Universes as Bigdata:

Superstrings, Calabi-Yau Manifolds and Machine-Learning

YANG-HUI HE

University of London
University of Oxford
School of Physics, NanKai University

Dept of Mathematics, City,
Merton College,

East Asian Strings 2020, (Virtual) Osaka-Seoul-Taipei

YANG-HUI HE (London/Oxford/Nankai)

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1984: 10 = 4 + 3 × 2

First String Revolution [Green-Schwarz] anomaly cancellation;
Heterotic string [Gross-Harvey-Martinec-Rohm]: E8 × E8 or SO(32), 1984 - 5String Phenomenology [Candelas-Horowitz-Strominger-Witten]: 1985

SU (3) × SU (2) × U (1) ⊂ SU (5) ⊂ SO(10) ⊂ E6 ⊂ E8Standard Solution (MANY more since): R3,1 × X , X Ricci-at, Kähler

mathematicians were independently thinking of the same problem:
Riemann Uniformization Theorem in dimC = 1: Trichotomy R < 0, = 0, > 0
Euler, Gauss, Riemann, Bourbaki, Atiyah-Singer . . .
χ(Σ) = 2 − 2g(Σ) = [c1 (Σ)] · [Σ] =

1
2π

Z
R=
Σ

2
X

(−1)i hi (Σ)

i=0Calabi-Yau
YANG-HUI HE (London/Oxford/Nankai)

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1984: 10 = 4 + 3 × 2

First String Revolution [Green-Schwarz] anomaly cancellation;
Heterotic string [Gross-Harvey-Martinec-Rohm]: E8 × E8 or SO(32), 1984 - 5String Phenomenology [Candelas-Horowitz-Strominger-Witten]: 1985

SU (3) × SU (2) × U (1) ⊂ SU (5) ⊂ SO(10) ⊂ E6 ⊂ E8Standard Solution (MANY more since): R3,1 × X , X Ricci-at, Kähler

mathematicians were independently thinking of the same problem:
Riemann Uniformization Theorem in dimC = 1: Trichotomy R < 0, = 0, > 0
Euler, Gauss, Riemann, Bourbaki, Atiyah-Singer . . .
χ(Σ) = 2 − 2g(Σ) = [c1 (Σ)] · [Σ] =

1
2π

Z
R=
Σ

2
X

(−1)i hi (Σ)

i=0Calabi-Yau
YANG-HUI HE (London/Oxford/Nankai)

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An Early Physical Challenge to Algebraic Geometry

CY3 X , tangent bundle SU (3) ⇒1

E6 GUT: commutant E8 → SU (3) × E6 , then2

Wilson-line/discrete symmetry to break E6 -GUT to some SUSY version ofStandard Model (generalize later)
Generation
n27 = h1 (X, T X) = h∂2,1 (X)
3 Particle Spectrum:
Anti-Generation n27 = h1 (X, T X ∗ ) = h∂1,1 (X)1,1Net-generation: χ = 2(h− h2,1 ) = Euler Number

1980s Question: Are there Calabi-Yau threefolds with Euler number ±6?
None of obvious ones

4e.g., Quintic Q in P§
h1,1 ,h2,1

is CY3 Qχ= Q1,101−200 so too may generations(even with quotient −200 6∈ 3Z)

YANG-HUI HE (London/Oxford/Nankai)

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An Early Physical Challenge to Algebraic Geometry

CY3 X , tangent bundle SU (3) ⇒1

E6 GUT: commutant E8 → SU (3) × E6 , then2

Wilson-line/discrete symmetry to break E6 -GUT to some SUSY version ofStandard Model (generalize later)
Generation
n27 = h1 (X, T X) = h∂2,1 (X)
3 Particle Spectrum:
Anti-Generation n27 = h1 (X, T X ∗ ) = h∂1,1 (X)1,1Net-generation: χ = 2(h− h2,1 ) = Euler Number

