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 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 resembling likelihood, information, biology, and desktop technological know-how. All contributions characteristic authors who attended the organization for ladies in arithmetic learn Symposium in 2015: this convention, the 3rd in a sequence of biennial meetings geared up through the organization, attracted over 330 members and showcased the study of ladies mathematicians from academia, undefined, and government.

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Extra resources for Advances in the Mathematical Sciences: Research from the 2015 Association for Women in Mathematics Symposium

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For Yij ∪Σj Yjl = Y˜ ij ∪Σ˜ j Y˜ jl , . . ∪Σ1 Yij ∪Σj Yjl ∪Σl . . = . . ∪Σ1 Y˜ il ∪Σl . . for Yij ∪Σj Yjl = Y˜ il , . . ∪Σ1 Yil ∪Σl . . = . . ∪Σ1 Y˜ ij ∪Σ˜ j Y˜ jl ∪Σl . . for Yil = Y˜ ij ∪Σ˜ j Y˜ jl . In the following, we will cast this notion—decompositions into simple pieces that are unique up to a set of moves—into more formal terms. For that purpose, we denote the union of all morphisms of a category C by Mor C := x1 ,x2 ∈ObjC Mor C (x1 , x2 ), and we denote all relations between composable chains14 of morphisms by (fi ), (gj ) ∈ (Mor C )k × (Mor C ) RelC := f1 ◦ .

H(n−1)n ∈ SMor such that m = h12 ◦ . . , any two presentations of the same morphism in terms of h12 , . . , h(n−1)n ∈ SMor and h˜ 12 , . . , h˜ (˜n−1)˜n ∈ SMor are related by a finite sequence15 of identities h12 ◦ . . ◦ h(n−1)n = h12 ◦ . . ◦ h(n −1)n = . . = h˜ 12 ◦ . . ◦ h˜ (˜n−1)˜n in which each equality replaces one subchain of simple morphisms by another, . . ◦ f12 ◦ . . ◦ f(k−1)k ◦ . . = . . ◦ g12 ◦ . . ◦ g( −1) according to a local Cerf move (f12 , . . , f(k−1)k ), (g12 , .

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 ] ◦ . . [L(k−1)k ] to [L01 ] ◦ . . [L(k −1)k ]. 4 Construction Principle for Floer Field Theories The algebraic background of Floer field theory is the following construction principle for functors between categories with Cerf decompositions.

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