Topological aspects of Boolean functions
We discuss ways in which tools from topology can be used to derive lower bounds for the circuit complexity of Boolean functions.
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arXiv subjects
Robert MacPherson
Publications and source records attributed to Robert MacPherson.
Asymptotic Betti numbers for hard squares in the homological liquid regime
We study configuration spaces $C(n; p, q)$ of $n$ ordered unit squares in a $p$ by $q$ rectangle. Our goal is to estimate the Betti numbers for large $n$, $j$, $p$, and $q$. We consider sequences of area-normalized coordinates, where $(\frac{n}{pq}, \frac{j}{pq})$ converges as $n$, $j$, $p$, and $q$ approach infinity. For every sequence that converges to a point in the "feasible region" in the $(x,y)$-plane, we show that the factorial growth rate of the Betti numbers is the same as the factorial growth rate of $n!$. This implies that (1) the Betti numbers are vastly larger than for the configuration space of $n$ ordered points in the plane, which have the factorial growth rate of $j!$, and (2) every point in the feasible region is eventually in the homological liquid regime.
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Homology of configuration spaces of hard squares in a rectangle
We study ordered configuration spaces $C(n;p,q)$ of $n$ hard squares in a $p \times q$ rectangle, a generalization of the well-known "15 Puzzle". Our main interest is in the topology of these spaces. Our first result is to describe a cubical cell complex and prove that is homotopy equivalent to the configuration space. We then focus on determining for which $n$, $j$, $p$, and $q$ the homology group $H_j [ C(n;p,q) ]$ is nontrivial. We prove three homology-vanishing theorems, based on discrete Morse theory on the cell complex. Then we describe several explicit families of nontrivial cycles, and a method for interpolating between parameters to fill in most of the picture for "large-scale" nontrivial homology.
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Persistent Local Systems
In this paper, we give lower bounds for the homology of the fibers of a map to a manifold. Using new sheaf theoretic methods, we show that these lower bounds persist over whole open sets of the manifold, and that they are stable under perturbations of the map. This generalizes certain ideas of persistent homology to higher dimensions.
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Computing Hecke Operators for Arithmetic Subgroups of General Linear Groups
We present an algorithm to compute the Hecke operators on the equivariant cohomology of an arithmetic subgroup $Γ$ of the general linear group $\mathrm{GL}_n$. This includes $\mathrm{GL}_n$ over a number field or a finite-dimensional division algebra. As coefficients, we may use any finite-dimensional local coefficient system. Unlike earlier methods, the algorithm works for the cohomology $H^i$ in all degrees $i$. It starts from the well-rounded retract $\tilde{W}$, a $Γ$-invariant cell complex which computes the cohomology. It extends $\tilde{W}$ to a new well-tempered complex $\tilde{W}^+$ of one higher real dimension, using a real parameter called the temperament. The algorithm has been coded up for $\mathrm{SL}_n(\mathbb{Z})$ for $n=2,3,4$; we present some results for congruence subgroups of $\mathrm{SL}_3(\mathbb{Z})$.
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Configuration spaces of disks in an infinite strip
We study the topology of the configuration spaces $C(n,w)$ of $n$ hard disks of unit diameter in an infinite strip of width $w$. We describe ranges of parameter or "regimes", where homology $H_j [C(n,w)]$ behaves in qualitatively different ways. We show that if $w \ge j+2$, then the homology $H_j[C(n, w)]$ is isomorphic to the homology of the configuration space of points in the plane, $H_j[C(n, \mathbb{R}^2)]$. The Betti numbers of $C(n, \mathbb{R}^2) $ were computed by Arnold, and so as a corollary of the isomorphism, $β_j[C(n,w)]$ is a polynomial in $n$ of degree $2j$. On the other hand, we show that if $2 \le w \le j+1$, then $β_j [ C(n,w) ]$ grows exponentially with $n$. Most of our work is in carefully estimating $β_j [ C(n,w) ]$ in this regime. We also illustrate, for every $n$, the homological "phase portrait" in the $(w,j)$-plane--- the parameter values where homology $H_j [C(n,w)]$ is trivial, nontrivial, and isomorphic with $H_j [C(n, \mathbb{R}^2)]$. Motivated by the notion of phase transitions for hard-spheres systems, we discuss these as the "homological solid, liquid, and gas" regimes.
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Topological Similarity of Random Cell Complexes and Applications
Although random cell complexes occur throughout the physical sciences, there does not appear to be a standard way to quantify their statistical similarities and differences. The various proposals in the literature are usually motivated by the analysis of particular physical systems and do not necessarily apply to general situations. The central concepts in this paper---the swatch and the cloth---provide a description of the local topology of a cell complex that is general (any physical system that may be represented as a cell complex is admissible) and complete (any statistical question about the local topology may be answered from the cloth). Furthermore, this approach allows a distance to be defined that measures the similarity of the local topology of two cell complexes. The distance is used to identify a steady state of a model dislocation network evolving by energy minimization, and then to rigorously quantify the approach of the simulation to this steady state.
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Measuring Shape with Topology
We propose a measure of shape which is appropriate for the study of a complicated geometric structure, defined using the topology of neighborhoods of the structure. One aspect of this measure gives a new notion of fractal dimension. We demonstrate the utility and computability of this measure by applying it to branched polymers, Brownian trees, and self-avoiding random walks.
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Discrete series characters and the Lefschetz formula for Hecke operators
This paper gives combinatorial formulas for discrete series constants, both stable and unstable, on real reductive groups. It also carries out one step of the comparison of the topological trace formula for Hecke operators with Arthur's trace formula for Hecke operators, the step in which the topological terms are combined in such a way that stable discrete series characters appear.
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Codimensions of root valuation strata
This paper defines and studies a stratification of the adjoint quotient of the Lie algebra of a reductive group over a Laurent power series field. The stratification arises naturally in the context of affine Springer fibers.
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Regular points in affine Springer fibers
Work of Kazhdan-Lusztig and Bezrukavnikov suggests the importance of points in affine Springer fibers for which the associated conjugacy class in the finite dimensional Lie algebra is regular. Such points are characterized in a different way in this paper, using the same kind of information as goes into the weight factors used to form weighted orbital integrals.
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Purity of equivalued affine Springer fibers
The affine Springer fiber corresponding to a regular integral equivalued semisimple element admits a paving by vector bundles over Hessenberg varieties and hence its (Borel-Moore) homology is "pure".
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Homology of affine Springer fibers in the unramified case
Assuming a certain "purity" conjecture, we derive a formula for the (complex) cohomology groups of the affine Springer fiber corresponding to any unramified regular semi-simple element. We use this calculation to present a complex analog of the fundamental lemma for function fields. We show that the "kappa" orbital integral which arises in the fundamental lemma is equal to the Lefschetz trace of Frobenius acting on the etale cohomology of a related variety.
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Errata to: Local contribution to the Lefschetz fixed point formula
This note contains a correction to the paper, ``Local contribution to the Lefschetz fixed point formula'', Inv. Math. 111 (1993), pp. 1-33.
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The topological trace formula
The Lefschetz formula for the action of a Hecke correspondence on the weighted cohomology of a locally symmetric space is derived. It is also proven that each Hecke correspondence on the reductive Borel-Serre compactification of the locally symmetric space is weakly hyperbolic.
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From moment graphs to intersection cohomology
We describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori, in terms of structure of the zero- and one-dimensional orbits. The class of varieties to which our formula applies includes Schubert varieties in flag varieties and affine flag varieties. We also prove a monotonicity result on local intersection cohomology stalks.
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