The identities for elliptic gamma functions discovered by Felder and Varchenko [8] are generalized to an infinite set of identities for elliptic gamma functions associated to pairs of planes in $3$-dimensional space. The language of stacks and gerbes gives a natural framework for a systematic description of these identities and their domain of validity. A triptic curve is the quotient of the complex plane by a subgroup of rank three. (It is a stack.) Our identities can be summarized by saying that elliptic gamma functions form a meromorphic section of a hermitian holomorphic abelian gerbe over the universal oriented triptic curve

We show that a reflexive subspace of the predual of a von Neumann algebra embeds into a noncommutative $L_p$-space for some $p>1$. This is a noncommutative version of Rosenthal's result for commutative $L_p$-spaces. Similarly for $1\le q<2$, an infinite-dimensional subspace $X$ of a noncommutative $L_q$-space either contains ${\ell }_q$ or embeds in $L_p$ for some $q<p<2$. The novelty in the noncommutative setting is a double-sided change of density

Let $P(x,z)=z^d+\sum _{i=1}^d{a}_i(x)z^{d-i}$ be a polynomial, where $a_i$ are real-analytic functions in an open subset $U$ of ${\mathbb{R}}^n$. If, for any $x\in U$, the polynomial $z\mapsto P(x,z)$ has only real roots, then we can write those roots as locally Lipschitz functions of $x$. Moreover, there exists a modification (a locally finite composition of blowups with smooth centers) $\sigma :W\to U$ such that the roots of the corresponding polynomial $\tilde{P}(w,z)=P(\sigma (w),z),w\in W$, can be written locally as analytic functions of $w$. Let $A(x),x\in U$, be an analytic family of symmetric matrices, where $U$ is open in ${\mathbb{R}}^n$. Then there exists a modification $\sigma :W\to U$ such that the corresponding family $\tilde{A}(w)=A(\sigma (w))$ can be locally diagonalized analytically (i.e., we can choose locally a basis of eigenvectors in an analytic way). This generalizes Rellich's well-known theorem (see [32]) from 1937 for $1$-parameter families. Similarly, for an analytic family $A(x),x\in U$, of antisymmetric matrices, there exists a modification $\sigma$ such that we can find locally a basis of proper subspaces in an analytic way

We show that the semistable conjecture of Fontaine and Jannsen (see [9]) is true for proper, vertical, fine, and saturated log-smooth families with reduction of Cartier type (e.g., proper schemes with simple semistable reduction). We derive it from Suslin's comparison theorem [31, Corollary 4.3] between motivic cohomology and étale cohomology. This gives a new proof of the semistable conjecture showing motivic character of p-adic period maps

Let $k$ be a field not of characteristic two, and let $\Lambda$ be a set consisting of almost all rational primes invertible in $k$. Suppose that we have a variety $X/k$ and strictly compatible system $\{M_{\ell }\to X:\ell \in \Lambda \}$ of constructible ${\mathbb{F}}_{\ell }$-sheaves. If the system is orthogonally or symplectically self-dual, then the geometric monodromy group of $M_{\ell }$ is a subgroup of a corresponding isometry group ${\Gamma }_{\ell }$ over ${\mathbb{F}}_{\ell }$, and we say that it has big monodromy if it contains the derived subgroup $D{\Gamma }_{\ell }$. We prove a theorem that gives sufficient conditions for $M_{\ell }$ to have big monodromy. We apply the theorem to explicit systems arising from the middle cohomology of families of hyperelliptic curves and elliptic surfaces to show that the monodromy is uniformly big as we vary $\ell$ and the system. We also show how it leads to new results for the inverse Galois problem

We prove the intersection of Igusa's conjecture in [10, Introduction] and the Denef-Sperber conjecture of [7, page 56] on nondegenerate exponential sums modulo $p^m$