We initiate the study of a class of polytopes, which we coin polypositroids, defined to be those polytopes that are simultaneously generalized permutohedra (or polymatroids) and alcoved polytopes. Whereas positroids are the matroids arising from the totally nonnegative Grassmannian, polypositroids are “positive” polymatroids. We parametrize polypositroids using Coxeter necklaces and balanced graphs, and describe the cone of polypositroids by extremal rays and facet inequalities. We introduce a notion of $(W,c)$-polypositroid for a finite Weyl group W and a choice of Coxeter element c. We connect the theory of $(W,c)$-polypositroids to cluster algebras of finite type and to generalized associahedra. We discuss membranes, which are certain triangulated 2-dimensional surfaces inside polypositroids. Membranes extend the notion of plabic graphs from positroids to polypositroids.

We study the ring of regular functions on the space of planar electrical networks, which we coin the grove algebra. This algebra is an electrical analog of the Plücker ring studied classically in invariant theory. We develop the combinatorics of double groves to study the grove algebra, and find a quadratic Gröbner basis for the grove ideal.

Mounting evidence suggests that the first few months of life are critical for the development of obesity. The relationships between the timing of solid food introduction and the risk of childhood obesity have been examined previously; however, evidence for the association of timing of infant formula introduction remains scarce. This study aimed to examine whether the timing of infant formula introduction is associated with growth z-scores and overweight at ages 1 and 3 years. This study included 5733 full-term (≥ 37 gestational weeks) and normal birth weight (≥ 2500 and < 4000 g) children in the Born in Guangzhou Cohort Study, a prospective cohort study with data collected at 6 weeks, 6, 12 and 36 months. Compared with infant formula introduction at 0–3 months, introduction at 4–6 months was associated with the lower BMI, weight-for-age and weight-for-length z-scores at 1 and 3 years old. Also, introduction at 4–6 months was associated with the lower odds of at-risk of overweight at age 1 (adjusted OR 0·72, 95 % CI 0·55, 0·94) and 3 years (adjusted OR 0·50, 95 % CI 0·30, 0·85). Introduction at 4–6 months also decreased the odds of overweight at age 1 year (adjusted OR 0·42, 95 % CI 0·21, 0·84) but not at age 3 years. Based on our findings, compared with introduction within the first 3 months, introduction at 4–6 months has a reduction on later high BMI risk and at-risk of overweight. However, these results need to be replicated in other well-designed studies before more firm recommendations can be made.

We study the back stable Schubert calculus of the infinite flag variety. Our main results are:

• – a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part;

• – a novel definition of double and triple Stanley symmetric functions;

• – a proof of the positivity of double Edelman–Greene coefficients generalizing the results of Edelman–Greene and Lascoux–Schützenberger;

• – the definition of a new class of bumpless pipedreams, giving new formulae for double Schubert polynomials, back stable double Schubert polynomials, and a new form of the Edelman–Greene insertion algorithm;

• – the construction of the Peterson subalgebra of the infinite nilHecke algebra, extending work of Peterson in the affine case;

• – equivariant Pieri rules for the homology of the infinite Grassmannian;

• – homology divided difference operators that create the equivariant homology Schubert classes of the infinite Grassmannian.

The (tree) amplituhedron $\mathcal {A}_{n,k,m}(Z)$ is a certain subset of the Grassmannian introduced by Arkani-Hamed and Trnka in 2013 in order to study scattering amplitudes in $N=4$ supersymmetric Yang–Mills theory. Confirming a conjecture of the first author, we show that when $m$ is even, a collection of affine permutations yields a triangulation of $\mathcal {A}_{n,k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(k+m,n)$ if and only if the collection of their inverses yields a triangulation of $\mathcal {A}_{n,n-m-k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(n-k,n)$. We prove this duality using the twist map of Marsh and Scott. We also show that this map preserves the canonical differential forms associated with the corresponding positroid cells, and hence obtain a parity duality for amplituhedron differential forms.