Configuration Space Of Points Research Articles

Generating Series of the Classes of Exotic Unordered Configuration Spaces

The concept of exotic (ordered) configuration spaces of points in a space was proposed by Baryshnikov. He obtained formulas for the (exponential) generating series of their Euler characteristics. We explore unordered analogs of the spaces. Considering a complex quasiprojective variety, we give a formula for the generating series of the classes of these configuration spaces in the Grothendieck ring of complex quasiprojective varieties. We state the result in terms of the natural power structure over this ring. This yields formulas of generating series of additive invariants of configuration spaces like the Hodge–Deligne polynomial and the Euler characteristic.

Configuration spaces and directed paths on the final precubical set

The main goal of this paper is to prove that the space of directed loops on the final precubical set is homotopy equivalent to the “total” configuration space of points on the plane; by “total” we mean that any finite number of points in the configuration

Correction to the article A spectral sequence for stratified spaces and configuration spaces of points

We correct some oversights in the paper "A spectral sequence for stratified spaces and configuration spaces of points" by the second named author. In particular we explain that an additional hypothesis should be added to Theorem 4.15 in said paper.

On factorization algebras arising in the quantum geometric Langlands theory

We study factorization algebras on configuration spaces of points on a curve, colored by elements of the root lattice. Our main result says that the factorization algebra attached to Lusztig's quantum group can be obtained as a direct image of a twisted Whittaker sheaf on the Zastava space.

Conformal correlation functions in four dimensions from quaternionic Lauricella system

Correlation functions in four-dimensional Euclidean conformal field theories are expressed in terms of representations of the conformal group SL(2,H), H being the field of quaternions, on the configuration space of points. The representations are obtained in terms of a Lauricella system derived using quaternions. It generalizes the two-dimensional case, wherein the N-point correlation function is expressed in terms of solutions of Lauricella system on the configuration space of N points on the complex plane, furnishing representation of the conformal group SL(2,C).

The local motivic DT/PT correspondence.

We show that the Quot scheme QLn=QuotA3(IL,n) parameterising length n quotients of the ideal sheaf of a line in A3 is a global critical locus, and calculate the resulting motivic partition function (varying n), in the ring of relative motives over the configuration space of points in A3. As in the work of Behrend–Bryan–Szendrői, this enables us to define a virtual motive for the Quot scheme of n points of the ideal sheaf IC⊂OY, where C⊂Y is a smooth curve embedded in a smooth 3‐fold Y, and we compute the associated motivic partition function. The result fits into a motivic wall‐crossing type formula, refining the relation between Behrend's virtual Euler characteristic of QuotY(IC,n) and of the symmetric product SymnC. Our ‘relative’ analysis leads to results and conjectures regarding the pushforward of the sheaf of vanishing cycles along the Hilbert–Chow map QLn→Symn(A3), and connections with cohomological Hall algebra representations.

Two-dimensional perturbative scalar QFT and Atiyah–Segal gluing

We study the perturbative quantization of 2-dimensional massive scalar field theory with polynomial (or power series) potential on manifolds with boundary. We prove that it fits into the functorial quantum field theory framework of Atiyah-Segal. In particular, we prove that the perturbative partition function defined in terms of integrals over configuration spaces of points on the surface satisfies an Atiyah-Segal type gluing formula. Tadpoles (short loops) behave nontrivially under gluing and play a crucial role in the result.

Hanany-Witten transition in quantum curves

It was known that the U(N )4 super Chern-Simons matrix model describing the worldvolume theory of D3-branes with two NS5-branes and two (1, k)5-branes in IIB brane configuration (dual to M2-branes after taking the T-duality and the M-theory lift) corresponds to the D5 quantum curve. For deformations of these two objects, on one hand the super Chern-Simons matrix model has three degrees of freedom (of relative rank defor- mations interpreted as fractional branes in brane configurations), while on the other hand the D5 curve has five degrees of freedom (characterized by point configurations of asymp- totic values). To identify the three-dimensional parameter space of brane configurations in the five-dimensional space of point configurations, we propose the necessity to cut the compact T-duality circle (or the circular quiver diagram) open, which is similar to the idea of “fixing a reference frame” or “fixing a local chart”. Since the parameter space of curves enjoys the D5 Weyl group beautifully, we are naturally led to conjecture that M2-branes are not only deformed by fractional branes but more obscure geometrical backgrounds.

