arXiv research

Classical Dedekind sums are connected to the modular group through the construction of a (Dedekind) symbol on the cusp set of the modular group. In this paper we study generalizations of Dedekind symbols and sums that can be associated to certain Fuchsian groups uniformizing 1-punctured tori.

Quantum modularity proved for SU(2) TQFT signature on genus 2 surfaces.

Study connects knot polynomials with number theory sums.

We derive a closed formula for the Heegaard Floer correction terms of lens spaces in terms of the classical Dedekind sum and its generalization, the Dedekind-Rademacher sum. Our proof relies on a reciprocity formula for the correction terms established by Ozsvath and Szabo. A consequence of our result is that the Casso…

We express the number of lattice points inside certain simplices via Dedekind-Rademacher sums. As an application, we prove a conjecture of Kronheimer and Mrowka in the special case of Brieskorn spheres (with at most 4 singular fibers). This conjecture relates the Euler characteristic of the Seiberg-Witten-Floer homolog…

Study G_2-manifolds from glued circles and Calabi-Yau manifolds.

We extend the adiabatic limit formula for eta-invariants by Bismut-Cheeger and Dai to Seifert fibrations. Our formula contains a new contribution from the singular fibres that takes the form of a generalised Dedekind sum. As an application, we compute the Eells-Kuiper and t-invariants of certain cohomogeneity one manif…

We compute the average Tristram---Levine signature of any graph link with positive weights in a three sphere, generalizing the results of Kirby and Melvin. The main tools are the Neumann's algorithm for computing the equivariant signatures of graph links and the Reciprocity Law for Dedekind sums.

We study properties of the signature function of the torus knot $T_{p,q}$. First we provide a very elementary proof of the formula for the integral of the signatures over the circle. We obtain also a closed formula for the Tristram--Levine signature of a torus knot in terms of Dedekind sums.

Formulas previously presented for the Casson-Walker invariant are generalized to Lescop's extension. These formulas in terms of linking numbers and surgery coefficients compute the change in Lescop's invariant under crossing changes in a framed link presenting a 3-manifold. This leads us to revisit an old formula for a…

We compute the Heegaard Floer homology of $S^3_1(K)$ (the (+1) surgery on the torus knot $T_{p,q}$) in terms of the semigroup generated by $p$ and $q$, and we find a compact formula (involving Dedekind sums) for the corresponding Ozsvath--Szabo d-invariant. We relate the result to known knot invariants of $T_{p,q}$ as …

We identify 998 closed hyperbolic 3-manifolds whose volumes are rationally related to Dedekind zeta values, with coprime integers $a$ and $b$ giving $a/b vol(M)=(-D)^{3/2}/(2π)^{2n-4} (ζ_K(2))/(2ζ(2))$ for a manifold M whose invariant trace field $K$ has a single complex place, discriminant $D$, degree $n$, and Dedekin…

Study on TQFT signatures converging to modular form.

This is the first of a series of articles in which we are going to study the regularized determinants of the Laplacians of Calabi Yau metrics acting on (0,q) forms on the moduli space of CY manifolds with a fixed polarization. It is well known that in case of the elliptic curves the Kronecker limit formula gives an exp…

Using adiabatic limits of Eta invariants, Rho invariants of the total space of a fiber bundle are investigated. One concern is to formulate the aspects of local index theory for families of Dirac operator in terms of the odd signature operator, and place known results in a context which permits the treatment of Rho inv…

The paper is concerned with the Kontsevich-Zagier formal power series $f(q)=\sum_{n=0}^\infty (1-q)... (1-q^n)$ and its analytic properties. To begin with, we give an explicit formula for the Borel transform of the associated formal power series $F(x)=e^{-1/(24x)}f(e^{-1/x})$ from which its analytic continuation, i…

We apply representation theory to study the homology of equivariant Dehn-fillings of a given finite, regular cover of a compact 3-manifold with boundary a torus. This yields a polynomial which gives the rank of the part of the homology carried by the solid tori used for Dehn-filling. The polynomial is a symmetrized for…

The difference between slice and doubly-slice knots is reflected in algebra by the difference between metabolic and hyperbolic Blanchfield linking forms. We exploit this algebraic distinction to refine the classical Witt group of linking forms by defining a `double Witt group' of linking forms. We calculate the double …

