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It is time for the legacy financial infrastructure to seamlessly connect with modern, decentralized infrastructure. Although it is increasingly evident that decentralized infrastructure for finance (namely distributed ledgers) will coexist with and complement legacy infrastructure, it is also clear that such interoperability efforts carry new risks and concerns. In particular, managing the range of heterogeneous (and not well-established) infrastructure brings security, privacy, and regulatory issues. The first step to overcome some of these challenges is to recognize that in many deployment instances using distributed ledgers, the purpose of the ledger is to share resources among the community members. The second step after recognizing that borders exist is to understand that interoperability across systems can be best achieved through the use of standardized service interfaces (or application programming interfaces (API)). In this paper we use the term ledger gateways (or simply gateways) to denote the computer and software systems that implement the standardized service interfaces into a distributed ledger. The main purpose of a gateway is to communicate with other peer gateways that implement the same standardized service interface. Among others, peer gateways perform the transfer of data and value across borders (legal or national borders). Gateways also become a mechanism to manage a permissioned environment, where abiding by laws and regulations is crucial for business compliance (e.g., EU General Data Protection Regulations (GDPR), EU MiCa regulation on digital assets, FAFT Recommendation 15, ISO 27001.

We prove that the trace of the Hecke operator $T_2$ acting on the vector space of cusp forms of level one takes no repeated values, except for 0, which only occurs when the space is trivial., Comment: To appear in Indagationes Mathematicae

The goal of this paper is to revise the Henderson-Chu approach by dividing the commuters into two income-based groups: the `rich' and the `poor'. The rich are clearly more flexible in their arrival times and would rather have more schedule delay cost instead of travel delay. The poor are quite inflexible as they have to arrive at work at a specified time -- travel delay cost is preferred over schedule delay cost. We combined multiple models of peak-load bottleneck congestion with and without toll-pricing to generate a Pareto improvement in Lexus lanes., Comment: Not complete. Needs revision

For each complex semisimple group $G_{\mathbb{C}}$, Moore and Tachikawa conjectured the existence of a certain two-dimensional topological quantum field theory $\eta_{G_{\mathbb{C}}} : \mathrm{Cob}_2 \to \mathrm{MT}$ whose target category has complex Lie groups as objects and holomorphic symplectic varieties with Hamiltonian actions of the groups as morphisms. The conjecture is motivated by string theory, where $\eta_{G_{\mathbb{C}}}$ is obtained by taking the Higgs branch of some supersymmetric quantum field theories (called theories of class $S$) depending on $G_{\mathbb{C}}$ and a Riemann surface. The goal of this paper is to define a real analogue $\mathrm{MT}_{\mathbb{R}}$ of the target category $\mathrm{MT}$ and to rigorously prove that it is a category., Comment: Not complete, needs revision

In this computational paper we verify a truncated version of the Buzzard-Calegari conjecture on the Newton polygon of the Hecke operator $T_2$ for all large enough weights. We first develop a formula for computing $p$-adic valuations of exponential sums, which we then implement to compute $2$-adic valuations of traces of Hecke operators acting on spaces of cusp forms. Finally, we verify that if Newton polygon of the Buzzard-Calegari polynomial has a vertex at $n\leq 15$, then it agrees with the Newton polygon of $T_2$ up to $n$., Comment: To appear in Annales math\'ematiques du Qu\'ebec

We prove a number of unconditional statistical results of the Hecke coefficients for unitary cuspidal representations of $\operatorname{GL}(2)$ over number fields. Using partial bounds on the size of the Hecke coefficients, instances of Langlands functoriality, and properties of Rankin-Selberg $L$-functions, we obtain bounds on the set of places where linear combinations of Hecke coefficients are negative. Under a mild functoriality assumption we extend these methods to $\operatorname{GL}(n)$. As an application, we obtain a result related to a question of Serre about the occurrence of large Hecke eigenvalues of Maass forms. Furthermore, in the cases where the Ramanujan conjecture is satisfied, we obtain distributional results of the Hecke coefficients at places varying in certain congruence or Galois classes.

Given two distinct newforms with real Fourier coefficients, we show that the set of primes where the Hecke eigenvalues of one of them dominate the Hecke eigenvalues of the other has density at least 1/16. Furthermore, if the two newforms do not have complex multiplication, and neither is a quadratic twist of the other, we also prove a similar result for the squares of their Hecke eigenvalues., Comment: To appear in Archiv der Mathematik

We construct an infinite family of representations of finite groups with an irreducible adjoint action and we give an application to the question of lacunary of Frobenius traces in Galois representations., Comment: To appear in the Journal of the Ramanujan Mathematical Society

We introduce a generalization of structured manifolds as the most general Riemannian metric g associated to an affinor (tensor field of (1,1)-type) F and initiate a study of their semi-invariant submanifolds. These submanifolds are generalization of CR-submanifolds of almost complex geometry and semi-invariant submanifolds of several interesting geometries (almost product, almost contact and others). We characterize the integrability of both invariant and anti-invariant distribution; the special case when F is covariant constant with respect to g gives major simplifications in computations.