arXiv research

Link homology compared with geometric link invariants using Bott-Samelson varieties.

New complex structures found on tangent bundles of Lie groups.

We consider the T-equivariant cohomology of Bott-Samelson desingularisations of Schubert varieties in the flag manifold of a connected semi-simple complex algebraic group of adjoint type with maximal torus T. We construct a combinatorially pure (in the sense of T. Braden and R. Macpherson) sheaf on the Bruhat graph of …

Bott and Samuelson constructed explicit cycles representing a basis of the Z_2-homology of the orbits of variationally complete representations of compact Lie groups. As a consequence, all those orbits are taut. We were able to show that an irreducible representation of a compact Lie group, all of whose orbits are taut…

Let $G$, $B$, and $H$ denote a complex semi-simple algebraic group, a Borel subgroup of $G$, and a maximal complex torus in $B$, respectively. Choose a compact real form $K$ of $G$ such that $T=K\cap H$ is a maximal torus in $T$. Then there are two models for the flag space of $G$: the complex quotient $X=G/B$ and the …

Let $G$ be a connected complex semi-simple Lie group, and let $Z_{\bf u}$ be an $n$-dimensional Bott-Samelson variety of $G$, where ${\bf u}$ is any sequence of simple reflections in the Weyl group of $G$. We study the Poisson structure $π_n$ on $Z_{\bf u}$ defined by a standard multiplicative Poisson structure $π_{\rm…

We give a construction of integrable complex structures on the total space of a smooth principal bundle over a complex manifold, with an even dimensional compact Lie group as structure group, under certain conditions. This generalizes the constructions of complex structure on compact Lie groups by Samelson and Wang, an…

We give an equivalent description of taut submanifolds of complete Riemannian manifolds as exactly those submanifolds whose normal exponential map has the property that every preimage of a point is a union of submanifolds. It turns out that every taut submanifold is also $\mathbb Z_2$-taut. We explicitely construct gen…

In this article we consider a variant of Rabinowitz Floer homology in order to define a homological count of discriminant points for paths of contactomorphisms. The growth rate of this count can be seen as an analogue of Givental's nonlinear Maslov index. As an application we prove a Bott-Samelson type obstruction theo…

Study of complex 3-folds with vanishing Bismut Ricci form via special Kähler geometry.

We introduce a new integral invariant for isometric actions of compact Lie groups, the copolarity. Roughly speaking, it measures how far from being polar the action is. We generalize some results about polar actions in this context. In particular, we develop some of the structural theory of copolarity k representations…

Let $(G,K)$ be a Riemannian symmetric pair of maximal rank, where $G$ is a compact simply connected Lie group and $K$ the fixed point set of an involutive automorphism $σ$. This induces an involutive automorphism $τ$ of the based loop space $Ω(G)$. There exists a maximal torus $T\subset G$ such that the canonical actio…

Study string topology on symmetric spaces, showing non-triviality and nilpotency results.

It was shown by Samelson and Wang that each compact Lie group K of even dimension admits left-invariant complex structures. When K has odd dimension it admits a left-invariant CR-structure of maximal dimension. This has been proved recently by Charbonnel and Khalgui who have also given a complete algebraic description …

The paper constructs a family of SKT metrics on the exceptional Lie group G2.

Develops Lie algebraic approach for compact complex homogeneous manifolds.

The paper connects different types of Lagrangian fillings to Legendrian weaves and their sheaf quantizations.

The paper examines SKT and CYT metrics on Lie groups.

The study characterizes compact homogeneous manifolds with Bismut parallel torsion.

The paper calculates quantum cohomology for coadjoint orbits and Hamiltonian groups.

The paper defines and explores vector bundles that can be turned and their properties.

Lectures on polar actions and their properties in Riemannian geometry.

Let $(X^{m+1}, g)$ be an $(m+1)$-dimensional globally hyperbolic spacetime with Cauchy surface $M^m$, and let $\widetilde M^m$ be the universal cover of the Cauchy surface. Let $\mathcal N_{X}$ be the contact manifold of all future directed unparameterized light rays in $X$ that we identify with the spherical cotangent…

We construct non-constructible simplicial $d$-spheres with $d+10$ vertices and non-constructible, non-realizable simplicial $d$-balls with $d+9$ vertices for $d\geq 3$.

A method to construct fractal surfaces by recurrent fractal curves is provided. First we construct fractal interpolation curves using a recurrent iterated functions system(RIFS) with function scaling factors and estimate their box-counting dimension. Then we present a method of construction of wider class of fractal su…

Modified curve shortening flow constructs $λ$-Angenent curve.

Article constructs coassociative submanifolds in Joyce's $G_2$-manifolds.

The twist construction is a geometric model of T-duality that includes constructions of nilmanifolds from tori. This paper shows how one-dimensional foliations on manifolds may be used in a shear construction, which in algebraic form builds certain solvable Lie groups from Abelian ones. We discuss other examples of geo…

We describe various constructions in Sasakian geometry. First we generalize the join construction of the first two authors to arbitrary Sasakian manifolds. We then give several examples, including ones which prove the existence of Sasakian-Einstein metrics on manifolds homeomorphic to $S^2\times S^5.$ Then we use a gen…

Recently, there has been emerging interest in constructing reproducing kernel Banach spaces (RKBS) for applied and theoretical purposes such as machine learning, sampling reconstruction, sparse approximation and functional analysis. Existing constructions include the reflexive RKBS via a bilinear form, the semi-inner-p…

We construct bi-invariant total orderings of residually torsion-free nilpotent groups by using Chen's iterated integrals. This construction can be seen as a generalization of the Magnus ordering of the free groups, and equivalent to the classical construction which uses an iteration of central extensions. Our geometric…

The paper constructs infinitely many surfaces with specific mean curvature.

Explains a 1978 construction for Yang-Mills instantons.

We construct several non-trivial examples of CAT(1) spaces by using the idea of free construction.

We study the performance of the adaptive construction scheme for a Bayesian inference on the Quadratic GARCH model which introduces the asymmetry in time series dynamics. In the adaptive construction scheme a proposal density in the Metropolis-Hastings algorithm is constructed adaptively by changing the parameters of t…

New pseudomodular groups constructed from jigsaw construction.

Study constructs associative submanifolds in $G_2$-manifolds from orbifolds.

Survey on rigorous construction of supersymmetric path integral.

Test-asset construction affects factor model performance.

We introduce the cutting construction of possibly non-compact symplectic toric manifolds, in particular, toric symplectic cones that correspond to a weakly convex good cone. Since the symplectization of a toric contact manifold is a toric symplectic cone, we can also construct toric contact manifolds that correspond to…

Extends knot surgery to exotic four-manifolds.

We construct calibrated submanifolds of R^7 and R^8 by viewing them as total spaces of vector bundles and taking appropriate sub-bundles which are naturally defined using certain surfaces in R^4. We construct examples of associative and coassociative submanifolds of R^7 and of Cayley submanifolds of R^8. This construct…

Researchers match complex affine structures in mirror constructions.

The paper constructs solutions to a critical Dirac equation on spheres.

In this paper, we construct the index bundle gerbe of a family of self-adjoint Dirac-type operators, refining a construction of Segal. In a special case, we construct a geometric bundle gerbe called the caloron bundle gerbe, which comes with a natural connection and curving, and show that it is isomorphic to the analyt…

Two constructions link path geometries to almost Grassmann structures.

New explicit constructions of unbalanced Ramanujan bipartite graphs.

Constructs subvarieties in translation surface strata using combinatorial input.