Algorithmic Semi-algebraic Geometry and Topology -- Recent Progress and Open Problems

Saugata Basu

Home/Archive/arXiv

arXiv · 0708.2854

Algorithmic Semi-algebraic Geometry and Topology -- Recent Progress and Open Problems

Saugata Basu

View original source ↗Read source document ↗Export metadata ↓

Abstract

We give a survey of algorithms for computing topological invariants of semi-algebraic sets with special emphasis on the more recent developments in designing algorithms for computing the Betti numbers of semi-algebraic sets. Aside from describing these results, we discuss briefly the background as well as the importance of these problems, and also describe the main tools from algorithmic semi-algebraic geometry, as well as algebraic topology, which make these advances possible. We end with a list of open problems.

Explore related subjects

math.GT cs.CC cs.CG math.AG math.AT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Saugata Basu. 2007-09-15. Algorithmic Semi-algebraic Geometry and Topology -- Recent Progress and Open Problems. https://arxiv.org/abs/0708.2854

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

3d Super Hyperbolic Geometry

Wigner's unitary representation of the Lorentz group is extended to a representation of the complex orthosymplectic Lie supergroup (\operatorname{OSp}_{\mathbb C}(1|2)) acting on Minkowski ((3,1|4))-dimensional superspace essentially by Hermitian conjugation. The invariant quadratic form is[x_1x_2-x\bar x+ϕψ+\barϕ\barψ]in Wigner's real coordinates (x_1,x_2) and complex coordinate (x), where (ϕ) and (ψ) are Dirac fermions. The extended action is linear in the superspace variables, but not quadratic in the odd group variables, and is described by an explicit even purely imaginary auxiliary parameter defined on the product of the Lie supergroup with its Lie algebra. This extension of Wigner's representation opens the door to the study of geometry in super hyperbolic three-space, initiated here with foundations on super geodesics, triangles, and tetrahedra, together with an analysis of ideal tetrahedra and a discussion of the nonzero fermionic correction to the Schläfli formula. This version includes a corrigendum that does not materially affect the main results, except for the claimed divergence of the volume of an ideal tetrahedron: the earlier divergence computation was correct as a calculation, but was carried out for an incorrect differential form.

math.GT↗

arXiv2026

Seifert cobordisms and the Chen-Yang volume conjecture

We study the large $r$ asymptotic behavior of the Turaev-Viro invariants $TV_r(M; e^{\frac{2πi}{r}})$ of 3-manifolds with toroidal boundary, under the operation of gluing a Seifert-fibered 3-manifold along a component of $\partial M$. We show that the Turaev-Viro invariants volume conjecture is closed under this operation. As an application we prove the volume conjecture for all Seifert fibered 3-manifolds with boundary and for large classes of graph 3-manifolds.

math.GT↗

arXiv2026

Equivariant lattice homology for Seifert homology 3-spheres

We develop an $S^1 \times \mathbb{Z}_p$-equivariant refinement of the lattice homotopy type for Seifert homology 3-spheres equipped with the Seifert $\mathbb{Z}_p$-action. We construct an equivariant comparison map to the $S^1 \times \mathbb{Z}_p$-equivariant Seiberg--Witten Floer spectrum that becomes an $S^1$-equivariant stable homotopy equivalence after forgetting the $\mathbb{Z}_p$-action, yielding a combinatorial description of its Borel equivariant cohomology.

math.GT↗