1980s Question: Are there Calabi-Yau threefolds with Euler number ±6?
None of obvious ones

4e.g., Quintic Q in P§
h1,1 ,h2,1

is CY3 Qχ= Q1,101−200 so too may generations(even with quotient −200 6∈ 3Z)

YANG-HUI HE (London/Oxford/Nankai)

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EAST, Jul, 2020

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An Early Physical Challenge to Algebraic Geometry

CY3 X , tangent bundle SU (3) ⇒1

E6 GUT: commutant E8 → SU (3) × E6 , then2

Wilson-line/discrete symmetry to break E6 -GUT to some SUSY version ofStandard Model (generalize later)
Generation
n27 = h1 (X, T X) = h∂2,1 (X)
3 Particle Spectrum:
Anti-Generation n27 = h1 (X, T X ∗ ) = h∂1,1 (X)1,1Net-generation: χ = 2(h− h2,1 ) = Euler Number

1980s Question: Are there Calabi-Yau threefolds with Euler number ±6?
None of obvious ones

4e.g., Quintic Q in P§
h1,1 ,h2,1

is CY3 Qχ= Q1,101−200 so too may generations(even with quotient −200 6∈ 3Z)

YANG-HUI HE (London/Oxford/Nankai)

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The First Data-sets in Mathematical Physics/Geometry[Candelas-A. He-Hübsch-Lutken-Schimmrigk-Berglund] (1986-1990)

CICYs (complete intersection CYs) multi-deg polys in products of CPni

CICYs

Problem: classify all conguration matrices; employed the best computers at
the time (CERN supercomputer); q.v. magnetic tape and dot-matrix printout in Philip's oce7890 matrices, 266 Hodge pairs (h1,1 , h2,1 ), 70 Euler χ ∈ [−200, 0][Candelas-Lynker-Schimmrigk, 1990]

Hypersurfaces in Weighted P47555 inequivalent 5-vectors wi , 2780 Hodge pairs, χ ∈ [−960, 960]
[Kreuzer-Skarke, mid-1990s - 2000]

Reexive Polytopes

Hypersurfaces in (Reexive, Gorenstein Fano) Toric 4-folds
6-month running time on dual Pentium SGI machineat least 473,800,776, with 30,108 distinct Hodge pairs, χ ∈ [−960, 960]
YANG-HUI HE (London/Oxford/Nankai)

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The age of data science in mathematicalphysics/string

theorynot as recent as you might thinkof course, experimental physics
had

been

decadesaheadindata-science/machine-learning

After

40

years

mathematiciansofresearch

andbyphysicists. . . ...

YANG-HUI HE (London/Oxford/Nankai)

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The Compact CY3 Landscape

The Calabi-Yau Landscape: from Geometry, to Physics, toMachine-Learning, 1812.02893, [Springer, to appear] ∼ 1010 data-points (andcf. YHH,

growing, still mined by many international collabs:

Some Recent Advances

Georgia O'Keefe on Kreuzer-Skarke
Vienna (KS, Knapp,. . . ), Penn (Ovrut, Cvetic, Donagi, Pantev . . . ), Oxford/London (Candelas, Constantin,

Lukas,

Mishra,YHH, . . . ),MIT (Taylor,

Johnson, Wang, . . . ), Northeastern/Wits (Halverson,
Long, Nelson, Jejjala, YHH), Virginia Tech (Anderson, Gray, SJ Lee, . . . ), Utrecht (Grimm . . . ), CERN
(Weigand, . . . ), Cornell (MacAllister, Stillman), Munich (Luest, Vaudravange), Uppsala (Larfors, Seong)...

