The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning

He, Yang-Hui

doi:10.48550/arxiv.1812.02893

Cited by 35 publications

(56 citation statements)

References 342 publications

(317 reference statements)

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“…Quiver gauge theories and their string theory constructions [1][2][3], as an extension to the AdS/CFT correspondence, have been a centre of intensive studies over the last two decades. While their holographic realization as brane tilings [4][5][6][7] turn out to be extremely useful and productive (see reviews in [8][9][10]), these combinatorial and geometrical constructions are somewhat limited to a kinematic level, meaning that other than the famous a-maximization/volume minimization [11][12][13][14], such key quantities as partition functions etc., have been relatively untouched. The purpose of the present work is to take the first step in a new direction which can be seen as an initial upgrade of the brane tilings paradigm to the dynamical level.…”

Section: Introductionmentioning

confidence: 99%

Mahler Measure for a Quiver Symphony

Bao,

He,

Zahabi

2021

Preprint

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Adopting the Mahler measure from number theory, we introduce it to toric quiver gauge theories, and study some of its salient features and physical implications. We propose that the Mahler measure is a universal measure for the quiver, encoding its dynamics with the monotonic behaviour along a so-called Mahler flow including two special points at isoradial and tropical limits. Along the flow, the amoeba, from tropical geometry, provides geometric interpretations for the dynamics of the quiver. In the isoradial limit, the maximization of Mahler measure is shown to be equivalent to a-maximization. The Mahler measure and its derivative are closely related to the master space, leading to the property that the specular duals have the same functions as coefficients in their expansions, hinting the emergence of a free theory in the tropical limit. Moreover, they indicate the existence of phase transition. We also find that the Mahler measure should be invariant under Seiberg duality.

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Section: Introductionmentioning

confidence: 99%

Mahler Measure for a Quiver Symphony

Bao,

He,

Zahabi

2021

Preprint

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“…Indeed, irrespective of ATP, the rôle of computers in mathematics is of increasing importance. From the explicit computations which helped resolving the 4-color theorem in 1976 [AHK], to the completion of the classification of the finite simple groups by Gorenstein et al from the mid-1950s to 2004 [Wil], to the vast number of software and databases emerging in the last decade or so to aid researchers in geometry [Sing,GRdB,HeCY,M2], number theory [MAG, LMFdB], representation theory [GAP], knot theory [KNOT], as well as the umbrella MathSage project [SAGE] etc., it is almost inconceivable that the younger generation of mathematicians would not find the computer as indispensable a tool as pen or chalk. The ICM panel of 2018 [ICM18] documents a lively and recent discussion on this progress in computer assisted mathematics.…”

Section: Bottom-upmentioning

confidence: 99%

Machine-Learning Mathematical Structures

He¹

2021

Preprint

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We review, for a general audience, a variety of recent experiments on extracting structure from machine-learning mathematical data that have been compiled over the years. Focusing on supervised machine-learning on labeled data from different fields ranging from geometry to representation theory, from combinatorics to number theory, we present a comparative study of the accuracies on different problems. The paradigm should be useful for conjecture formulation, finding more efficient methods of computation, as well as probing into certain hierarchy of structures in mathematics. Based on various colloquia, seminars and conference talks in 2020, this is a contribution to the launch of the journal "Data Science in the Mathematical Sciences."

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“…The works [22,23] are in a sense part of a more general program of applying methods from data science and machine learning to study problems in mathematical physics, see [24,25] for a pedagogical treatment. By now, machine learning and data science have been used to learn various aspects of number theory [26,27,28], quiver gauge theories and cluster algebras [29], knot theory [30,31,32] as well as graph theory [33] and commutative algebra [34].…”

Section: Introductionmentioning

confidence: 99%