arXiv research

This paper has been withdrawn due to a error in Theorem 3.1.

We give a short proof of the Morse index theorem for geodesics in semi-Riemannian manifolds by using K-theory. This makes the Morse index theorem reminiscent of the Atiyah-Singer index theorem for families of selfadjoint elliptic operators.

Proves Poincaré surgery theorem using homotopy theory.

New Lie theoretic proof for complex homogeneous manifolds.

We clarify measurability assumptions in the agnostic PAC learning theorem.

We prove a discrete Gauss-Bonnet-Chern theorem which states where summing the curvature over all vertices of a finite graph G=(V,E) gives the Euler characteristic of G.

Proves a lattice version of the Atiyah-Singer index theorem.

Proves a generalized vanishing theorem for quasi-smooth stacks, with applications in K-theory and birational geometry.

Unified approach to totally ramified values in various surface theories.

Stokes' theorem's boundary maximizes entropy.

Proof of Donaldson's theorem using Seiberg-Witten equations for multiple spinors.

This paper has been withdrawn by the author.

Mathematical framework for minimum enclosing ball problem.

Graph theory connects automorphisms to cohomology.

Following a line of reasoning suggested by Eliashberg, we prove Cerf's theorem that any diffeomorphism of the 3-sphere extends over the 4-ball. To this end we develop a moduli-theoretic version of Eliashberg's filling-with-holomorphic-discs method.

New proofs and refined theorems on bounded cohomology.

The Burde--de Rham theorem is extended to finitely presented pro-$p$ groups with specific conditions.

In this paper we explain how Morse theory for the Yang-Mills functional can be used to prove an analogue, for surface groups, of the Atiyah-Segal theorem. Classically, the Atiyah-Segal theorem relates the representation ring R(Γ) of a compact Lie group $Γ$ to the complex K-theory of the classifying space $BΓ$. For infi…

This is the third of three papers about the Compression Theorem: if M^m is embedded in Q^q X R with a normal vector field and if q-m > 0, then the given vector field can be straightened (ie, made parallel to the given R direction) by an isotopy of M and normal field in Q X R. The theorem can be deduced from Gromov's th…

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…

The exchange algorithm is studied for its convergence and asymptotic variance.

We formulate and establish a generalization of Kollár's injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Kollár's torsion-freeness, Kollár's vanishing theorem, and a generic vanishing theorem for pseudo-effective line bundles. Our approach is not Hodge…

A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. We compute a mapping class group of a hypekahler manifold $M$, showing that it is commensurable to an arithmetic subgroup in SO(3, b_2-3). A Teichmuller space of $M$ is a space of complex structures on $M$ up to is…

A classical theorem due to Quillen (1969) identifies the unitary bordism ring with the Lazard ring, which classifies the universal one-dimensional commutative formal group law. We prove an equivariant generalization of this result by identifying the homotopy theoretic $\mathbb{Z}/2$-equivariant unitary bordism ring, in…

We have been studying the index theory for some special infinite-dimensional manifolds with a "proper cocompact" actions of the loop group LT of the circle T, from the viewpoint of the noncommutative geometry. In this paper, we will introduce the LT-equivariant KK-theory and we will construct three KK-elements: the ind…

In the article arXiv:1108.5443 we established a general group-theoretical approach to the construction of Bäcklund transformations. We then showed how this construction can be applied to construct Bäcklund transformation between equations which are Darboux integrable. Here we give a number of detailed examples and new …

A key challenge for modern Bayesian statistics is how to perform scalable inference of posterior distributions. To address this challenge, variational Bayes (VB) methods have emerged as a popular alternative to the classical Markov chain Monte Carlo (MCMC) methods. VB methods tend to be faster while achieving comparabl…

Equivariant cohomology simplifies symplectic manifold integrals with group actions.

We investigate index theory in the context of Dirac operators coupled to superconnections. In particular, we prove a local index theorem for such operators, and for families of such operators. We investigate eta-invariants and prove an APS-theorem, and construct a geometric determinant line bundle for families of such …

In this thesis, we study value distribution theoretical properties of the Gauss map of pseudo-algebraic minimal surfaces in n-dimensional Euclidean space. After reviewing basic facts, we give estimates for the number of exceptional values and the totally ramified value numbers and the corresponding unicity theorems for…

The paper proves a smooth Birman-Hilden theorem for hyperkähler manifolds.

The Noether theorem is extended to stochastic control problems using contact symmetries.

Measures time-delay embedding for noisy, sparse data.

New obstructions for embedding one compact oriented 3-manifold in another are given. A theorem of D. Krebes concerning 4-tangles embedded in links arises as a special case. Algebraic and skein-theoretic generalizations for 2n-tangles provide invariants that persist in the corresponding invariants of links in which they…

Survey various symmetry notions for toric varieties.

We define the LS-category cat_g by means of covers of a space by general subsets, and show that this definition coincides with the classical Lusternik-Schnirelmann category for compact metric ANR spaces. We apply this result to give short dimension theoretic proofs of the Grossman-Whitehead theorem and Dranishnikov's t…

In this paper, we prove a Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on compact manifolds with or without boundary. As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action on boundary. We also compute the spectral action for Dirac operators with two-fo…

The grassmannian of hermitian lagrangian spaces in $\mathbb{C}^n\oplus \mathbb{C}^n$ is a natural compactification of the space of hermitian $n\times n$ matrices. We describe a Schubert-like, Whitney regular stratification on this space which has a Morse theoretic origin. We prove that these strata define closed subana…

This paper investigates Lie Quandles and Leibniz Racks, extending Noether's first theorem.

Study on nonlinear elliptic equations with variable exponents, proving existence and multiplicity of solutions.

By proving graph theoretical versions of Green-Stokes, Gauss-Bonnet and Poincare-Hopf, core ideas of undergraduate mathematics can be illustrated in a simple graph theoretical setting. In this pedagogical exposition we present the main proofs on a single page and add illustrations. While discrete Stokes is is old, the …

The paper establishes a Poisson integral formula for bounded pluriharmonic functions on Teichmüller space.

Simplified proof of Wang's theorem on complex homogeneous manifolds.

Notes on embedding criteria for smooth manifolds.

Graph Shift (GS) algorithms are recently focused as a promising approach for discovering dense subgraphs in noisy data. However, there are no theoretical foundations for proving the convergence of the GS Algorithm. In this paper, we propose a generic theoretical framework consisting of three key GS components: simplex …

Study smooth embeddings of line configurations in complex projective plane.

The article discusses extensions of Harish-Chandra's admissibility theorem.

Anabelian geometry reformulated using Hodge theory for hyperbolic curves.