YANG-HUI HE (London/Oxford/Nankai)

1,1Horizontal χ = 2(hML Mathematical Structures

1,1
2
−h2,1
) vs.Jul,
VerticalEAST,2020 h 6 /+h34

Triadophilia
Exact (MS)SM Particle Content from String Compactication[Braun-YHH-Ovrut-Pantev, Bouchard-Cvetic-Donagi 2005] rst exact MSSM
[Anderson-Gray-YHH-Lukas, 2007-] use alg./comp. algebraic geo & siftAnderson-Gray-Lukas-Ovrut-Palti ∼ 200 in 1010 MSSM Stable Sum of Line Bundles
over CICYs (Oxford-Penn-Virginia 2012-)
Constantin-YHH-Lukas '19: 1023 exact MSSMs (by extrapolation on above set)?
A Special Corner[New Scientist, Jan, 5, 2008 feature]

P. Candelas, X. de la Ossa, YHH,
and B. Szendroi
Triadophilia:

A

SpecialCorneroftheLandscape ATMP, 2008

YANG-HUI HE (London/Oxford/Nankai)

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The Landscape Explosion & Vacuum Degeneracy Problem

meanwhile . . . LANDSCAPE grew rapidly with
D-branes Polchinski 1995
M-Theory/G2 Witten, 1995
F-Theory/4-folds Katz-Morrison-Vafa, 1996
AdS/CFT Maldacena 1998

Alg Geo of AdS/CFT

Flux-compactication Kachru-Kallosh-Linde-Trivedi, 2003, Denef-Douglas

500

2005-6: 10

possibilities . . .

String theory trades one hard-problem [quantization of gravity] by another[looking for the right compactication] (in many ways a richer and moreinteresting problem, especially for the string/maths community)

YANG-HUI HE (London/Oxford/Nankai)

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Where we stand . . .

The Good

Last 10-15 years: large collaborations of physicists, computational
mathematicians (cf. SageMATH, GAP, Bertini, MAGMA,
Macaulay2, Singular) have bitten the bullet computed manygeometrical/physical quantities and compiled them into various
databases Landscape Data (10The Bad9∼10

entries typically)

Generic computation HARD: dual cone algorithm (exponential),
triangulation (exponential), Gröbner basis (double-exponential). . . e.g., how to construct stable bundles over the  473 million KS
CY3? Sifting through for SM computationally impossible . . .

The ???

Borrow new techniques from Big Data revolution

YANG-HUI HE (London/Oxford/Nankai)

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A Wild Question

Typical Problem in String Theory/Algebraic Geometry:

IN P U T

OU T P U T

integer tensor −→ integer

Q: Can (classes of problems in computational) Algebraic Geometry belearned by AI ? , i.e., can we machine-learn the landscape?"[YHH 1706.02714] Deep-Learning the Landscape,(PLB 774, 2017

Science, Aug, vol 365 issue 6452, 2019 ): Experimentally, it seems so formany situations in geometry and beyond.
2017
YHH (1706.02714), Seong-Kre (1706.03346), Ruehle (1706.07024),
Cario-Halverson-Krioukov-Nelson (1707.00655)

YANG-HUI HE (London/Oxford/Nankai)

ML Mathematical Structures

Progress in String Theory
EAST, Jul, 202010 / 34

A Prototypical Question

Hand-writing Recognition, e.g., my 0 to 9 is dierent from yours:

How to set up a bijection that takes these to {1, 2, . . . , 9, 0}? Find a clever
Morse function? Compute persistent homology? Find topological invariants?
ALL are inecient and too sensitive to variation.
What does your iPhone/tablet do? What does Google do? Machine-Learn

Take large sample, take a few hundred thousand (e.g. NIST database)...

28 × 28 × (RGB)Supervised ML in 1 min
YANG-HUI HE (London/Oxford/Nankai)

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EAST, Jul, 202011 / 34

NN Doesn't Care/Know about Alg. Geometry (1706.02714)

Hodge Number of a Complete Intersection CY is the association rule, e.g.



