Volume 2017, Issue 732

Suppose N is an affine SL⁢(2,ℝ){\mathrm{SL}(2,{\mathbb{R}})}-invariant submanifold of the moduli space of pairs (M,ω)(M,\omega), where M is a curve, and ω is a holomorphic 1-form on M . We show that the Forni bundle of N (i.e. the maximal SL⁢(2,ℝ){\mathrm{SL}(2,{\mathbb{R}})}-invariant isometric subbundle of the Hodge bundle of N ) is always flat and is always orthogonal to the tangent space of N . As a corollary, it follows that the Hodge bundle of N is semisimple.

We develop an analytic theory of operator-valued additive free convolution in terms of subordination functions. In contrast to earlier investigations our functions are not just given by power series expansions, but are defined as Fréchet analytic functions in all of the operator upper half plane. Furthermore, we do not have to assume that our state is tracial. Combining this new analytic theory of operator-valued free convolution with Anderson’s selfadjoint version of the linearization trick we are able to provide a solution to the following general random matrix problem: Let X1(N),…,Xn(N){X_{1}^{(N)},\dots,X_{n}^{(N)}} be selfadjoint N×N{N\times N} random matrices which are, for N→∞{N\to\infty}, asymptotically free. Consider a selfadjoint polynomial p in n non-commuting variables and let P(N){P^{(N)}} be the element P(N)=p⁢(X1(N),…,Xn(N)){P^{(N)}=p(X_{1}^{(N)},\dots,X_{n}^{(N)})}. How can we calculate the asymptotic eigenvalue distribution of P(N){P^{(N)}} out of the asymptotic eigenvalue distributions of X1(N),…,Xn(N){X_{1}^{(N)},\dots,X_{n}^{(N)}}?

We establish several close links between the Galois structures of a range of arithmetic modules including certain natural families of ray class groups, the values at strictly positive integers of p -adic Artin L -series, the Shafarevich–Weil Theorem and the conjectural surjectivity of certain norm maps in cyclotomic ℤp{\mathbb{Z}_{p}}-extensions. Non-commutative Iwasawa theory and the theory of organising matrices play a key role in our approach.

We prove in this paper the Ax–Lindemann–Weierstraß theorem for all mixed Shimura varieties and discuss the lower bounds for Galois orbits of special points of mixed Shimura varieties. In particular, we reprove a result of Silverberg [57] in a different approach. Then combining these results we prove the André–Oort conjecture unconditionally for any mixed Shimura variety whose pure part is a subvariety of 𝒜6n{\mathcal{A}_{6}^{n}} and under the Generalized Riemann Hypothesis for all mixed Shimura varieties of abelian type.

We prove an explicit formula for the total Chern character of the Verlinde bundle of conformal blocks over ℳ¯g,n\overline{\mathcal{M}}_{g,n} in terms of tautological classes. The Chern characters of the Verlinde bundles define a semisimple CohFT (the ranks, given by the Verlinde formula, determine a semisimple fusion algebra). According to Teleman’s classification of semisimple CohFTs, there exists an element of Givental’s group transforming the fusion algebra into the CohFT. We determine the element using the first Chern class of the Verlinde bundle on the interior ℳg,n{\mathcal{M}}_{g,n} and the projective flatness of the Hitchin connection.

Given a locally compact quantum group 𝔾{\mathbb{G}}, we define and study representations and C∗{\mathrm{C}^{\ast}}-completions of the convolution algebra L1⁢(𝔾){L_{1}(\mathbb{G})} associated with various linear subspaces of the multiplier algebra Cb⁢(𝔾){C_{b}(\mathbb{G})}. For discrete quantum groups 𝔾{\mathbb{G}}, we investigate the left regular representation, amenability and the Haagerup property in this framework. When 𝔾{\mathbb{G}} is unimodular and discrete, we study in detail the C∗{\mathrm{C}^{\ast}}-completions of L1⁢(𝔾){L_{1}(\mathbb{G})} associated with the non-commutative Lp{L_{p}}-spaces Lp⁢(𝔾){L_{p}(\mathbb{G})}. As an application of this theory, we characterize (for each p∈[1,∞){p\in[1,\infty)}) the positive definite functions on unimodular orthogonal and unitary free quantum groups 𝔾{\mathbb{G}} that extend to states on the Lp{L_{p}}-C∗{\mathrm{C}^{\ast}}-algebra of 𝔾{\mathbb{G}}. Using this result, we construct uncountably many new examples of exotic quantum group norms for compact quantum groups.

Mirror symmetry relates Gromov–Witten invariants of an elliptic curve with certain integrals over Feynman graphs [10]. We prove a tropical generalization of mirror symmetry for elliptic curves, i.e., a statement relating certain labeled Gromov–Witten invariants of a tropical elliptic curve to more refined Feynman integrals. This result easily implies the tropical analogue of the mirror symmetry statement mentioned above and, using the necessary Correspondence Theorem, also the mirror symmetry statement itself. In this way, our tropical generalization leads to an alternative proof of mirror symmetry for elliptic curves. We believe that our approach via tropical mirror symmetry naturally carries the potential of being generalized to more adventurous situations of mirror symmetry. Moreover, our tropical approach has the advantage that all involved invariants are easy to compute. Furthermore, we can use the techniques for computing Feynman integrals to prove that they are quasimodular forms. Also, as a side product, we can give a combinatorial characterization of Feynman graphs for which the corresponding integrals are zero. More generally, the tropical mirror symmetry theorem gives a natural interpretation of the A-model side (i.e., the generating function of Gromov–Witten invariants) in terms of a sum over Feynman graphs. Hence our quasimodularity result becomes meaningful on the A-model side as well. Our theoretical results are complemented by a Singular package including several procedures that can be used to compute Hurwitz numbers of the elliptic curve as integrals over Feynman graphs.

The main result of this paper is a formula, in terms of characters, for the number of elements of a normal subgroup H of a finite group G which are not commutators in G .

We compute the K -theory of quantum automorphism groups of finite-dimensional C*{C^{*}}-algebras in the sense of Wang. The results show in particular that the reduced C*{C^{*}}-algebras of functions on the quantum permutation groups Sn+{S_{n}^{+}} are pairwise non-isomorphic for different values of n . Along the way we discuss some general facts regarding torsion in discrete quantum groups. In fact, the duals of quantum automorphism groups are the most basic examples of discrete quantum groups exhibiting genuine quantum torsion phenomena.