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As an analogue of a link group, we consider the Galois group of the maximal pro-$p$-extension of a number field with restricted ramification which is cyclotomically ramified at $p$, i.e, tamely ramified over the intermediate cyclotomic…

For many finite groups, the Inverse Galois Problem can be approached through modular/automorphic Galois representations. This is a report explaining the basic strategy, ideas and methods behind some recent results. It focusses mostly on the…

This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. Algorithms are described that compute the maximum integer modulo which two monic coprime integral polynomials…

We discuss the existence of Galois relations obeyed by certain link invariants. Some of these relations have recently been identified and exploited within the context of CFT and Lie/Kac-Moody representation theory. These relations should…

We propose a topological paradigm in alterfold topological quantum field theory to explore various concepts, including modular invariants, $\alpha$-induction and connections in Morita contexts within a modular fusion category of non-zero…

We study numerical invariants associated with the reduction of singularities of holomorphic foliation germs on $(\mathbb{C}^2, 0)$. Building on our previous work on generalized curve foliations, we extend explicit formulas for several…

We introduce a method that generates invariant functions from perturbative classical field theories depending on external parameters. Applying our methods to several field theories such as abelian BF, Chern-Simons and 2-dimensional…

We apply the methods of Fukaya, Kato and Sharifi to refine Mazur's study of the Eisenstein ideal. Given prime numbers $N$ and $p\geq 5$ such that $p\mid \varphi(N)$, we study the quotient of the cohomology group of modular curve $X_{0}(N)$…

The Milnor conjecture has been a driving force in the theory of quadratic forms over fields, guiding the development of the theory of cohomological invariants, ushering in the theory of motivic cohomology, and touching on questions ranging…

We extend Milnor's mu-invariants of link homotopy to ordered (classical or virtual) tangles. Simple combinatorial formulas for mu-invariants are given in terms of counting trees in Gauss diagrams. Invariance under Reidemeister moves…

Let $q$ be a prime power. This paper provides a new class of linear codes that arises from the action of the alternating group on $\mathbb F_q[x_1,\dots,x_m]$ combined with the ideas in (M. Datta and T. Johnsen, 2022). Compared with…

We define invariants $\operatorname{inv}_1,\dots,\operatorname{inv}_m$ of Galois extensions of number fields with a fixed Galois group. Then, we propose a heuristic in the spirit of Malle's conjecture which asymptotically predicts the…

We establish a Galois connection between sub-monads of an augmented monad and sub-functors of the forgetful functor from its Eilenberg-Moore category. This connection is given in terms of invariants and stabilizers defined through universal…

We prove a modularity lifting theorem for minimally ramified deformations of two-dimensional odd Galois representations, over an arbitrary number field. The main ingredient is a generalization of the Taylor-Wiles method in which we patch…

We discuss and relate finiteness conditions for certain field invariants which are studied in quadratic form theory. This includes the $u$-invariant, the reduced stability index and the symbol lengths for Galois cohomology groups with…

We construct a Hennings type logarithmic invariant for restricted quantum $\mathfrak{sl}(2)$ at a $2\mathsf{p}$-th root of unity. This quantum group $U$ is not braided, but factorizable. The invariant is defined for a pair: a 3-manifold $M$…

Let p be a prime number. The Hasse invariant is a modular form modulo p that is often used to produce congruences between modular forms of different weights. We show how to produce such congruences between forms of weights 2 and p+1, in…

We publish a table of primitive finite-type invariants of order less than or equal to six, for knots of ten or fewer crossings. We note certain mod-2 congruences, one of which leads to a chirality criterion in the Alexander polynomial. We…

We introduce a new graph invariant of finite groups that provides a complete characterization of the splitting types of unramified prime ideals in normal number field extensions entirely in terms of the Galois group. In particular, each…

Minhyong Kim introduced arithmetic Chern-Simons invariants for totally imaginary number fields as arithmetic analogues of the Chern-Simons invariants for 3-manifolds. In this paper, we extend Kim's definition for any number field, by using…