1 1 0 0 0 0 0 0

1 0 1 0 0 0 0 0
0 0 0 1 0 1 0 0
0 0 0 0 1 0 1 0
X=
0 0 0 0 0 0 2 0 ,
0 1 1 0 0 0 0 1

h1,1 (X) = 8

;

−→ 8

1 0 0 0 0 1 1 0
0 0 0 1 1 0 0 1

CICY is 12 × 15 integer matrix with entries ∈ [0, 5] is simply represented as a

12 × 15 pixel image of 6 colours Proper Way ; ML in matter of seconds/minutes

1,1


- Take samples of X → h
Cross-Validation:- train a NN, or SVMYANG-HUI HE (London/Oxford/Nankai)- Validation on

unseen X → h1,1

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Deep-Learning Algebraic Geometry

YHH (1706.02714) Bull-YHH-Jejjala-Mishra (1806.03121, 1903.03113), Krippendorf-Syvaeri[2003.13679] Erbin-Finotello (2007.13379; Google Inception NN) YHH-Lukas [2009.02544]

Hodge Number - Validation Learning Curves

Learning Hodge Number

0.9

Accuracy0.8

h1,1 ∈ [0, 19] so can set up 20-

0.7

channel NN classifer, regressor, as

0.6
0.5
0.4

SVM Classifier Validation Accuracy
Neural Net Regressor, Validation Accuracy
Neural Net Classifier, Validation Accuracy
0.2

well as SVM,
bypass exact sequences

0.4
0.6
0.8
Fraction of data used for training

YHH-SJ Lee (1904.08530): Distinguishing Elliptic Fibrations in CY3Accuracy
1.0
0.9

bypass Oguiso-Kollar-Wilson

0.8

Matthews ϕPrecision
Theorem/Conjecture

0.7
0.6
0.5

learning curves for precision and Matthews φ
20

40

60

YANG-HUI HE (London/Oxford/Nankai)

80

Training %100

ML Mathematical Structures

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More Success Stories in Algebraic Geometry/Strings

Ruehle '17: genetic algorithm for bundle cohomology
Halverson, Nelson, Long et al `17- programme of ML of KS data
Brodie-Constantin-Lukas '19: EXACT formulae for line-bundle coho /complex surfaces Interpolation vs Extrapolation ; Conjecture Formulation
Ashmore-YHH-Ovrut '19: ML Calabi-Yau metric: improves Donaldson
alg. for numerical CY metric by 10-100 times
Deen-YHH-Lee-Lukas '20: Distinguishing Heterotic SMs from thesum-line-bundle database and extrapolating beyond
q.v. K. Hashimoto `18: AdS/CFT = Boltzmann Machine;
Halverson-Anindita 2008.08601 NN=QFT; Vanchurin 2008.01540
NN=Spacetime; 2× de Mello-Koch, Cheng 1906.05212 RL = RG

YANG-HUI HE (London/Oxford/Nankai)

ML Mathematical Structures

EAST, Jul, 202014 / 34

]=

Representation/Group Theory
Why stop at string/geometry?
[YHH-MH. Kim 1905.02263] Machine-Learning Algebraic Structures(GAP nite groups/ring DB)
When is a Latin Square (Sudoku) the Cayley (multiplication) table of a nitegroup? (rmk: there is a known quadrangle-thm to test this) NN/SVM nd to 94.9%(φ = 0.90) at 25-75 cross-validation.
Can one look at the Cayley table and recognize a nite simple group?
Accuracy
1.00

bypass Sylow and Noether Thm

0.95

SVM: space of nite-groups

Precision (% agreed)
Matthews Phi(point-cloud of Cayley tables), ?∃0.90

hypersurface separating

0.85

0.80

0.05

0.10

0.15

YANG-HUI HE (London/Oxford/Nankai)

Training %0.20

simple/non-simple?

