arXiv research

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…

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

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 …

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.

Minimal triangulations for 229 hyperbolic census knots discovered.

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.

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,…

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…

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…

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.

This paper contains a generalization of the convex ideal case of the Thurston-Andreev theorem when the genus is greater than 1. The heart of the paper concerns taking formal angle data on a surface and ``conformally flowing'' this formal angle data to uniquely associated uniform angle data. This flow turns out to be th…

We propose studies of special Riemannian geometries with structure groups $H_1=SO(3)\subset SO(5)$, $H_2=SU(3)\subset SO(8)$, $H_3=Sp(3)\subset SO(14)$ and $H_4=F_4\subset SO(26)$ in respective dimensions 5, 8, 14 and 26. These geometries, have torsionless models with symmetry groups $G_1=SU(3)$, $G_2=SU(3)\times SU(3)…

New parameterization of Teichmüller spaces for complex Lie groups.

Ridge regularization simplifies model complexity in data science.

We give a complete description of exceptional surgeries on pretzel knots of type $(-2, p, p)$ with $p \ge 5$. It is known that such a knot admits a unique toroidal surgery yielding a toroidal manifold with a unique incompressible torus. By cutting along the torus, we obtain two connected components, one of which is a t…

Let $δ_{g,n}$ be the minimal dilatation of pseudo-Anosovs defined on an orientable surface of genus $g$ with $n$ punctures. Tsai proved that for any fixed $g \ge 2$, the logarithm of the minimal dilatation $\log δ_{g,n}$ is on the order of $\frac{\log n}{n}$. The main result of this paper is that if $2g+1$ is relativel…

Using nonlinear pde techniques, we construct a new family of globally smooth tt* structures. This includes tt* structures associated to the (orbifold) quantum cohomology of a finite number of complex projective spaces and weighted projective spaces. The existence of such "magical solutions" of the tt* equations, namely…

This paper concerns the set $\hat{\mathcal{M}}$ of pseudo-Anosovs which occur as monodromies of fibrations on manifolds obtained from the magic 3-manifold $N$ by Dehn filling three cusps with a mild restriction. We prove that for each $g$ (resp. $g \not\equiv 0 \pmod{6}$), the minimum among dilatations of elements (res…

Our main aim is to provide a uniform geometric characterization of the analogues over arbitrary fields of the four complex Severi varieties, i.e.~the quadric Veronese varieties in 5-dimensional projective spaces, the Segre varieties in 8-di\-men\-sional projective spaces, the line Grassmannians in 14-dimensional projec…

We discuss the local and global problems for the equivalence of geometric structures of an arbitrary order and, in later sections, attention is given to what really matters, namely the equivalence with respect to transformations belonging to a given pseudo-group of transformations. We first give attention to general pr…

It is known that any target function is realized in a sufficiently small neighborhood of any randomly connected deep network, provided the width (the number of neurons in a layer) is sufficiently large. There are sophisticated theories and discussions concerning this striking fact, but rigorous theories are very compli…

To promote economic stability, finance should be studied as a hard science, where scientific methods apply. When a trading strategy is proposed, the underlying model should be transparent and defined robustly to allow other researchers to understand and examine it thoroughly. Like any hard sciences, results must be rep…

Principal component analysis (PCA) is a mainstay of modern data analysis - a black box that is widely used but (sometimes) poorly understood. The goal of this paper is to dispel the magic behind this black box. This manuscript focuses on building a solid intuition for how and why principal component analysis works. Thi…

The Sampled Gaussian Mechanism's noise level decreases with larger subsampling rates, improving privacy-utility trade-offs.

Explains classic quantitative strategies and their workings.