A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
We develop the general theory for the construction of Extended Topological Quantum Field Theories (ETQFTs) associated with the Costantino-Geer-Patureau quantum invariants of closed 3-manifolds. In order to do so, we introduce relative modular categories, a class of ribbon categories which are modeled on representations…
New invariants for 3-manifolds derived from supergroup representations.
problem Developing invariants for 3-manifolds using supergroup analogues.
method Introducing supergroup analogues of 3-manifold invariants for superunitary groups, focusing on SU(2|1). Calculating q-series for specific 3-manifolds and studying their properties.
result Explicit calculation and study of q-series for certain 3-manifolds, providing a formula relating new invariants to quantum invariants.
Invariants of 3-manifolds from a non semi-simple category of modules over a version of quantum sl(2) were obtained by the last three authors in arXiv:1202.3553 . They are invariants of 3-manifolds together with a cohomology class which can be interpreted as a line bundle with flat connection. In arXiv:1404.7289 we …
We study relationships between the restricted unrolled quantum group UqH(sl2) at 2r-th root of unity q=eπi/r,r≥2, and the singlet vertex operator algebra M(r). We use deformable families of modules to efficiently compute (1,1)-tangle invariants colored with projecti…
Constructs TQFTs for cobordisms with cohomology class decorations.
problem Creating TQFTs for cobordisms with cohomology class decorations.
method Starting from an abelian group G and a factorizable ribbon Hopf G-bialgebra H, constructs a TQFT JH for connected framed cobordisms between connected surfaces with connected boundary decorated with cohomology classes with coefficients in G.
result Our functor recovers a special case of Kerler-Lyubashenko TQFTs when restricted to trivial decorations.
This paper analyzes the generalization risk of unrolled neural networks using Stein's Unbiased Risk Estimator.
problem Analyzing the generalization risk of unrolled neural networks and its relationship to network design and train sample size.
method Using Stein's Unbiased Risk Estimator (SURE), the paper analyzes the generalization risk with bias and variance components for recurrent unrolled networks, focusing on the degrees-of-freedom (DOF) component and the trace of the end-to-end network Jacobian.
result DOF is well-approximated by the weighted path sparsity of the network under incoherence conditions on the trained weights, and DOF increases with train sample size and converges to the generalization risk for both recurrent and non-recurrent schemes.
This paper introduces Non-Autonomous Input-Output Stable Network(NAIS-Net), a very deep architecture where each stacked processing block is derived from a time-invariant non-autonomous dynamical system. Non-autonomy is implemented by skip connections from the block input to each of the unrolled processing stages and al…
We discuss an approach to quantum gerbes over quantum groups in terms of q-deformation of transition functions for a loop group bundle. The case of the quantum group SUq(2) is treated in some detail.
We introduce a method to stabilize Generative Adversarial Networks (GANs) by defining the generator objective with respect to an unrolled optimization of the discriminator. This allows training to be adjusted between using the optimal discriminator in the generator's objective, which is ideal but infeasible in practice…
It is shown that there is a C∗-algebraic quantum group related to any double Lie group. An algebra underlying this quantum group is an algebra of a differential groupoid naturally associated with a double Lie group
Quantum theory constructs a group and skein module for knot complements.
problem Understanding the fundamental group of knot complements using quantum methods.
method Using bottom tangles, the universal space of quantum representations is constructed, then factored by the skein relation to get the skein module.
result Derives recurrence relation for the colored Jones polynomial, known as Aq polynomial.
Quantum affine bundles are quantum principal bundles with affine quantum structure groups. A general theory of quantum affine bundles is presented. In particular, a detailed analysis of differential calculi over these bundles is performed, including the description of a natural differential calculus over the structure …
If a compact quantum group acts faithfully and smoothly (in the sense of Goswami 2009) on a smooth, compact, oriented, connected Riemannian manifold such that the action induces a natural bimodule morphism on the module of sections of the co-tangent bundle, then it is proved that the quantum group is necessarily commut…
We generalize the asymptotic faithfulness of the skein quantum SU(2) representations of mapping class groups of orientable closed surfaces to skein SU(3). Skein quantum representations of mapping class groups are different from the Reshetikin-Turaev ones from quantum groups or geometric quantization because they ar…
We expose a K-theoretic approach to study group C*-algebras and C*-algebraic compact quantum groups: 1. The conception of multidimensional geometric quantization and the index of group C*-algebras; 2. the entire homology of noncommutative de Rham currents and the noncommutative Chern characters, and their computation f…
Suppose that a compact quantum group Q acts faithfully and isomet- rically (in the sense of [10]) on a smooth compact, oriented, connected Riemannian manifold M . If the manifold is stably parallelizable then it is shown that the compact quantum group is necessarily commutative as a C \ast algebra i.e. Q = C(G) for som…