We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular hermitian line bundles of finite height. In particular, the theorem applies to the log-singular metrics of Burgos-Kramer-Kühn. Our theorem is thus suitable for application to some non-compact Shimura varieties with their bundles of cusp forms…

It is proved that the Hasse-Weil zeta functions of the canonical components of the ${\rm SL}_2$ (${\rm PSL}_2$)-character varieties of closed orientable complete hyperbolic $3$-manifolds of finite volume are equal to the Dedekind zeta functions of their trace fields (invariant trace fields). When the closed $3$-manifol…

Develops a new non-abelian framework for Riemann surfaces and differential equations.

This is a survey of several approaches to the framework for working with infinitesimals and infinite numbers, originally developed by Abraham Robinson in the 1960s, and their constructive engagement with the Cantor-Dedekind postulate and the Intended Interpretation hypothesis. We highlight some applications including (…

We prove a new structural result for the spherical Tits building attached to SL_n(K) for many number fields K, and more generally for the fraction fields of many Dedekind domains O: the Steinberg module St_n(K) is generated by integral apartments if and only if the ideal class group cl(O) is trivial. We deduce this int…

Machine learning predicts properties of number fields with high accuracy.

We define the generalized connected sum for generic closed plane curves, generalizing the strange sum defined by Arnold, and completely describe how the Arnold invariants $J^{\pm}$ and $\mathit{St}$ behave under the generalized connected sums.

We propose a generalization of the classical notions of plumbing and Murasugi summing operations to smooth manifolds of arbitrary dimensions, so that in this general context Gabai's credo "the Murasugi sum is a natural geometric operation" holds. In particular, we prove that the sum of the pages of two open books is ag…

Genus 3 Heegaard groups of lens space connected sums are finitely generated.

Proves a general connected sum formula for families Seiberg-Witten invariants.

Can the Minkowski sum of two compact convex bodies be made smoother by rotating one of them? We construct two infinitely differentiable strictly convex plane bodies such that after any generic rotation (in the Baire category sense) of one of the summands the Minkowski sum is not five times differentiable. On the other …

Gradient methods converge exponentially in concave network games.

Just as war is sometimes fallaciously represented as a zero sum game -- when in fact war is a negative sum game - stock market trading, a positive sum game over time, is often erroneously represented as a zero sum game. This is called the "zero sum fallacy" -- the erroneous belief that one trader in a stock market exch…

Summing over 3-manifolds using TQFT partition functions.

We prove that transversal non-simplicity is preserved under taking connect sum, generalizing Vertesi's result.

Derives exact formula for Minkowski sum of ellipsoids in N-space.

Gradient Descent Ascent converges to von-Neumann solution in hidden zero-sum games.

Study end sum for surfaces and prove uniqueness results.

Study Nash equilibrium in non-zero-sum game with Bermudan strategies.

New methods optimize sums of bivariate functions on finite domains.

We consider the question, asked by Friedl, Livingston and Zentner, of which sums of torus knots are concordant to alternating knots. After a brief analysis of the problem in its full generality, we focus on sums of two torus knots. We describe some effective obstructions based on Heegaard Floer homology.

Study symmetry groups and curves from sums of exponentials.

New computations show symplectic groups and mapping class groups have different properties regarding torsion.

ELBO converges to a sum of entropies for many generative models.

Proves a formula for a special invariant of 4-manifolds.

The study allows for connected sums in manifolds with positive intermediate Ricci curvature.

Characterizes Stein surfaces with finite homotopy rank-sum.

We study Legendrian singular links up to contact isotopy. Using a special property of the singular points, we define the singular connected sum of Legendrian singular links. This concept is a generalization of the connected sum and can be interpreted as a tangle replacement, which provides a way to classify Legendrian …

The paper studies Gauss sums and their applications in algebra and topology.

High dimensional superposition models characterize observations using parameters which can be written as a sum of multiple component parameters, each with its own structure, e.g., sum of low rank and sparse matrices, sum of sparse and rotated sparse vectors, etc. In this paper, we consider general superposition models …