arXiv research

Classifies limits of groups of involutions in SL(2,F) over local fields.

Define web algebras for annular SL(2) and SL(3) using foam TQFTs.

Classifies actions of SL(n,R) and SL(n,Z) on closed n-manifolds.

Study of Lagrangian submanifolds in pseudo-nearly Kähler SL(2,R)×SL(2,R).

We show the correspondence between left invariant flat projective structures on Lie groups and certain prehomogeneous vector spaces. Moreover by using the classification theory of prehomogeneous vector spaces, we classify complex Lie groups admitting irreducible left invariant flat complex projective structures. As a r…

Explicitly found generators of cohomology for SL_n(Z) using sharbly cycles and cosharbly cocycles.

Extends $sl(3)$ invariant to tangles via webs and functors.

We construct an equivariant colored sl(N)-homology for links, which generalizes both the colored sl(N)-homology defined by the author and the equivariant sl(N)-homology defined by Krasner. The construction is a straightforward generalization of that of the colored sl(N)-homology. The proof of invariance is based on a s…

The paper classifies topological properties of specific geometric forms on manifolds.

New sl(2) action defined on a mathematical module.

It is showed that many examples of AMD submanifolds of higher dimensions come from SL normal bundles. A symmetry property of SL submanifolds and Björling type problem for SL normal bundles are also mentioned.

Classifies real-analytic SL(n,R) actions on closed manifolds.

We construct a categorification of the quantum sl_3 projectors, the sl_3 analog of the Jones-Wenzl projectors, as the stable limit of the complexes assigned to k-twist torus braids (as k goes to infinity) in a suitably shifted version of Morrison and Nieh's geometric formulation of sl_3 link homology (math.GT/0612754).…

We study the Chabauty compactification of two families of closed subgroups of $SL(n,\mathbb{Q}_p)$. The first family is the set of all parahoric subgroups of $SL(n,\mathbb{Q}_p)$. Although the Chabauty compactification of parahoric subgroups is well studied, we give a different and more geometric proof using various Le…

Decomposes $SL_3$ skein algebras for surfaces.

We define two functors from Elias and Khovanov's diagrammatic Soergel category, one targeting Clark-Morrison-Walker's category of disoriented sl(2) cobordisms and the other the category of (universal) sl(3) foams.

The study proves non-existence of hypercomplex structures on SL(3,R) and finds one on SL(2n+1,C).

We classify SL(2;C)-representations of a Brieskorn homology 3-sphere. We show any irreducible representation into SL(2;C) is conjugate to that into either SU(2) or SL(2;R). We also give a construction of SL(2;R)-representations for a Brieskorn homology 3-sphere from PSL(2;R)-representations of the base orbifold fundame…

Odd-dimensional SL(n,Q) contains dense surface subgroups.

Two specific Einstein metrics found on a product of SL(2,R) groups.

The paper calculates colored Jones polynomials for specific link configurations.

Given a finite volume hyperbolic 3-manifold, we compose a lift of the holonomy in SL(2,C) with the n-dimensional irreducible representation of SL(2,C) in SL(n,C). In this paper we give local coordinates of the SL(n,C)-character variety around this representation. As a corollary, this representation is isolated among al…

Study local limits of subgroups in $SL_3(\mathbb{R})$.

We introduce two invariants called sl(3) Khovanov module and pointed sl(3) Khovanov homology for spatial webs (bipartite trivalent graphs). Those invariants are related to Kronheimer-Mrowka's instanton invariants $J^\sharp$ and $I^\sharp$ for spatial webs by two spectral sequences. As an application of the spectral seq…

The paper constructs bases for cluster varieties using ${ m SL}_3$-webs and laminations.

Study on cohomology of SL_n(Z) for n>=3, proving vanishing of certain cohomology groups.

SL(n) covariant valuations on Orlicz spaces are represented and characterized.

Lectures introduce evaluation of SL(3) foams and link homology.

The paper classifies totally geodesic Lagrangian submanifolds in a specific pseudo-nearly Kähler space.

Study SL(2,C) character schemes for finitely generated groups.

Extends knot invariant computation to symmetrically colored sl_N.

Paper proves zero stability for one-row colored sl₃-Jones polynomials.

New operator reveals unique features of sl(N) link homology.

In this paper I define certain interesting 2-functors from the Khovanov-Lauda 2-category which categorifies quantum sl(k), for any k>1, to a 2-category of universal sl(3) foams with corners. For want of a better name I use the term "foamation" to indicate those 2-functors. I conjecture the existence of similar 2-functo…

New central elements found in a quantum algebra related to knot theory.

Quantum cluster algebra constructed from web skein relations on surfaces.

Combining known spectral sequences with a new spectral sequence relating reduced and unreduced sl(N)-homology yields a relationship between the Homflypt-homology of a knot and its sl(N)-concordance invariants. As an application, some of the sl(N)-concordance invariants are shown to be linearly independent.

We find two different families of $Sp(2,R)$ symmetric $G_2$ structures in seven dimensions. These are $G_2$ structures with $G_2$ being the split real form of the simple exceptional complex Lie group $G_2$. The first family has $τ_2\equiv 0$, while the second family has $τ_1\equivτ_2\equiv 0$. The families are differen…

Synthetic lethality (SL) is a promising concept for novel discovery of anti-cancer drug targets. However, wet-lab experiments for detecting SLs are faced with various challenges, such as high cost, low consistency across platforms or cell lines. Therefore, computational prediction methods are needed to address these is…

SL(2,Z) forms lead to new anomaly formulas.

The main purpose of this paper is calculation of differential invariants which arise from prolonged actions of two Lie groups SL(2) and SL(3) on the $n$th jet space of $R^2$. It is necessary to calculate $n$th prolonged infenitesimal generators of the action.

Researchers calculate colored Jones polynomials for pretzel links using Kuperberg's theory.

Quantum Frobenius map for $SL_3$ skein modules constructed and described.

A new invariant for links generalizes Alexander polynomial for sl_3.

We compute the fundamental class (in the extended Bloch group) for representations of fundamental groups of 3-manifolds to SL(4,R) that factor over SL(2,C), in particular for those factoring over the isomorphism PSL(2,C) = S0(3,1). We also discuss consequences for the number of connected components of SL(4,R)-character…

Study reveals $\mathfrak{sl}_2$ action on link homology of specific torus links.

We characterize groups admitting Anosov representations into $\mathsf{SL}(3,\mathbb R)$, projective Anosov representations into $\mathsf{SL}(4,\mathbb R)$, and Borel Anosov representations into $\mathsf{SL}(4,\mathbb R)$. More generally, we obtain bounds on the cohomological dimension of groups admitting $P_k$-Anosov r…

This is the fourth in a series of papers math.DG/0008021, math.DG/0008155, math.DG/0010036 constructing explicit examples of special Lagrangian submanifolds (SL m-folds) in C^m. A submanifold of C^m is ruled if it is fibred by a family of real straight lines in C^m. This paper studies ruled special Lagrangian 3-folds i…