arXiv research

Researchers create coordinates for hyperbolic surfaces, proving a magic formula.

We calculate the higher derivatives of length functions on Teichmuller space along earthquake deformations. This generalizes the cosine formula for the first derivative by Kerckhoff and Wolpert and the sine formula for second derivative by Wolpert.

A well-known theorem of Wolpert shows that the Weil-Petersson symplectic form on Teichmüller space, computed on two infinitesimal twists along simple closed geodesics on a fixed hyperbolic surface, equals the sum of the cosines of the intersection angles. We define an infinitesimal deformation starting from a more gene…

We present a brief but nearly self-contained proof of a formula for the Weil-Petersson Hessian of the geodesic length of a closed curve (either simple or not simple) on a hyperbolic surface. The formula is the sum of the integrals of two naturally defined positive functions over the geodesic, proving convexity of this …

Study character variety of a special 3-manifold, finding a polynomial formula.

We give a computer free proof of the Deligne, Cohen and deMan formulas for the dimensions of the irreducible $g$-modules appearing in the tensor powers of $g$, where $g$ ranges over the exceptional complex simple Lie algebras. We give additional dimension formulas for the exceptional series, as well as uniform dimensio…

Topological proof of Weil-Petersson symplectic form using Fenchel-Nielsen coordinates.

By formulating N = 1, 2, 4, 8, D = 3, Yang-Mills with a single Lagrangian and single set of transformation rules, but with fields valued respectively in R,C,H,O, it was recently shown that tensoring left and right multiplets yields a Freudenthal-Rosenfeld-Tits magic square of D = 3 supergravities. This was subsequently…

A soap film is actually a thin solid fluid bounded by two surfaces of opposite orientation. It is natural to model the film using one polyhedron for each side. Two problems are to get the polyhedra for both sides to be in the same place without canceling each other out and to model triple junctions without introducing …

Symplectic structures on Teichmüller spaces for surfaces with ideal boundary.

We prove that a certain series defines a constant function using Wolpert's formula for the variation of the length of a geodesic along a Fenchel Nielsen twist. Subsequently we determine the value viewing it as function on the the Deligne Mumford compactification and evaluating it at the stable curve at infinity.

Wolpert's cosine formula on Teichmüller space gives the Weil-Petersson Poisson bracket $\{l_α, l_β\}$ for geodesic length functions $l_α,l_β$ of closed curves $α,β$ as the sum of the cosines of the angle of intersection of the associated geodesics. This was recently generalized to Hitchin representations by Labourie. I…

This work estimates edge weights of edge-reinforced random walks using observed data.

Magic braids can be made with leatherworking technique.

MAGIC-Flow generates and classifies medical images with interpretability.

We study compact hyperbolic surface laminations. These are a generalization of closed hyperbolic surfaces which appear to be more suited to the study of Teichmüller theory than arbitrary non-compact surfaces. We show that the Teichmüller space of any non-trivial hyperbolic surface lamination is infinite dimensional. In…

We define an associative algebra AS_h(S) generated by framed arcs and links over a punctured surface S which is a quantization of the Poisson algebra C(S) of arcs and curves on S. We then construct a Poisson algebra homomorphism from C(S) to the space of smooth functions on the decorated Teichmuller space endowed with …

Minimal triangulations for 229 hyperbolic census knots discovered.

We prove that the universal Teichmuller space T(1) carries a new structure of a complex Hilbert manifold. We show that the connected component of the identity of T(1), the Hilbert submanifold T_{0}(1), is a topological group. We define a Weil-Petersson metric on T(1) by Hilbert space inner products on tangent spaces, c…

We define a Poisson Algebra called the {\em swapping algebra} using the intersection of curves in the disk. We interpret a subalgebra of the fraction algebra of the swapping algebra -- called the {\em algebra of multifractions} -- as an algebra of functions on the space of cross ratios and thus as an algebra of functio…

We investigate nonintegrable Riemannian geometries modelled after certain symmetric spaces related to the Freudenthal-Tits Magic Square. The collection of four such structures found by Nurowski is extended by further eight. A focus is given to those admitting a compatible connection with completely skew torsion.

Unified framework for imputation and prediction in healthcare time series.

We prove all knots can be transformed into a trefoil using special diagrams.

