Related papers: Bordism of constrained Morse functions

Let x be a random vector coming from any k-wise independent distribution over {-1,1}^n. For an n-variate degree-2 polynomial p, we prove that E[sgn(p(x))] is determined up to an additive epsilon for k = poly(1/epsilon). This answers an open…

We restrict geometric tangential equivariant complex $T^n$-bordism to torus manifolds and provide a complete combinatorial description of the appropriate non-commutative ring. We discover, using equivariant $K$-theory characteristic…

We give a Morse-theoretic characterization of simple closed geodesics on Riemannian $2$-spheres. On any Riemannian $2$-sphere endowed with a generic metric, we show there exists a simple closed geodesic with Morse index $1$, $2$ and $3$. In…

We prove a Morse index theorem for action functionals on paths that are allowed to reflect at a hypersurface (either in the interior or at the boundary of a manifold). Both fixed and periodic boundary conditions are treated.

In this paper and in the forthcoming Part II we introduce a Morse complex for a class of functions f defined on an infinite dimensional Hilbert manifold M, possibly having critical points of infinite Morse index and coindex. The idea is to…

We study the cobordism of manifolds with boundary, and its applications to codimension 2 embeddings $M^m\subset N^{m+2}$, using the method of the algebraic theory of surgery. The first main result is a splitting theorem for cobordisms of…

We prove cobordism index invariance for pseudo-differential elliptic operators on closed orbifolds with $K$--theoretical methods.

According to Kiyoshi Igusa a generalized Morse function on an n-dimensional manifold M is a smooth function with only Morse and birth-death singularities and a framed function is a generalized Morse function with an additional structure: a…

We pose some open problems related to boundedness of real-valued functions on balleans and coarse spaces. Also we prove that the Bergman property of groups is a coarse invariant. A special attention is payed to balleans on groups.

Let $X$ be a complex affine variety in $\mathbb{C}^N$, and let $f:\mathbb{C}^N\to \mathbb{C}$ be a polynomial function whose restriction to $X$ is nonconstant. For $g:\mathbb{C}^N \to \mathbb{C}$ a general linear function, we study the…

For a stably framed Liouville manifold X , we construct a "Donaldson-Fukaya category over the sphere spectrum" F(X; S). The objects are closed exact Lagrangians whose Gauss maps are nullhomotopic compatibly with the ambient stable framing,…

In this note we consider Boolean functions defined on the discrete cube equipped with a biased product probability measure. We prove that if the spectrum of such a function is concentrated on the first two Fourier levels, then the function…

We find a recursive expression for the Bessel function of S. I. Gelfand for irreducible generic representations of $\operatorname{GL}_n\left(\mathbb{F}_q\right)$. We show that special values of the Bessel function can be realized as the…

We construct a new family, indexed by the odd integers $N\geq 1$, of $(2+1)$-dimensional quantum field theories called {\it quantum hyperbolic field theories} (QHFT), and we study its main structural properties. The QHFT are defined for…

In 1998, Mukherjee and Sankaran posed two problems concerning the algebraic structure of the equivariant bordism ring of smooth closed $(\mathbb{Z}_2)^k$-manifolds with only isolated fixed points. One is the property of being finitely…

Recently the author has introduced cobordism-like modules induced from generic maps whose codimensions are negative. They are generalizations of cobordism modules of manifolds. They have been introduced in generalizing the following theorem…

Let $X$ be a reduced connected $k$-scheme pointed at a rational point $x \in X(k)$. By using tannakian techniques we construct the Galois closure of an essentially finite $k$-morphism $f:Y\to X$ satisfying the condition…

Given a simplicial complex $K$ and an injective function $f$ from the vertices of $K$ to $\mathbb{R}$, we consider algorithms that extend $f$ to a discrete Morse function on $K$. We show that an algorithm of King, Knudson and Mramor can be…

The main goal of this paper is to give a unified treatment to many known cuplength estimates. As the base case, we prove that for $C^0$-perturbations of a function which is Morse-Bott along a closed submanifold, the number of critical…

In this paper we introduce Morse Lie groupoid morphisms and study their main properties. We show that this notion is Morita invariant which gives rise to a well defined notion of Morse function on differentiable stacks. We show a groupoid…