Algebraic quantum field theory is an approach to relativistic quantum physics, notably the theory of elementary particles, which complements other modern developments in this field. It is particularly powerful for structural analysis but has also proven to be useful in the rigorous treatment of models. In this contribution a non–technical survey is given with emphasis on interesting recent deve...

Given a Cylindrical Algebraic Decomposition of an implicit algebraic curve, visualizing distinct curve arcs is not as easy as it stands because, despite the absence of singularities in the interior, the arcs can pass arbitrary close to each other. We present an algorithm to visualize distinct connected arcs of an algebraic curve efficiently and precise (at a given resolution), irrespective of h...

In 1979, Richard Stanley made the following conjecture: Every Cohen–Macaulay simplicial complex is partitionable. Motivated by questions in the theory of face numbers of simplicial complexes, the Partitionability Conjecture sought to connect a purely combinatorial condition (partitionability) with an algebraic condition (Cohen–Macaulayness). The algebraic combinatorics community widely believed...

We investigate the space complexity of refuting 3-CNFs in Resolution and algebraic systems. No lower bound for refuting any family of 3-CNFs was previously known for the total space in resolution or for the monomial space in algebraic systems. Using the framework of [10], we prove that every Polynomial Calculus with Resolution refutation of a random 3-CNF φ in n variables requires, with high pr...

Deligne’s regularity criterion for an integrable connection ∇ on a smooth complex algebraic variety X says that ∇ is regular along the irreducible divisors at infinity in some fixed normal compactification of X if and only if the restriction of ∇ to every smooth curve on X is fuchsian (i. e. has only regular singularities at infinity). The “only if” part is the difficult implication. Deligne’s ...

This is draft “version 0.2” of these notes. Please send comments to zteitler@ tamu. edu . To a polynomial f ∈ C[z1, . . . , zn], one can associate a family of ideals J(f, t) ⊂ C[z1, . . . , zn], for t ∈ R, t ≥ 0, called the multiplier ideals of f . The purpose of this note is to define these ideals in terms of local integrability (as in analysis) and in terms of resolution of singularities (as ...

Reactive transport equations are used in numerous application fields: CO2 or nuclear waste storage monitoring, separation process in chemical engineering. We present a general method to account robustly for instantaneous equilibriums in reactive transport. This method is adapted to all kind of hydraulic transport models including 1D to 3D convection-diffusion equations. This leads to the resolu...

In this paper we study Schlichting's K-theory groups of the Buchweitz-Orlov singularity category $\mathcal{D}^{sg}(X)$ a quasi-projective algebraic scheme $X/k$ with applications to Algebraic K-theory. We prove that for isolated quotient singularities $\mathrm{K}_0(\mathcal{D}^{sg}(X))$ is finite torsion, and $\mathrm{K}_1(\mathcal{D}^{sg}(X)) = 0$. One main varieties satisfy rational Poincare ...

We study the problem of reconstructing a positive discrete measure on compact set K?Rn from finite moments (possibly known only approximately) via convex optimization. give new uniqueness results, quantitative estimates for approximate recovery and sum-of-squares based hierarchy super-resolution semi-algebraic sets.