3-manifolds are CR uniformized on spheres, proving a conjecture.

MAGIC method optimally estimates model predictions changes.

New criterion for bi-ordering free groups under automorphisms.

We connect the algebraic geometry and representation theory associated to Freudenthal's magic square. We give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations, and interpreting them in terms of composition algebras. In particular, we show h…

Two multifidelity trust-region methods use low-fidelity models for efficient optimization.

Topological theory for qLDPC codes enables non-Clifford gates and magic state injection.

The paper studies the center of the Goldman Lie algebra and its properties.

For various series of complex semi-simple Lie algebras $\fg (t)$ equipped with irreducible representations $V(t)$, we decompose the tensor powers of $V(t)$ into irreducible factors in a uniform manner, using a tool we call {\it diagram induction}. In particular, we interpret the decompostion formulas of Deligne \cite{d…

In this note, we use Chern's magic form $Φ_k$ in his famous proof of the Gauss-Bonnet theorem to define a mass for asymptotically flat manifolds. It turns out that the new defined mass is equivalent to the one that we introduced recently by using the Gauss-Bonnet-Chern curvature $L_k$. Moreover, this equivalence implie…

We consider a hyperbolic surface bundle over the circle with the smallest known volume among hyperbolic manifolds having 3 cusps, so called "the magic manifold". We compute the entropy function on the fiber face of the unit ball with respect to the Thurston norm, determine homology classes whose representatives are gen…

MAGIC uncovers disease heterogeneity across brain scales.

We study the magic manifold $N$ which is a hyperbolic and fibered $3$-manifold. We give an explicit construction of a fiber $F_a$ and its monodromy $:F_a \rightarrow F_a$ of the fibration associated to each fibered class $a$ of $N$. Let $δ_g$ (resp. $δ_g^+$) be the minimal dilatation of pseudo-Anosovs (resp. pseudo-Ano…

The split version of the Freudenthal-Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [3, 17, 18]. The geometries appearing in the second row are Severi-Brauer varieties [20]. We provide an easy uniform axiomatization of these geometries and related ones,…

Proofs for spectral and geometric properties of hyperbolic surfaces.

The statistical complexity of quantum circuits is studied using Rademacher complexity.

We propose an offline-online procedure for Fourier transform based option pricing. The method supports the acceleration of such essential tasks of mathematical finance as model calibration, real-time pricing, and, more generally, risk assessment and parameter risk estimation. We adapt the empirical magic point interpol…

the main theorem gives a sufficient condition for a n elements of SL(2,R) to generate a free group.The idea behind it is to use a nonorientable version of the Dehn-Wolpert-Goldman twist and to sew it with the original representation of a free group to get representation of the closed surfase group and then to apply Gol…

Baker showed that 10 of the 12 classes of Berge knots are obtained by surgery on the minimally twisted 5-chain link. In this article we enumerate all hyperbolic knots in S^3 obtained by surgery on the minimally twisted 5 chain link that realize the maximal known distances between slopes corresponding to exceptional (le…

This paper adapts Thurston's earthquake metric to Riemann surfaces with marked points.

We obtain the classical Hanner inequalities by the Bellman function method. These inequalities give sharp estimates for the moduli of convexity of Lebesgue spaces. Easy ideas from differential geometry help us to find the Bellman function using neither "magic guesses" nor calculations.

New quantum code breaks distance barrier with transversal non-Clifford gates.

Let M be a manifold and g a Lie algebra acting on M. Differential forms Omega(M) carry a natural action of Lie derivatives L(x) and contractions I(x) of fundamental vector fields for x \in g. Contractions (anti-) commute with each other, [I(x), I(y)]=0. Together with the de Rham differential, they satisfy the Cartan's …

By using two different invariants for the Rubik's Magic puzzle, one of metric type, the other of topological type, we can dramatically reduce the universe of constructible configurations of the puzzle. Finding the set of actually constructible shapes remains however a challenging task, that we tackle by first reducing …

Historically, games of all kinds have often been the subject of study in scientific works of Computer Science, including the field of machine learning. By using machine learning techniques and applying them to a game with defined rules or a structured dataset, it's possible to learn and improve on the already existing …

Natural gradient descent avoids the magic of model parametrization, leading to different optimization outcomes.