We give a formula of the colored Alexander invariant in terms of the homological representation of the braid groups which we call truncated Lawrence's representation. This formula generalizes the famous Burau representation formula of the Alexander polynomial.
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Abstract: Characterizes frontals and wavefronts with formulas.
problem Characterizing frontals and wavefronts.
method Representation formulas for frontals and wavefronts.
result Representation formulas for various types of frontals and wavefronts.
We construct a new representation formula for indefinite improper affine spheres in terms of two para-holomorphic functions and study singularities which appear in this representation formula. As a result, it follows that cuspidal cross caps never appear as the singularities on indefinite improper affine spheres and so…
Study finds formula for non-conformal harmonic surfaces.
problem Non-conformal harmonic surfaces in with prescribed ()transforms.
method Representation formula derived for non-conformal harmonic surfaces.
result Representation formula for non-conformal harmonic surfaces in .
A holomorphic representation formula for special parabolic hyperspheres is given.
Derives integral formula for ReLU networks with limited weights.
problem Finding optimal neural network weights with limited -norm.
method Derives integral representation formula for shallow ReLU networks under -norm constraint.
result Explicitly solves the least -norm neural network representation for a given function.
New parametrizations for minimal timelike surfaces discovered.
problem Finding parametrizations for minimal timelike surfaces in specific spaces.
method Derived representation formulas for null curves leading to parametrizations of minimal timelike surfaces.
result Examples of minimal timelike surfaces constructed.
Formula derived for discrete improper affine spheres.
problem Constructing discrete improper affine spheres.
method Loop group factorizations and Birkhoff decomposition.
result Representation formula for discrete indefinite affine spheres.
Holomorphic volume form on circle representations generalizes Witten's formula for surfaces with boundary.
problem Generalizing Witten's formula to surfaces with boundary.
method Introducing an holomorphic volume form on the space of representations of the circle.
result Holomorphic volume form appears as a peripheral term in the generalized Witten's formula.
Geometric properties near singular points of surfaces given by certain representation formulaemath.DG
Study geometric properties of surfaces with specific formulae.
problem Calculate curvature of surfaces with singular points.
method Investigate singular curvature and limiting normal curvature of surfaces.
result Determine geometric properties of surfaces with singular points.
New proof and formula linking fusion trees to quantum knot invariants.
problem Quantum knot invariants encoding in non-semisimple TQC.
method Connection between fusion trees and Lawrence representations, using graphical calculus.
result Explicit encoding of quantum knot invariants via fusion trees.
We provide an explicit algorithm to calculate invariant tensors for the adjoint representation of the simple Lie algebra , as well as arbitrary representation in terms of roots. We also obtain explicit formulae for the adjoint representations of the orthogonal and symplectic Lie algebras and .
We prove that an isometric immersion of a simply connected Lorentzian surface in is equivalent to a normalised spinor field solution of a Dirac equation on the surface. Using the quaternions and the Lorentz numbers, we also obtain an explicit representation formula of the immersion in terms of the sp…
We give a topological formula of the loop expansion of the colored Jones polynomials by using identification of generic quantum sl2 representation with homological representations. This gives a direct topological proof of the Melvin-Morton-Rozansky conjecture, and a connection between entropy of braids and quantum repr…
In this paper we give a geometrically invariant spinorial representation of surfaces in four-dimensional space forms. In the Euclidean space, we obtain a representation formula which generalizes the Weierstrass representation formula of minimal surfaces. We also obtain as particular cases the spinorial characterization…
The purpose of this short note is to relate a representation formula due to the Author and P. Romon for Lagrangian surfaces (see math.DG/0009202) to a more general Weierstrass representation type formula found by Konopelchenko for surfaces in 4-dimensional space (see math.DG/9807129). Simplifications are pointed out.
Study topological components of surface group representations into SL(2,R) and PSL(2,R).
problem Understanding the structure of representation spaces of surface groups.
method Utilized the signature formula to determine connected components of representation spaces.
result Determined the number of connected components for different types of holonomies.
