Related papers: Toda brackets and congruences of modular forms
In this paper we prove the laws of Toda brackets on the homotopy groups of a connective ring spectrum and the laws of the cup-one square in the homotopy groups of a commutative connective ring spectrum.
The classical modular equations involve bivariate polynomials that can be seen to be univariate with coefficients in the modular invariant $j$. Kiepert found modular equations relating some $\eta$-quotients and the Weber functions…
The Tutte polynomial is the most general invariant of matroids and graphs that can be computed recursively by deleting and contracting edges. We generalize this invariant to any class of combinatorial objects with deletion and contraction…
We survey various classical results on invariants of polynomials, or equivalently, of binary forms, focussing on explicit calculations for invariants of polynomials of degrees 2, 3, 4.
Analogues of invariant theory's well-known Roberts theorem are proved for ternary forms. We established that covariants, contravariants and mixed concomitants of a ternary form are uniquely determined by their lead coefficients.
Using deformation quantization and suitable 2 by 2 quantum $R$-matrices we show that a list of Toda like classical integrable systems given by Y.B.Suris is quantum integrable in the sense that the classical conserved quantities (which are…
The Hofstadter model allows to describe and understand several phenomena in condensed matter such as the quantum Hall effect, Anderson localization, charge pumping, and flat-bands in quasiperiodic structures, and is a rare example of…
We give a complete classification of all simple current modular invariants, extending previous results for $(\Zbf_p)^k$ to arbitrary centers. We obtain a simple explicit formula for the most general case. Using orbifold techniques to this…
The homology cobordism group of homology cylinders is a generalization of both the mapping class group of surfaces and the string link concordance group. We consider extensions of Johnson homomorphisms of a mapping class group, Milnor…
We show that there exist infinitely many pairs of non-homeomorphic closed oriented SOL torus bundles with the same quantum (TQFT) invariants. This follows from the arithmetic behind the conjugacy problem in $SL(2,\Z)$ and its congruence…
We prove various congruences for Catalan and Motzkin numbers as well as related sequences. The common thread is that all these sequences can be expressed in terms of binomial coefficients. Our techniques are combinatorial and algebraic:…
In this article we study abstract and embedded invariants of reduced curve germs via topological techniques. One of the most important numerical analytic invariants of an abstract curve is its delta invariant. Our primary goal is to develop…
In 1973, Swinnerton-Dyer completely classified all congruences for coefficients of normalized eigenforms in weights $k \in \{12, 16, 18, 20, 22, 26\}$ on $\Gamma_{0}(1) = \operatorname{SL}_{2}(\mathbb{Z})$ using the theory of modular Galois…
The universal Vassiliev-Kontsevich invariant is a functor from the category of tangles to a certain graded category of chord diagrams, compatible with the Vassiliev filtration and whose associated graded is an isomorphism. The Vassiliev…
It is known that if A and B are two n-by-n complex matrices and (A,A^T) is simultaneously equivalent to (B,B^T), then A is congruent to B. We extend this statement to multilinear forms.
New integrable lattice systems are introduced, their different integrable discretization are obtained. B\"acklund transformations between these new systems and the relativistic Toda lattice (in the both continuous and discrete time…
In (B-Gran, 2004), was given a categorical formulation of the Shifting Lemma which is a characterization of the Congruence Modular Varieties among all the variety of Universal Algebra, introduced in (Gumm, 1983). Starting from a…
Given a Tyurin degeneration of a Calabi-Yau complete intersection in a toric variety, we prove gluing formulas relating the generalized functional invariants, periods, and $I$-functions of the mirror Calabi-Yau family and those of the two…
We study the dependence of the eta invariant $\eta_D$ on the spin structure, where $D$ is a twisted Dirac operator on a (4k+3)-dimensional spin manifold. The difference between the eta invariants for two spin structures related by a…
We determine the quantum filtration structure of the Lee homology of all torus links. In particular, this determines the $s$-invariant of a torus link equipped with any orientation. In the special case $T(n,n)$, our result confirms a…