The full moment problem on subsets of probabilities and point configurations

The aim of this paper is to study the full K−moment problem for measures supported on some particular non-linear subsets K of an infinite dimensional vector space. We focus on the case of random measures, that is K is a subset of all non-negative Radon measures on Rd. We consider as K the space of sub-probabilities, probabilities and point configurations on Rd. For each of these spaces we provide at least one representation as a generalized basic closed semi-algebraic set to apply the main result in Infusino et al. (2014) [20]. We demonstrate that this main result can be significantly improved by further considerations based on the particular chosen representation of K. In the case when K is a space of point configurations, the correlation functions (also known as factorial moment functions) are easier to handle than the ordinary moment functions. Hence, we additionally express the main results in terms of correlation functions.

Configuration Spaces of Graphs with Certain Permitted Collisions

If G is a graph with vertex set V, let $${{\mathrm{Conf}}}_n^{{{\mathrm{sink}}}}(G,V)$$ be the space of n-tuples of points on G, which are only allowed to overlap on elements of V. We think of $${{\mathrm{Conf}}}_n^{{{\mathrm{sink}}}}(G,V)$$ as a configuration space of points on G, where points are allowed to collide on vertices. In this paper, we attempt to understand these spaces from two separate, but closely related, perspectives. Using techniques of combinatorial topology we compute the fundamental groups and homology groups of $${{\mathrm{Conf}}}_n^{{{\mathrm{sink}}}}(G,V)$$ in the case where G is a tree. Next, we use techniques of asymptotic algebra to prove statements about $${{\mathrm{Conf}}}_n^{{{\mathrm{sink}}}}(G,V)$$ , for general graphs G, whenever n is sufficiently large. It is proven that, for general graphs, the homology groups exhibit generalized representation stability in the sense of Ramos ( arXiv:1606.02673 , 2016).

Unwinding the amplituhedron in binary

We present new, fundamentally combinatorial and topological characterizations of the amplituhedron. Upon projecting external data through the amplituhedron, the resulting configuration of points has a specified (and maximal) generalized “winding number”. Equivalently, the amplituhedron can be fully described in binary: canonical projections of the geometry down to one dimension have a specified (and maximal) number of “sign flips” of the projected data. The locality and unitarity of scattering amplitudes are easily derived as elementary consequences of this binary code. Minimal winding defines a natural “dual” of the amplituhedron. This picture gives us an avatar of the amplituhedron purely in the configuration space of points in vector space (momentum-twistor space in the physics), a new interpretation of the canonical amplituhedron form, and a direct bosonic understanding of the scattering super-amplitude in planar mathcal{N} = 4 SYM as a differential form on the space of physical kinematical data.

The Lp–diameter of the group of area-preserving diffeomorphisms of S2

We show that for each $p \geq 1,$ the $L^p$-metric on the group of area-preserving diffeomorphisms of the two-sphere has infinite diameter. This solves the last open case of a conjecture of Shnirelman from 1985. Our methods extend to yield stronger results on the large-scale geometry of the corresponding metric space, completing an answer to a question of Kapovich from 2012. Our proof uses configuration spaces of points on the two-sphere, quasi-morphisms, optimally chosen braid diagrams, and, as a key element, the cross-ratio map $X_4(\mathbb{C} P^1) \to \mathcal{M}_{0,4} \cong \mathbb{C} P^1 \setminus \{\infty,0,1\}$ from the configuration space of $4$ points on $\mathbb{C} P^1$ to the moduli space of complex rational curves with $4$ marked points.

Linear configurations of complete graphs K4 and K5 in ℝ3, and higher dimensional analogs

We investigate the space [Formula: see text] of images of linearly embedded finite simplicial complexes in [Formula: see text] isomorphic to a given complex [Formula: see text], focusing on two special cases: [Formula: see text] is the [Formula: see text]-skeleton [Formula: see text] of an [Formula: see text]-simplex, and [Formula: see text] is the [Formula: see text]-skeleton [Formula: see text] of an [Formula: see text]-simplex, so [Formula: see text] has codimension 2 in [Formula: see text], in both cases. The main result is that for [Formula: see text], [Formula: see text] (for either [Formula: see text]) deformation retracts to a subspace homeomorphic to the double mapping cylinder [Formula: see text] where [Formula: see text] is the alternating group and [Formula: see text] the symmetric group. The resulting fundamental group provides an example of a generalization of the braid group, which is the fundamental group of the configuration space of points in the plane.

Logarithms and deformation quantization

We prove the statement/conjecture of M. Kontsevich on the existence of the logarithmic formality morphism \(\mathcal {U}^{\mathrm{log}}\). This question was open since 1999, and the main obstacle was the presence of dr / r type singularities near the boundary \(r=0\) in the integrals over compactified configuration spaces. The novelty of our approach is the use of local torus actions on configuration spaces of points in the upper half-plane. It gives rise to a version of Stokes’ formula for differential forms with singularities at the boundary which implies the formality property of \(\mathcal {U}^{\mathrm{log}}\). We also show that the logarithmic formality morphism admits a globalization from \({\mathbb {R}}^d\) to an arbitrary smooth manifold.

