By Buekenhout F., Huybrechts C.

We turn out the lifestyles of a rank 3 geometry admitting the Hall-Janko workforce J2 as flag-transitive automorphism team and Aut(J2) as complete automorphism crew. This geometry belongs to the diagram (c·L*) and its nontrivial residues are whole graphs of measurement 10 and twin Hermitian unitals of order three.

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As a result of the existence of compatible complex structures on ι∗ P , lagrangian subbundles of ι∗ P complementary to the image of ι∗ : T L → ι∗ T P always exist. 19 implies the following nonlinear “normal form” result, which states that, near any of its lagrangian submanifolds L, a symplectic manifold looks like a neighborhood of the zero section in T ∗ L. 20 If L is a lagrangian submanifold of P , then a neighborhood of L in P is symplectomorphic to a neighborhood of the zero section in T ∗ L, by a map which is the identity on L.

E. pulled-back) to yield a second-order approximate solution (πL−1 )∗ eiφ/ a to the Schr¨odinger equation on M , as above. This interpretation of the WKB approximation leads us to consider a semi-classical state as a quadruple (L, ι, φ, a) comprised of a projectable, exact lagrangian embedding ι : L → T ∗ M , a “generalized” phase function φ : L → R satisfying dφ = ι∗ αM , and a half-density a on L. Our correspondence table now assumes the form 34 Object Classical version Quantum version basic space T ∗M HM state (L, ι, φ, a) as above on M time-evolution Hamilton’s equations Schr¨odinger equation generator of evolution function H on T ∗ M stationary state ˆ on HM operator H ˆ state (L, ι, φ, a) such that eigenvector of H H ◦ ι = E and LXH,ι a = 0 Since the Hamilton-Jacobi and transport equations above make sense for any triple (L, ι, a) consisting of an arbitrary lagrangian immersion ι : L → T ∗ M and a half-density a on L, it is tempting to regard (L, ι, a) as a further generalization of the concept of semi-classical state in T ∗ M , in which we drop the conditions of projectability and exactness.

In other words, downward motion to the right of the fiber is positive, while the sign changes if either the direction of motion or the side of the fiber is reversed, but not if both are. A circle in the phase plane has a right and a left singular point, both of which are positive according our rule when the circle is traversed counterclockwise. The Maslov index of the circle therefore equals 2. In fact, the same is true of any closed embedded curve traversed counterclockwise. On the other hand, a figure-eight has Maslov index zero.

### A (c. L* )-Geometry for the Sporadic Group J2 by Buekenhout F., Huybrechts C.

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