arXiv research

Geometrically decomposes Kähler functions on toric manifolds.

Geometrically computes superpotentials for certain 4D N=2 theories.

Root Laplacian Eigenmaps help in spectral embedding of graphs.

A theory of feature geometry using spectral analysis of weight matrices.

We study the relationship between Bar-Natan's perturbation in Khovanov homology and Szabo's geometric spectral sequence, and construct a link invariant that generalizes both into a common theory. We study a few properties of the new invariant, and introduce a family of s-invariants from the new theory in the same spiri…

New approach to geometric quantization for symplectic manifolds.

Paper explores SNN for learning spectral geometric info from data.

This paper develops a new theory for ensemble learning beyond variance reduction.

Unified view of spectral networks linking geometry and gauge theory.

New spectral sequences derived from shellable tilings.

The aim of this paper is to suggest a new viewpoint to study qualitative properties of solutions of semilinear elliptic PDE's defined outside a compact set. The relevant tools come from spectral theory and from a combination of stochastic properties of the relevant differential operators. Possible links between spectra…

A simple equation explains standard model coupled to gravity.

Computes immersions of $C_2$-projective spaces using K-theory.

We introduce the notion of a Khovanov-Floer theory. Roughly, such a theory assigns a filtered chain complex over Z/2 to a link diagram such that (1) the E_2 page of the resulting spectral sequence is naturally isomorphic to the Khovanov homology of the link; (2) this filtered complex behaves nicely under planar isotopy…

Survey of analytic and geometric results on fibred cusp spaces.

Harmonic gauge simplifies geometric analysis of Riemannian metrics.

We give a geometric characterisation of the topological invariants associated to SO(m,m+1)-Higgs bundles through KO-theory and the Langlands correspondence between orthogonal and symplectic Hitchin systems. By defining the split orthogonal spectral data, we obtain a natural grading of the moduli space of SO(m,m+1)-Higg…

The paper generalizes index theory for periodic manifolds and proves equivalence of signatures for tori.

Enhanced spectral clustering for geometric graphs improves clustering accuracy.

Study geometric quantization on K3 surfaces, showing spectral convergence.

Study spectral functionals on manifolds with torsion.

In the framework of the theory of differential coverings \cite{KV}, we discuss a general geometric construction that serves the base for the so-called Lax pairs containing differentiation with respect to the spectral parameter \cite{OS}. Such kind of objects arise, for example, when studying integrability properties of…

Cappell and Shaneson pointed out in 1978 interesting properties of Browder - Livesay invariants which are similar to differentials in some spectral sequence. Such spectral sequence was constructed in 1991 by Hambleton and Kharshiladze. This spectral sequence is closely related to a problem of realization of elements of…

Let $G$ be a finite group. Noncommutative geometry of unital $G$-algebras is studied. A geometric structure is determined by a spectral triple on the crossed product algebra associated with the group action. This structure is to be viewed as a representative of a noncommutative orbifold. Based on a study of classical o…

New theorem shows gaps in magnetic Schrödinger operator spectra for large coupling.

Overview of geometric analysis for manifold learning.

We provide a unified framework for proving Reidemeister-invariance and functoriality for a wide range of link homology theories. These include Lee homology, Heegaard Floer homology of branched double covers, singular instanton homology, and \Szabo's geometric link homology theory. We follow Baldwin, Hedden, and Lobb (a…

The paper proves spectral convergence for a specific type of geometric quantization.

The aim of this paper is to introduce and study a geometric spectral sequence in Khovanov homology. The construction was motivated by a similar spectral sequence from Khovanov homology to Heegaard Floer homology.

We study the generalization of the Willmore functional for surfaces in the three-Heisenberg group. Its construction is based on the spectral theory of the Dirac operator coming to the Weierstrass representation of surfaces (see math.DG/0503707). By using surfaces of revolution we demonstrate that it resembles the Willm…

These lecture notes are a systematic and self-contained exposition of the cohomological theories naturally related to partial differential equations: the Vinogradov C-spectral sequence and the C-cohomology, including the formulation in terms of the horizontal (characteristic) cohomology. Applications to computing invar…

We provide two examples of spectral analysis techniques of Schroedinger operators applied to geometric Laplacians. In particular we show how to adapt the method of analytic dilation to Laplacians on complete manifolds with corners of codimension 2 finding the absence of singular continuous spectrum for these operators,…

We provide a systematic approach to describing the Ramond-Ramond (RR) fields as elements in twisted differential K-theory. This builds on a series of constructions by the authors on geometric and computational aspects of twisted differential K-theory, which to a large extent were originally motivated by this problem. I…

Study connects spectral and algebraic torsion in geometric contexts.

Study spectral and index properties of Hodge-Dirac operator on compact manifolds.

Khovanov-Floer theories are a class of homological link invariants which admit spectral sequences from Khovanov homology. They include Khovanov homology, Szab{ó}'s geometric link homology, singular instanton homology, and various Floer theories applied to branched double covers. In this short note we show that certain …

We discuss generalizations of Ozsvath-Szabo's spectral sequence relating Khovanov homology and Heegaard Floer homology, focusing attention on an explicit relationship between natural Z (resp., 1/2 Z) gradings appearing in the two theories. These two gradings have simple representation-theoretic (resp., geometric) inter…

Using the notion of vacuum pairs we show how the (square of the) mass matrix of the fermions can be considered geometrically as curvature. This curvature together with the curvature of space-time, defines the total curvature of the Clifford module bundle representing a ``free'' fermion within the geometrical setup of s…

This paper is part of a series of articles on noncommutative geometry and conformal geometry. In this paper, we reformulate the local index formula in conformal geometry in such a way to take into account of the action of conformal diffeomorphisms. We also construct and compute a whole new family of geometric conformal…

Proves new inequality linking spectral numbers of Lagrangians and their reductions.

We study the spectral theory of asymptotically hyperbolic manifolds with ends of warped product type. Our main result is an upper bound on the resonance counting function with a geometric constant expressed in terms of the respective Weyl constants for the core of the manifold and the base manifold defining the ends.

In this paper we prove a tertiary index theorem which relates a spectral geometric and a homotopy theoretic invariant of an almost complex manifold with framed boundary. It is derived from the index theoretic and homotopy theoretic versions of a complex elliptic genus and interestingly related with the structure of the…

Study Floer theory of hyperbolic three-manifolds using Dirac spectral flow.

Ozsvath and Szabo have established an algebraic relationship, in the form of a spectral sequence, between the reduced Khovanov homology of (the mirror of) a link L in S^3 and the Heegaard Floer homology of its double-branched cover. This relationship has since been recast by the authors as a specific instance of a broa…

Constructs QFT on curved surfaces, proving axioms and calculating entropy.

We study the spectral gap of the Erdős--Rényi random graph through the connectivity threshold. In particular, we show that for any fixed $δ> 0$ if $p \ge \frac{(1/2 + δ) \log n}{n},$ then the normalized graph Laplacian of an Erdős--Rényi graph has all of its nonzero eigenvalues tightly concentrated around $1$. We est…

Geometric Arbitrage Theory reformulates a generic asset model possibly allowing for arbitrage by packaging all assets and their forwards dynamics into a stochastic principal fibre bundle, with a connection whose parallel transport encodes discounting and portfolio rebalancing, and whose curvature measures, in this geom…

New construction of Fukaya-Seidel categories using complex gradient flow equation.