The Orlik-Solomon model for hypersurface arrangements

We develop a model for the cohomology of the complement of a hypersurface arrangement inside a smooth projective complex variety. This generalizes the case of normal crossing divisors, discovered by P. Deligne in the context of the mixed Hodge theory of smooth complex varieties. Our model is a global version of the Orlik-Solomon algebra, which computes the cohomology of the complement of a union of hyperplanes in an affine space. The main tool is the complex of logarithmic forms along a hypersurface arrangement, and its weight filtration. Connections with wonderful compactifications and the configuration spaces of points on curves are also studied.

On homological stability for configuration spaces on closed background manifolds

We introduce a new map between configuration spaces of points in a background manifold - the replication map - and prove that it is a homology isomorphism in a range with certain coefficients. This is particularly of interest when the background manifold is closed, in which case the classical stabilisation map does not exist. We then establish conditions on the manifold and on the coefficients under which homological stability holds for configuration spaces on closed manifolds. These conditions are sharp when the background manifold is a two-dimensional sphere, the classical counterexample in the field. For field coefficients this extends results of Church ( Church, 2012) and Randal-Williams (Randal-Williams, 2013a) to the case of odd charac- teristic, and for p-local coefficients it improves results of Bendersky- Miller (Bendersky and Miller, 2014).

On a Quantum Model of Brain Activities — Distribution of the Outcomes of EEG-Measurements and Random Point Fields

Considering models based on classical probability theory, states of signals in the brain should be identified with probability distributions of certain random point fields representing the configuration of excited neurons. Then the outcomes of EEG-measurements can be considered as random variables being certain functions of that random point field. In practice, specialists use certain statistical methods evaluating the outcomes of the sequence of these measurements. To make these statistical investigations precise, one should know the distribution of the stochastic process on the space of point configurations representing the time evolution of the configuration of excited neurons in the brain. Up to now that distribution is totally unknown. In this paper we consider time evolutions of random point fields as well as the distribution of the outcomes of EEG-measurements related to unitary evolutions of certain quantum states used in [4, 5, 10 – 14] in order to describe activities of the brain.

Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes

The infinite-dimensional unitary group U(∞) is the inductive limit of growing compact unitary groups U(N). In this paper we solve a problem of harmonic analysis on U(∞) stated in [Ol3]. The problem consists in computing spectral decomposition for a remarkable 4-parameter family of characters of U(∞). These characters generate representations which should be viewed as analogs of nonexisting regular representation of U(∞). The spectral decomposition of a character of U(∞) is described by the spectral measure which lives on an infinite-dimensional space Ω of indecomposable characters. The key idea which allows us to solve the problem is to embed Ω into the space of point configurations on the real line without two points. This turns the spectral measure into a stochastic point process on the real line. The main result of the paper is a complete description of the processes corresponding to our concrete family of characters. We prove that each of the processes is a determinantal point process. That is, its correlation functions have determinantal form with a certain kernel. Our kernels have a special ‘integrable’ form and are expressed through the Gauss hypergeometric function. From the analytic point of view, the problem of computing the correlation kernels can be reduced to a problem of evaluating uniform asymptotics of certain discrete orthogonal polynomials studied earlier by Richard Askey and Peter Lesky. One difficulty lies in the fact that we need to compute the asymptotics in the oscillatory regime with the period of oscillations tending to 0. We do this by expressing the polynomials in terms of a solution of a discrete Riemann-Hilbert problem and computing the (nonoscillatory) asymptotics of this solution. From the point of view of statistical physics, we study thermodynamic limit of a discrete log-gas system. An interesting feature of this log-gas is that its density function is asymptotically equal to the characteristic function of an interval. Our point processes describe how different the random particle configuration is from the typical ‘densely packed’ configuration. In simpler situations of harmonic analysis on infinite symmetric groups and harmonic analysis of unitarily invariant measures on infinite hermitian matrices, similar results were obtained in our papers [BO1], [BO2], [BO4].

Periodic trajectories in 3-dimensional convex billiards

We give a lower bound on the number of periodic billiard trajectories inside a generic smooth strictly convex closed surface in 3-space: for odd n, there are at least 2(n-1) such trajectories. Convex plane billiards were studied by G. Birkhoff, and the case of higher dimensional billiards is considered in our previous papers. We apply a topological approach based on the calculation of cohomology of certain configuration spaces of points on 2-sphere.

The intersection pairings on the configuration spaces of points in the projective line

We compute the intersection pairings on the configuration spaces of points in $\mathbf{CP}^{1}$ with arbitrary weight from an algebro-geometric point of view. We also derive their generating function, which was originally proved by Takakura based on symplectic technique and representation theory.