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

Offering the newest findings in issues from around the mathematical spectrum, this quantity comprises ends up in natural arithmetic in addition to a variety of new advances and novel purposes to different fields akin to likelihood, records, biology, and computing device technological know-how. All contributions characteristic authors who attended the organization for girls in arithmetic examine Symposium in 2015: this convention, the 3rd in a chain of biennial meetings prepared via the organization, attracted over 330 members and showcased the study of ladies mathematicians from academia, undefined, and government.

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**Example text**

L(k−1)k ] = [L01 ] ◦ . . ◦ [L(k −1)k ] in Mor Symp means by definition that the corresponding morphisms in Symp# are equivalent (L01 , . . , L(k−1)k ) ∼ (L01 , . . 2. Recall that this relation is generated by the geometric composition moves Comp ⊂ Mor Symp# × Mor Symp# , so that there is a sequence of moves from (L01 , . . , L(k−1)k ) to (L01 , . . , L(k −1)k ) in which adjacent pairs are replaced by their embedded geometric composition. Our definition of Cerf ⊂ RelSymp by moves on equivalence classes encoded by Comp translates this into a sequence of Cerf moves from [L01 ] ◦ .

Since its first announcement in [80], this Floer field philosophy has been applied to obtain various proposals for 2 + 1 field theories, which are inspired from various gauge theories. 8. Instead of discussing the technicalities and possible obstructions, this section focusses on the motivations, and thus presents both intuitive and naive reasonings why theories along these lines are to be expected. The intuitive reason for an intimate connection between symplectic geometry and gauge theory in dimensions 2 + 1 is the following example of a partial functor from Bor conn 2+1 to a category of infinite dimensional symplectic Banach spaces and Lagrangian Banach-submanifolds.

6 below to induce a functor Bor conn d+1 → Cat, as claimed. Here, the existence of the Yoneda functor follows from the fact that Sympτ extends to a 2-category. A formal notion of d + 1 Floer field theory should also include a notion of duality. However, the abstract categorical notion of duality requires a monoidal structure— roughly speaking, an associative multiplication of objects that extends to a bifunctor. While in the bordism category Bor d+1 a monoidal structure is naturally given by disjoint unions of objects and morphisms, an extension of the gauge theoretic examples in Sect.