arXiv research

By adapting methods of \cite{AC} we prove a sharp estimate on the expansion modulus of the gradient of the log of the parabolic kernel to the Schördinger operator with convex potential, which improves an earlier work of Brascamp-Lieb. We also include alternate proofs to the improved log-concavity estimate, and to the f…

The paper discusses rigidity results for inequalities on weighted Riemannian manifolds.

Information concentration of probability measures have important implications in learning theory. Recently, it is discovered that the information content of a log-concave distribution concentrates around their differential entropy, albeit with an unpleasant dependence on the ambient dimension. In this work, we prove th…

The SCMU algorithm computes cone factorizations for symmetric cones, improving upon existing methods.

We extend the Framization of the Temperley-Lieb algebra to Coxeter systems of type $\mathtt{B}$. We first define a natural extension of the classical Temperley-Lieb algebra to Coxeter systems of type $\mathtt{B}$ and prove that such an extension supports a unique linear Markov trace function. We then introduce the Fram…

In this paper we introduce the Yokonuma-Temperley-Lieb algebra as a quotient of the Yokonuma-Hecke algebra over a two-sided ideal generated by an expression analogous to the one of the classical Temperley-Lieb algebra. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yok…

An algorithm for computing positive semidefinite factorizations of matrices.

We propose a framization of the Temperley-Lieb algebra. The framization is a procedure that can briefly be described as the adding of framing to a known knot algebra in a way that is both algebraically consistent and topologically meaningful. Our framization of the Temperley-Lieb algebra is defined as a quotient of the…

Sharp comparison theorems are derived for all eigenvalues of the (weighted) Laplacian, for various classes of weighted-manifolds (i.e. Riemannian manifolds endowed with a smooth positive density). Examples include Euclidean space endowed with strongly log-concave and log-convex densities, extensions to $p$-exponential …

Simplified proof for Frank and Lieb's inequality on Heisenberg group.

Projectors defined in virtual Temperley-Lieb algebra with properties similar to Jones-Wenzl projectors.

The study proves Lieb-Thirring inequalities on hyperbolic manifolds.

The Temperley-Lieb algebra is a fundamental component of SU(2) topological quantum field theories. We construct chain complexes corresponding to minimal idempotents in the Temperley-Lieb algebra. Our results apply to the framework which determines Khovanov homology. Consequences of our work include semi-orthogonal deco…

New equivariant version of Khovanov homology for annuli.

The paper proves various inequalities on gradient shrinking Ricci solitons.

We show that the Hilbert space formed from a block spin renormalization construction of a cyclic quantum spin chain (based on the Temperley-Lieb algebra) does not support a chiral conformal field theory whose Hamiltonian generates translation on the circle as a continuous limit of the rotations on the lattice.

New invariants detect exotic smooth structures in 4-manifolds.

In this paper, we derive from the supersymmetry of the Witten Laplacian Brascamp-Lieb's type inequalities for general differential forms on compact Riemannian manifolds with boundary. In addition to the supersymmetry, our results essentially follow from suitable decompositions of the quadratic forms associated with the…

We use the Jones-Wenzl idempotents to construct a basis of Temperley-Lieb algebra TL_n. This allows a short calculation for a Gram determinant of Lickorish's bilinear form on the Temperley-Lieb algebra.

The paper proves extension theorems for complex manifolds with Levi $q$-concave domains.

In this paper we show that the matrix of chromatic joins and the Gram matrix of the Temperley-Lieb algebra are similar (after rescaling), with the change of basis given by diagonal matrices.

The faithfulness of the orthogonal group case of Brauer's representation of the Brauer centralizer algebras restricted to their Temperley-Lieb subalgebras, which was established by Vaughan Jones, is here proved in a new, elementary and self-contained, manner.

The study proves non-existence of concave functions on specific metric spaces.

The paper explores inequalities on weighted Riemannian manifolds with boundary.

Proves a synthetic Lorentzian Cartan-Hadamard theorem.

