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 comprises ends up in natural arithmetic in addition to a number of new advances and novel functions to different fields reminiscent of likelihood, records, biology, and desktop technology. 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 geared up through the organization, attracted over 330 members and showcased the learn of ladies mathematicians from academia, undefined, and government.

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

**Sample text**

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 ◦ . . ◦ fk = g1 ◦ .

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 , . . , g( ◦ ... −1) ) ∈ Cerf. The bordism categories Bor d+1 are the motivating example of categories with Cerf decompositions, with SMor and Cerf given by the simple cobordisms and Cerf 14 Throughout, we will use the term “composable chain” to denote ordered tuples of morphisms, in which each consecutive pair is composable, so that the entire tuple—by associativity of composition—has a well defined composition.

Composition is by gluing via boundary identifications as in Bor d+1 . If we allow Σ = ∅ as object, then closed, connected, oriented d + 1-manifolds are contained in this category as morphisms from ∅ to ∅. In this language, the Cerf decomposition theorem for 3-manifolds—in the connected case proven in [26] and reviewed in [27]—can be stated as in the following theorem, and is illustrated in Fig. 5. Here, in strict categorical language, a 3-cobordism from Σ− to Σ+ is an equivalence class [(Y , ι− , ι+ )] of 3-cobordisms and embeddings ι± : Σ± → ∂Y modulo diffeomorphisms relative to the boundary identifications ι± .