rn−l = Rn /Rl . Similarly if A = AR ⊂ G2n,n is the set of totally real n-dimensional subspaces in R2n = Cn , then AR -directed embeddings are exactly totally real ones, i.e., whose tangent spaces contain no complex lines. The key role in the proof of the corresponding h-principle for A-directed embeddings plays the following lemma about approximation of a tangential homotopy. Lemma 1 Let i : V → Rn be a k-dimensional, k < n, submanifold in Rn , and K ⊂