Advances in the Mathematical Sciences: Research from the by Gail Letzter, Kristin Lauter, Erin Chambers, Nancy Flournoy,

By Gail Letzter, Kristin Lauter, Erin Chambers, Nancy Flournoy, Julia Elisenda Grigsby, Carla Martin, Kathleen Ryan, Konstantina Trivisa

Proposing the most recent findings in themes from around the mathematical spectrum, this quantity contains ends up in natural arithmetic in addition to various new advances and novel functions to different fields akin to chance, data, biology, and laptop technology. All contributions function authors who attended the organization for girls in arithmetic study Symposium in 2015: this convention, the 3rd in a sequence of biennial meetings geared up by means of the organization, attracted over 330 members and showcased the study of ladies mathematicians from academia, undefined, and government.

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G. 4. , a natural factorization of the category CM×M that is associated to a Cartesian product of symplectic manifolds. However, our construction of the symplectic 2-category and the induced functor in Sect. 5 is such that the objects of CM×M are general Lagrangian submanifolds of M × M , not just split Lagrangians L × L ⊂ M × M arising from objects L ∈ ObjCM and L ∈ ObjCM in the categories associated to the factors of the Cartesian product. Homological algebra allows one to formulate a sense in which refined versions of these categories may be equivalent, CM×M ∼ CM ⊗ CM , but it would likely require significant restrictions on the geometry of the symplectic manifolds M, M .

1, functoriality then requires F ([Y ]) = Lι−Y , LY01 , LY12 , . . , LY(n−1)n , L(ι+Y )−1 to be given by the algebraic composition in Symp of the corresponding Lagrangian submanifolds. 1—but also allow for a diffeomorphism Ψ : Y → Z that intertwines ± boundary identifications, Ψ ◦ ι± Y = ιZ . The latter induces a Cerf decomposition Z = Ψ (Y01 ) ∪Ψ (Σ1 ) Ψ (Y12 ) ∪ . . ∪Ψ (Σn−1 ) Ψ (Y(n−1)n ) with Σi := Y(i−1)i ∩ Yi(i+1) ⊂ Y , whose value under F is F ([Z]) = = LΨ |∂ − Y ◦ι−Y , LΨ (Y01 ) , LΨ (Y12 ) , .

While step 1 fixes the functor F on all objects, steps 2 and 3 fix explicit Lagrangians F ([Y ]) = L Y only for simple morphisms Y as LZφ = Lφ for cylindrical cobordisms, LYα = Lα for 2-handle attachments, and LYα− = LαT for their adjoint 1-handle attachments. To determine the value of the functor F ([Y ]) = [L Y ] on a general cobordism Y ∈ Mor Bor2+1 (Σ, Σ ), we choose a Cerf decomposition Y = Y01 ∪Σ1 Y12 . . ∪Σk−1 Y(k−1)k into a composable chain of simple morphisms Yij ∈ Mor Bor2+1 (Σi , Σj ) from Σ0 = Σ to Σk = Σ .

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