The monoids of simplicial endomorphisms, i.e. the monoids of endomorphisms in the simplicial category, are submonoids of monoids one finds in Temperley-Lieb algebras, and as the monoids of Temperley-Lieb algebras are linked to situations where an endofunctor is adjoint to itself, so the monoids of simplicial endomorphi…

A function is exponentially concave if its exponential is concave. We consider exponentially concave functions on the unit simplex. In a previous paper we showed that gradient maps of exponentially concave functions provide solutions to a Monge-Kantorovich optimal transport problem and give a better gradient approximat…

In this survey we collect all results regarding the construction of the Framization of the Temperley-Lieb algebra of type $A$ as a quotient algebra of the Yokonuma-Hecke algebra of type $A$. More precisely, we present all three possible quotient algebras the emerged during this construction and we discuss their dimensi…

We determine the image of the braid groups inside the Temperley-Lieb algebras, defined over finite field, in the semisimple case, and for suitably large (but controlable) order of the defining (quantum) parameter. We also prove that, under natural conditions on this parameter, the representations of the Hecke algebras …

Let $n$ be a nonnegative integer, we use ribbon $n-$graph diagrams and the Yamada polynomial skein relations to construct an algebra ${\mathcal Y}_n$ which is shown to be closely related to the Temerley-Lieb Algebra. We prove that the algebra ${\mathcal Y}_2$ is isomorphic to some quotient of a three variables polynomi…

New saddle network architectures preserve convex-concave geometry in optimization problems.

The paper studies combinatorics of injective words in the context of Temperley-Lieb algebras.

In this paper, we proved a rigidity theorem of the Hodge metric for concave horizontal slices and a local rigidity theorem for the monodromy representation.

In this paper we first present the construction of the new 2-variable classical link invariants arising from the Yokonuma-Hecke algebras ${\rm Y}_{d,n}(q)$, which are not topologically equivalent to the Homflypt polynomial. We then present the algebra ${\rm FTL}_{d,n}(q)$ which is the appropriate Temperley-Lieb analogu…

Proves Calabi-Yau theorem for certain nonnegative curvature manifolds.

Paper studies homology and cohomology of Temperley-Lieb algebra TL_n(a).

Derives concavity inequality and estimates for $k$-Hessian equations.

The paper explores handlebody versions of various diagram algebras.

For $φ$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $\int_X e^{-φ}.$ We prove that the logarithm of the volume is concave along continuous geodesics in the space of positively curved metrics on $-K_X$ and that the concavity is strict unless the geodesic comes f…

New discrete cobordism category for nested manifolds and relations to algebraic structures.

It is known that by dualizing the Bochner-Lichnerowicz-Weitzenböck formula, one obtains Poincaré-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry-Émery Curvature-Dimension condition (combining a lower bound on its generalized Ricci curvature and an upper bound on its generalize…

New invariants derived from Kauffman bracket for 3-manifolds.

Estimates volume of convex Alexandrov spaces with boundary.

We use tools from $n$-dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold $M$. On one hand we extend a theorem of Lieb and prove that any nodal domain $Ω_λ$ almost fully contains a ball of radius $\sim \frac{1}{\sqrtλ}$. …

Researchers found counterexamples to conjectures about optimal transport maps on curved spaces.

This note is devoted to Keller-Lieb-Thirring spectral estimates for Schrödinger operators on infinite cylinders: the absolute value of the ground state level is bounded by a function of a norm of the potential. Optimal potentials with small norms are shown to depend on a single variable. The proof is a perturbation arg…

Frank and Lieb gave a new, rearrangement-free, proof of the sharp Hardy-Littlewood-Sobolev inequalities by exploiting their conformal covariance. Using this they gave new proofs of sharp Sobolev inequalities for the embeddings $W^{k,2}(\mathbb{R}^n)\hookrightarrow L^{\frac{2n}{n-2k}}(\mathbb{R}^n)$. We show that their …

This paper gives a self-contained and complete proof of the isomorphism of freely generated monoids extracted from Temperley-Lieb algebras with monoids made of Kauffman's diagrams.