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We review basic notions in the field of information geometry such as Fisher metric on statistical manifold, α-connection and corresponding curvature following Amari's work [1,2,3]. We show application of information geometry to asymptotic statistical inference.

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We formalize a framework for coordinating the funding of projects and sharing the costs among agents with quasi-linear utility functions and individual budgets. Our model contains the classical discrete participatory budgeting model as a special case, while capturing other well-motivated problems. We propose several important axioms and objectives and study how well they can be simultaneously satisfied.One of our main results is that whereas welfare maximization admits an FPTAS, welfare maximization subject to a well-motivated and very weak participation requirement leads to a strong inapproximability result. We show that this result is bypassed if we consider some natural restricted valuations or when we take an average-case heuristic approach.

Let G be a finite connected simple graph. We define the moduli space of conformal structures on G. We propose a definition of conformally covariant operators on graphs, motivated by [25]. We provide examples of conformally covariant operators, which include the edge Laplacian and the adjacency matrix on graphs. In the case where such an operator has a nontrivial kernel, we construct conformal invariants, providing discrete counterparts of several results in [11,12] established for Riemannian manifolds. In particular, we show that the nodal sets and nodal domains of null eigenvectors are conformal invariants. Proposition 6.9. For each (i 1 , i 2 ) ∈ I ×I, Λ i1,i2 (J, F, w) is conformally covariant.Proof. Let D i1 J (u) be the matrix obtained from D J (u) by removing the i 1 st row and i 1 st column, and let D i2 J (u) be the matrix obtained from D J (u) by removing the i 2 nd row and i 2 nd column. Then, it follows from Lemma 6.8 that Λ i1,i2 (J, F,w) = D i1 J (u) · Λ i1,i2 (J, F, w) · D i2 J (u).