Paper proves a fixed point formula and applies it to a new proof of Harish-Chandra's character formula.
problem Proving a fixed point formula for equivariant indices of elliptic differential operators.
method Fixed point formula for proper actions by connected semisimple Lie groups on manifolds.
result New proof of Harish-Chandra's character formula for discrete series representations.
The paper derives formulas for symplectic volume forms on surface representation varieties.
problem Calculating symplectic volume forms on surface representation varieties.
method Multiplicative gluing formulas and Heusener-Porti results.
result Symplectic volume form on is a product of forms on and .
Like all other knot polynomials, the superpolynomials should be defined in arbitrary representation R of the gauge group in (refined) Chern-Simons theory. However, not a single example is yet known of a superpolynomial beyond symmetric or antisymmetric representations. We consider the expansion of the superpolynomial a…
Direct formula found for ADO invariants from homological representations.
problem Computing ADO invariants from quantum group representations.
method Direct homological formula for ADO invariants using partial traces of homological representations.
result Direct formula for ADO invariants without further truncations.
We study the conditions for a nilpotent Lie group to be foliated into subgroups that have square integrable (relative discrete series) unitary representations, that fit together to form a filtration by normal subgroups. Then we use that filtration to construct a class of "stepwise square integrable" representations on …
We give a new proof of the Jantzen sum formula for integral representations of Chevalley schemes over Spec Z. This is done by applying the fixed point formula of Lefschetz type in Arakelov geometry to generalized flag varieties. Our proof involves the computation of the equivariant Ray-Singer torsion for all equivarian…
We prove a quasi-Poisson bracket formula for the space of representations of the fundamental groupoid of a surface with boundary, which generalizes Goldman's Poisson bracket formula. We also deduce a similar formula for quasi-Poisson cross-sections.
A formula connects discrete harmonic surfaces to holomorphic functions.
problem Creating smooth discrete harmonic surfaces from holomorphic data.
method Weierstrass representation formula for discrete harmonic surfaces.
result Smooth converging sequence of discrete harmonic surfaces converges to a minimal surface.
Bismut and Zhang computed the ratio of the Ray-Singer and the combinatorial torsions corresponding to non-unitary representations of the fundamental group. In this note we show that for representations which belong to a connected component containing a unitary representation the Bismut-Zhang formula follows rather easi…
We give an efficient simplicial formula for the volume and Chern-Simons invariant of a boundary-parabolic PSL(2,C)-representation of a tame 3-manifold. If the representation is the geometric representation of a hyperbolic 3-manifold, our formula computes the volume and Chern-Simons invariant directly from an ideal tria…
We give an immersion formula, the Sym-Bobenko formula, for minimal surfaces in the 3-dimensional Heisenberg space. Such a formula can be used to give a generalized Weierstrass type representation and construct explicit examples of minimal surfaces.
The paper calculates asymptotic Betti numbers and homology multiplicities for graph configuration spaces.
problem Understanding the homology of ordered configuration spaces of graphs.
method Explicit formulas for asymptotic Betti numbers and homology multiplicities in characteristic zero.
result Explicit formulas for asymptotic multiplicities in homology of irreducible representations of the symmetric group.
In this note we prove a Weierstrass representation formula for pluriminimal submanifolds of euclidean spaces. We use this formula to produce new families of examples of pluriminimal submanifolds. We also prove that any affine algebraic manifold can be pluriminimally embedded into some euclidean space in a non holomorph…
Paper provides a formula for translating solitons and singular minimal surfaces.
problem Representing translating solitons and singular minimal surfaces in 3D space.
method Develops a Weierstrass representation formula.
result Solves a general Cauchy problem for the class of surfaces.
We provide a formula describing the G-module structure of the Hurwitz-Hodge bundle for admissible G-covers in terms of the Hodge bundle of the base curve, and more generally, for describing the G-module structure of the push-forward to the base of any sheaf on a family of admissible G-covers. This formula can be interp…
A relation between the Goldstein-Petrich hierarchy for plane curves and the Toda lattice hierarchy is investigated. A representation formula for plane curves is given in terms of a special class of -functions of the Toda lattice hierarchy. A representation formula for discretized plane curves is also discussed.
