arXiv research

Classifies extended Abelian Chern-Simons theories using quadratic modules.

Chern-Simons and Reshetikhin-Turaev theories are shown equivalent for U(1) gauge group.

The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a non-degenerate invariant symmetric bilinear form. We show that any metric Lie algebra without simple ideals has the structure of a so called balanced quadratic extension of a…

We prove that the Steinberg module of the special linear group of a quadratic imaginary number ring which is not Euclidean is not generated by integral apartments. Assuming the generalized Riemann hypothesis, this shows that the Steinberg module of a number ring is generated by integral apartments if and only if the ri…

Study of cubic skein modules in 3-sphere and arbitrary 3-manifolds.

New techniques prove bounded acyclicity results for semi-simplicial sets.

Proves finiteness and holonomicity of skein modules for 3-manifolds.

Study disproves finiteness conjecture for a specific link's skein module.

Following our approach to metric Lie algebras developed in math.DG/0312243 we propose a way of understanding pseudo-Riemannian symmetric spaces which are not semi-simple. We introduce cohomology sets (called quadratic cohomology) associated with orthogonal modules of Lie algebras with involution. Then we construct a fu…

Proves finiteness of Kauffman bracket skein modules for 3-manifolds.

Finite intersection numbers between horizontal foliations of quadratic differentials.

Proves transitivity of a specific class of quadratic polynomials.

Study Kauffman bracket skein modules of Seifert fibered spaces.

We give a new, algebraically computable formula for skein modules of closed 3-manifolds via Heegaard splittings. As an application, we prove that skein modules of closed 3-manifolds are finite-dimensional, resolving in the affirmative a conjecture of Witten.

Projective resolves symplectic Steinberg module for number rings.

FI-modules were introduced by the first three authors in [CEF] to encode sequences of representations of symmetric groups. Over a field of characteristic 0, finite generation of an FI-module implies representation stability for the corresponding sequence of S_n-representations. In this paper we prove the Noetherian pro…

We define an invariant of tangles and framed tangles given a finite crossed module and a pair of functions, called a Reidemeister pair, satisfying natural properties. We give several examples of Reidemeister pairs derived from racks, quandles, rack and quandle cocycles, 2-crossed modules and braided crossed modules. We…

Chern-Simons and Reshetikhin-Turaev theories are shown equivalent.

Classifies modules of surface-knots in terms of their properties.

In this paper, we propose one index $i_1(f)-i_2(f)$ which measures how well-behaved a given finitely determined multigerm $f: (\mathbb{K}^n,S)\to (\mathbb{K}^p,0)$ $(n\le p)$ of corank at most one is from the viewpoint of liftable vector fields; and we answer the following problems when the index indicates that the giv…

Extends Serre-Swan theorem to all finitely generated modules over smooth functions.

We construct a finitely presented group $G$ with non-quadratic Dehn function $f$ majorizable by a quadratic function on arbitrary long intervals.

We exhibit a Poisson module restoring a twisted Poincare duality between Poisson homology and cohomology for the polynomial algebra R=C[X_1,...,X_n] endowed with Poisson bracket arising from a uniparametrised quantum affine space. This Poisson module is obtained as the semiclassical limit of the dualising bimodule for …

Researchers computed Kauffman bracket skein modules of specific Seifert manifolds.

Cochran defined the nth-order integral Alexander module of a knot in the three sphere as the first homology group of the knot's (n+1)th-iterated abelian cover. The case n=0 gives the classical Alexander module (and polynomial). After a localization, one can get a finitely presented module over a principal ideal domain,…

This expository survey describes how holomorphic quadratic differentials arise in several aspects of Teichmüller theory, highlighting their relation with various geometric structures on surfaces. The final section summarizes results for non-compact surfaces of finite type, when the quadratic differential has poles of f…

Novel method computes Kauffman bracket skein modules of small 3-manifolds.

Study variance-optimal hedging of forward curve derivatives under stochastic volatility.

Proves finiteness for 3-manifolds from handlebodies with 2-handles.

A streaming algorithm estimates quadratic covariation from financial data efficiently.

A sequence of rational functions in a variable $q$ is $q$-holonomic if it satisfies a linear recursion with coefficients polynomials in $q$ and $q^n$. We prove that the degree of a $q$-holonomic sequence is eventually a quadratic quasi-polynomial. Our proof uses differential Galois theory (adapting proofs regarding hol…

New knot theory module shows torsion-ness in number theory.

Photonic chip speeds up option pricing with GAN for financial efficiency.

We study the structure of symplectic quandles, quandles which are also R-modules equipped with an antisymmetric bilinear form. We show that every finite dimensional symplectic quandle over a finite field F or arbitrary field F of characteristic other than 2 is a disjoint union of a trivial quandle and a connected quand…

Hilbert(ian) A-modules over finite von Neumann algebras A with a faithful normal trace state (from global analysis) and Hilbert W*-modules over A (from operator algebra theory) are compared, and a categorical equivalence is established. The correspondence between these two structures sheds new light on basic results in…

We identify branched coverings (continuous open surjections p:Y->X of Hausdorff spaces with uniformly bounded number of pre-images) with Hilbert C*-modules C(Y) over C(X) and with faithful unital positive conditional expectations E:C(Y)->C(X) topologically of index-finite type. The case of non-branched coverings corres…

Extends Manin triples to Lie bialgebroids over Lie groupoids.

New quantum knot invariants derived from Verma modules.

The paper shows geometric realisation over specific groups and knots.

The classical Serre-Swan's theorem defines a bijective correspondence between vector bundles and finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an et…

Study Swan modules and homotopy types, resolving Wall and Dyer questions.

We show that the module of lowerable vector fields for a finitely L-determined multigerm is finitely generated in a constructive way.

We define for arbitrary modules over a finite von Neumann algebra $\cala$ a dimension taking values in $[0,\infty]$ which extends the classical notion of von Neumann dimension for finitely generated projective $\cala$-modules and inherits all its useful properties such as additivity, cofinality and continuity. This all…

A finitely generated module over the ring L=Z[t, t^{-1}] of integer Laurent polynomials that has no Z-torsion is determined by a pair of sub-lattices of L^d. Their indices are the absolute values of the leading and trailing coefficients of the order of the module. This description has applications in knot theory.

We derive the price of a spread option based on two assets which follow a bivariate volatility modulated Volterra process dynamics. Such a price dynamics is particularly relevant in energy markets, modelling for example the spot price of power and gas. Volatility modulated Volterra processes are in general not semimart…

4-manifolds with specific groups have unique homotopy types.

If an augmented algebra K over Q is filtered by powers of its augmentation ideal I, the associated graded algebra grK need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this paper we give a sufficient criterion (called the …

Study bounds variance modulation function for K-spider distributions.