Indian Institute of Technology Palakkad&nbsp;(IIT Palakkad or IIT PKD) is a public autonomous&nbsp;engineering&nbsp;and&nbsp;research&nbsp;institute located in&nbsp;Palakkad,&nbsp;Kerala. It was&nbsp;one of the five new&nbsp;IITs&nbsp;proposed in the&nbsp;2014 Union budget of India. The campus was inaugurated on 3 August 2015. The institute has 94 faculty, 978 students, and 63 non-teaching staffs.

IT Palakkad currently offers four-year Bachelor of Technology (B.Tech) and two years MS programmes in 8 different engineering disciplines including&nbsp;Computer Science, Electrical, Mechanical, Civil among others.

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IIT Palakkad

A Lie bracket defined on the linear span of the free homotopy classes of undirected closed curves on surfaces was discovered in stages passing through Thurston’s earthquake deformations, Wolpert’s corresponding calculations with Hamiltonian vector fields and Goldman’s algebraic treatment of the latter leading to a Lie bracket on the span of directed closed curves. The purpose of this work is to deepen the understanding of the former Lie bracket which will be referred to as the Thurston-Wolpert-Goldman Lie bracket or, briefly, the TWG bracket.We give a local direct geometric definition of the TWG bracket and use this geometric point of view to prove three results: firstly, the center of the TWG bracket is the Lie sub algebra generated by the class of the trivial loop and the classes of loops parallel to boundary components or punctures; secondly the analogous result holds for the centers of the universal enveloping algebra and of the symmetric algebra determined by the TWG Lie algebra; and thirdly, in terms of the natural basis, the TWG bracket of two non-central curves is always a linear combination of non-central curves. These three results hold for surfaces with or without boundary. We also give a brief and more illuminating proof of a known result, namely, the TWG bracket counts intersection between a simple closed curve and any other closed curve.We conclude by discussing substantial computer evidence suggesting an unexpected and strong conjectural statement relating the intersection structure of curves and the TWG bracket, namely, if the TWG bracket of two distinct undirected curves is zero then these curve classes have disjoint representatives. The computer experiments were performed on curves on surfaces with boundary.The main tools are basic hyperbolic geometry and Thurston’s earthquake theory.

World University root node

Department of Mathematics

American Mathematical Society

The Lie bracket of undirected closed curves on a surface

Transactions of the American Mathematical Society

The Lie bracket of undirected curves on a surface

arXiv preprint arXiv:2011.

Poisson algebras of loops on surfaces and skein algebras associated to their quantization

Fulltext

Journal of Topology and Analysis

On the entropy norm on the group of diffeomorphisms of closed oriented surface

We first give a new direct geometric definition of a Lie bracket of two undirected curves on a surface which was found originally by Wolpert and Goldman in the eighties. and which is referred to here as the GW bracket. The geometric picture reveals readily certain nontrivial known results, unveils new ones and also motivates an unexpected but likely true conjecture. The conjecture reduces a hard to obtain geometric property, disjointness, to a structured algebraic calculation. The new result for the GW-bracket is that its center consists precisely of the obvious central elements, namely the Lie sub algebra generated by the class of the trivial loop and the classes of loops parallel to the boundary components or punctures. We also prove in two short steps that there is no cancellation of geometric terms in the GW bracket of two curves if one curve is simple: firstly by showing that a cancelling pair of terms must have supplementary angles between geodesic representatives, by noting secondly, earthquaking along the simple geodesic changes these angles monotonically, this leading to the contradiction. Concerning the conjecture, we have substantial computer evidence suggesting that if the GW-bracket of two undirected curves is zero then these curves have disjoint representatives. This possibility was unanticipated by the history. The GW-bracket and an initial non-cancellation result go back to more algebraic work of first Wolpert on Poisson Lie algebras and then Goldman on his Lie algebras of directed and of undirected curves, all in the 80's.

arXiv e-prints, arXiv: 1910.

The Goldman-Wolpert Lie algebra of undirected curves

We study the geometric properties of the terms of the Goldman bracket between two free homotopy classes of oriented closed curves in a hyperbolic surface. We provide an obstruction for the equality of two terms in the Goldman bracket, namely if two terms in the Goldman bracket are equal to each other then for every hyperbolic metric, the angles corresponding to the intersection points are equal to each other. As a consequence, we obtain an alternative proof of a theorem of Chas, i.e., if one of the free homotopy classes contains a simple representative then the geometric intersection number and the number of terms (counted with multiplicity) in the Goldman bracket are the same. © 2018, University at Albany. All rights reserved.

New York Journal of Mathematics

Equal angles of intersecting geodesics for every hyperbolic metric

Irreducible Artin groups of finite type can be parametrized via their associated Coxeter diagrams into six sporadic examples and four infinite families, each of which is further parametrized by the natural numbers. Within each of these four infinite families, we investigate the relationship between elementary equivalence and isomorphism. For three out of the four families, we show that two groups in the same family are equivalent if and only if they are isomorphic; a positive, but weaker, result is also attained for the fourth family. In particular, we show that two braid groups are elementarily equivalent if and only if they are isomorphic. The 1 fragment suffices to distinguish the elementary theories of the groups in question. As a consequence of our work, we prove that there are infinitely many elementary equivalence classes of irreducible Artin groups of finite type. We also show that mapping class groups of closed surfaces - a geometric analogue of braid groups - are elementarily equivalent if and only if they are isomorphic. © 2018 World Scientific Publishing Company.

International Journal of Algebra and Computation

Elementary equivalence in Artin groups of finite type

Goldman Bracket: Center, Geometric Intersection Number &amp; Length Equivalent Curves

We show that the center of the Goldman Lie algebra associated to a closed orientable surface is generated by the class of the trivial loop. For an orientable nonclosed surface of finite type, the center is generated by closed curves which are either homotopically trivial or homotopic to boundary components or punctures. © 2016, Mathematical Sciences Publishers. All rights reserved.

Algebraic and Geometric Topology

Center of the Goldman Lie Algebra

arXiv preprint arXiv:1609.

A NON-CANCELLATION LEMMA FOR GOLDMAN BRACKET

A pair of free homotopy classes of closed curves in an orientable surface $F$ with negative Euler characteristic are said to be length equivalent if for any hyperbolic structure on $F$, the length of the geodesic representative of one class is equal to the length of the geodesic representative of the other class. Given an orientable hyperbolic surface $F$ with negative Euler characteristic, any self intersecting geodesic $\alpha$ on $F$ and any self-intersection point $P$ of $\alpha$, we construct infinitely many pairs of length equivalent curves in $F$. We also relate these examples with the terms of Goldman bracket.