New method constructs Conway's potential function using braids and Gassner representation.
problem Computing and resolving sign ambiguity in multivariable Alexander-Conway polynomial.
method Using braids and the reduced Gassner representation.
result Efficient method of computing Conway's potential function and resolving sign ambiguity.
New formula calculates volumes and Chern-Simons invariants for closed 3-manifolds.
problem Computing volumes and Chern-Simons invariants for non-parabolic representations.
method Introducing deformed Ptolemy varieties to extend Zickert's formula.
result Volume and Chern-Simons invariants computed for closed 3-manifolds.
We derive an integral representation which encodes all coefficients of the Riemann normal coordinate expansion, and also a closed formula for those coefficients.
Extends Clark-Ocone theorem to non-Malliavin differentiable random variables using Ito's formula.
problem Extending Clark-Ocone theorem to non-Malliavin differentiable random variables.
method Uses Ito's formula instead of Malliavin calculus.
result Explicit representation of locally risk-minimizing strategies for digital options in Levy models.
The paper develops a new formula for financial pricing under multiple interest rates and collateralization.
problem Financial pricing under multiple interest rates and collateralization.
method Derives a change of measure formula for recursive conditional expectations in a jump-diffusion setting.
result Generalizes the change of numéraire technique for multiple interest rates and collateralization.
The paper develops a new approach to conditional risk measures using modular convex analysis.
problem Developing a new method for conditional risk measures.
method Random modular approach to conditional certainty equivalents and niveloids in the conditional -space.
result Retrieves a conditional variational formula for optimized certainty equivalents and applies it to the conditional entropic risk measure.
Rosso and Jones gave a formula for the colored Jones polynomial of a torus knot, colored by an irreducible representation of a simple Lie algebra. The Rosso-Jones formula involves a plethysm function, unknown in general. We provide an explicit formula for the second plethysm of an arbitrary representation of $\fsl_3$, …
Diagrammatic calculus proves Alexander polynomial formulas.
problem Proving formulas for Alexander polynomial.
method Diagrammatic calculus of quantum representations.
result Quantum invariant determines Alexander polynomial.
We obtain a family of matrix integrals which decompose to a product of Gamma-functions (they have some relations with S.G.Gindikin 'Beta', but generally speaking essentially differ from it). We obtain Plancherel formula for Berezin representations for all series of classical groups (for large values of parameters of re…
On the Skew Stickiness Ratioq-fin.MF
Derives formula for skew stickiness ratio in asset price and volatility dynamics.
problem Capturing joint dynamics of asset price and volatility.
method Uses Itô-Wentzell and Clark-Ocone formulae to derive representation.
result Derives asymptotics of skew stickiness ratio under stochastic volatility models.
Abstract: Formula for Lorentzian surfaces in R^(2,2)
problem Representing Lorentzian surfaces in R^(2,2)
method Spinor and Lorentz number approach
result New spinor representation formula for flat Lorentzian surfaces in Anti-de Sitter space
Study calculates volume variations for hyperbolic 3-manifolds.
problem Calculating volume variations for hyperbolic 3-manifolds.
method Uses Lie algebra cohomology and variation strategy similar to Reznikov's rigidity.
result Derives a variation formula for the volume of representations in SL_n(C).
Paper studies Iwasawa invariants for 3-manifolds, proving a formula similar to Kida's.
problem Analogizing Iwasawa invariants to 3-dimensional topology.
method Using -adic representations of a finite group and parallel to Iwasawa's second proof.
result Proves an analogue of Kida's formula for -invariants in -extensions of -fields for 3-manifolds.
Cataclysm deformations study Anosov representations, leading to new formulas and non-open sets.
problem Understanding Anosov representations and their deformations.
method Cataclysm deformations based on twisted transverse cocycles.
result Uniform convergence of cataclysm deformations on compact sets.
G2SAT learns to generate SAT formulas from real-world examples.
problem Lack of diverse real-world SAT formulas for testing and benchmarking.
method Generative neural network that learns to transform real-world SAT formulas into latent graph representations.
result G2SAT generates SAT formulas that closely resemble real-world instances and improves solver performance.