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History and Overview

New submissions

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New submissions for Tue, 20 Mar 18

[1]  arXiv:1803.06610 [pdf, ps, other]
Title: Can You Pave the Plane Nicely with Identical Tiles
Authors: Chuanming Zong
Comments: 10 pages, 16 figures
Subjects: Metric Geometry (math.MG); History and Overview (math.HO)

Every body knows that identical regular triangles or squares can tile the whole plane. Many people know that identical regular hexagons can tile the plane properly as well. In fact, even the bees know and use this fact! Is there any other convex domain which can tile the Euclidean plane? Yes, there is a long list of them! To find the list and to show the completeness of the list is a unique drama in mathematics, which has lasted for more than one century and the completeness of the list has been mistakenly announced not only once! Up to now, the list consists of triangles, quadrilaterals, three types of hexagons, and fifteen types of pentagons. In 2017, Michael Rao announced a computer proof for the completeness of the list. Meanwhile, Qi Yang and Chuanming Zong made a series of unexpected discoveries in multiple tilings in the Euclidean plane. For examples, besides parallelograms and centrally symmetric hexagons, there is no other convex domain which can form any two-, three- or four-fold translative tiling in the plane; there are only two types of octagons and one type of decagons which can form five-fold translative tilings.

[2]  arXiv:1803.06636 [pdf, ps, other]
Title: Complexity problems in enumerative combinatorics
Authors: Igor Pak
Comments: 30 pages; an expanded version of the ICM 2018 paper (Section 4 added, refs expanded)
Subjects: Combinatorics (math.CO); Computational Complexity (cs.CC); Discrete Mathematics (cs.DM); History and Overview (math.HO); Probability (math.PR)

We give a broad survey of recent results in Enumerative Combinatorics and their complexity aspects.

[3]  arXiv:1803.06714 [pdf, other]
Title: Mathematics for cryo-electron microscopy
Authors: Amit Singer
Comments: Proceedings of the International Congress of Mathematicians 2018
Subjects: Computational Physics (physics.comp-ph); History and Overview (math.HO)

Single-particle cryo-electron microscopy (cryo-EM) has recently joined X-ray crystallography and NMR spectroscopy as a high-resolution structural method for biological macromolecules. Cryo-EM was selected by Nature Methods as Method of the Year 2015, large scale investments in cryo-EM facilities are being made all over the world, and the Nobel Prize in Chemistry 2017 was awarded to Jacques Dubochet, Joachim Frank and Richard Henderson "for developing cryo-electron microscopy for the high-resolution structure determination of biomolecules in solution". This paper focuses on the mathematical principles underlying existing algorithms for structure determination using single particle cryo-EM.

Replacements for Tue, 20 Mar 18

[4]  arXiv:1605.07543 (replaced) [pdf, other]
Title: On tea, donuts and non-commutative geometry
Authors: Igor Nikolaev
Comments: to appear Surveys in Mathematics and its Applications
Subjects: Algebraic Geometry (math.AG); History and Overview (math.HO); Operator Algebras (math.OA)
[ total of 4 entries: 1-4 ]
[ showing up to 2000 entries per page: fewer | more ]

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