The structure of T is determined in a simple way by the complex of bounded faces of the real part of the arrangement. On the one hand, this facilitates a geometric analysis of the second homotopy group. At the same time, this universal cover complex yields an algebraic description of the homology of abelian covers in terms of modules over polynomial rings.

The study of free arrangements will center on the various notions of formality introduced by the investigator and his co- workers.

## Bibliography

The complement of an arrangement of hyperplanes in n-space is just the collection of chunks into which n-space falls when cleaved by a number of n-1 -spaces. This seemingly innocuous structure has called forth a surprising amount of heavy machinery from various parts of algebra and topology and led to some gratifyingly attractive analysis and general combinatorial principles. Although the work is theoretical and leads to rigorous proofs, at intermediate stages it is useful to experiment and test examples.

Some machine computation is indispensable at this point, particularly using such symbolic manipulators as Cayley, for which a Sun SPARCstation is available to the investigator. Please report errors in award information by writing to: awardsearch nsf.

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## Combinatorial Foundation of Homology and Homotopy

How to Manage Your Award. Grant Policy Manual. Marks, B. Seward, R. Tucker-Drob Hyperfiniteness and Borel combinatorics rec. Cortazar, M. Musso Green's function and infinite-time bubbling in the critical nonlinear heat equation rec. Curien, T. Hutchcroft, A. Nachmias Geometric and spectral properties of causal maps rec. Dang, G. Riviere Pollicott Ruelle spectrum and Witten Laplacians rec. Dang, B. Zhang Renormalization of Feynman amplitudes on manifolds by spectral zeta regularization and blow-ups.

Diaconu, I. Dill Unlikely intersections between isogeny orbits and curves rec.

Dirksen, S. Druet, P. Duits, A. Kuijlaars The two periodic Aztec diamond and matrix valued orthogonal polynomials rec. Duminil-Copin, A. Glazman, R. Peled, Y. Spinka Macroscopic loops in the loop O n model at Nienhuis' critical point rec. Efimov Homotopy finiteness of some DG categories from algebraic geometry rec. Elias, G. Williamson Relative hard Lefschetz for Soergel bimodules rec.

Erbar, K. Sturm Rigidity of cones with bounded Ricci curvature rec. Fintzen On the Moy-Prasad filtration rec.

## Quillen cohomology and Baues-Wirsching cohomology of algebraic, theories

Fischler, T. Rivoal Linear independence of values of G-functions rec. Fisher, J. Lafont, N. Miller, M. Frank, D.

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Putman The high-dimensional cohomology of the moduli space of curves with level structures rec. Furusawa, K. Morimoto Refined global Gross-Prasad conjecture on special Bessel periods and Boecherer's conjecture rec. Gandini, A. Maffei The Bruhat order on Hermitian symmetric varieties and on abelian nilradicals rec. Gess Optimal regularity for the porous medium equation rec. Gigli, L. Glutsyuk On polynomially integrable Birkhoff billiards on surfaces of constant curvature rec. Goodearl, M. Yakimov The Berenstein--Zelevinsky quantum cluster algebra conjecture rec.

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Goujard, M. Hartmann, K. Kellay, M. Tucsnak From the reachable space of the heat equation to Hilbert spaces of holomorphic functions rec. He, N. Hedenmalm, A.

https://rikalipermea.tk Herron, S. Huang, D. Milicic, B. Sun Harish-Chandra's orthogonality relations for admissible representations rec. Imbert, L. Silvestre The Weak Harnack inequality for the Boltzmann equation without cut-off rec. Izquierdo, G. Lucchini Arteche Homogeneous spaces, algebraic K-theory and cohomological dimension of fields rec. Jiang, N. Leung, Y. Jiang, L. Kang, G.