ML Mathematical Structures

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Combinatorics, Graph/Quivers, Symmetries[YHH-ST. Yau 2006.16619] (Wolfram Finite simple graphs DB)

ML standard graph properties: acyclic? (0.95); planar? (0.8); genus>, =, < 0? (0.8); ?∃ Hamilton/Euler cycles (∼ 0.8)

spectral bounds (R2 ∼ 0.9) . . .
Recognition of Ricci-Flatness (0.9) (todo: nd new Ricci-at graphs);[Bao-Franco-YHH-Hirst-Musiker-Xiao 2006.10783]: categorizing dierent
quiver mutation (Seiberg-dual) classes (0.9 - 1.0)
[Chen-YHH-Lal-Zas 2006.16114]: even/odd/reection sym (>0.99);
distinguishing CFT 3pt functions (>0.99); Fourier coecients / conformal
block presence (>0.97) . . . (q.v. [Krippendorf-Syvaeri 2003.13679])
NB. Only solving the likes of traveling salesman (similarly Groebner bases ingeom) stochastically

YANG-HUI HE (London/Oxford/Nankai)

ML Mathematical Structures

EAST, Jul, 202016 / 34

Number Theory: A Reprobate[YHH 1706.02714, 1812.02893:] Predicting primes tried supervised ML on

2 → 3, 2, 3 → 5, 2, 3, 5 → 7; xed window of (yes/no)1,2,...,k to(yes/no)k+1 , no breaking banks yet (expect same for Riemann zeroes)[Hirst-YHH-Peterken 2004.05218]: adjacency of dessin d'enfants(Grothendieck's Esquisse for Abs. Galois) → transcendental degree (<0.3)[Alessandretti-Baronchelli-YHH 1911.02008] (LMFdb/Cremona Database)
ML/TDA@Birch-Swinnerton-Dyer

New Scientist feature Dec 9 (BSD:

L-function L(s, E) of elliptic curve E has L(s → 1, E) given in terms of precise quantities: rank r ,torsion T , period Ω, Tate-Shaferevich group X, conductor N , regulator R, Tamagawa number c );X and Ω ok with regression and boosted decision trees: RMS < 0.1;
Weierstrass → rank: random

YANG-HUI HE (London/Oxford/Nankai)

ML Mathematical Structures

EAST, Jul, 202017 / 34

Meta-mathematics/physics?
[YHH-Jejjala-Nelson ] hep-th 1807.00735
Word2Vec: [Mikolov et al., '13] NN which maps words in sentences to avector space by context (much better than word-frequency, quickly adopted
by Google); maximize (partition function) over all words with sliding window(W1,2 weights of 2 layers, Cα window size, D # windows )Z(W1 , W2 ) :=

|D|
Cα
Y
1 X
exp([~xc ]T · W1 · W2 )log
VP
|D| α=1c=1
exp([~xc ]T · W1 · W2 )j=1

6

We downloaded all ∼ 10

titles of hep-th, hep-ph, gr-qc, math-ph, hep-lat

from ArXiv since the beginning (1989) till end of 2017

Word Cloud(rmk: Ginzparg has been doing a version of linguistic ML on ArXiv)(rmk: abs and full texts in future)

YANG-HUI HE (London/Oxford/Nankai)

ML Mathematical Structures

EAST, Jul, 202018 / 34

Subelds on ArXiv has own linguistic particulars

Linear Syntactical Identities

bosonic + string-theory = open-string
holography + quantum + string + ads = extremal-black-holestring-theory + calabi-yau = m-theory + g2
space + black-hole = geometry + gravity . . .

binary classication (Word2Vec + SVM) of formal (hep-th, math-ph, gr-qc)
vs phenomenological (hep-ph, hep-lat) : 87.1% accuracy (5-fold classication
65.1% accuracy).
Cf.

ArXiv classications

Tshitoyan et al., Unsupervised word embeddings capture latentknowledge from materials science literature, Nature July, 2019: 3.3. million
materials-science abstracts; uncovers structure of periodic table, predicts discoveries of new
thermoelectric materials years in advance, and suggests as-yet unknown materials

YANG-HUI HE (London/Oxford/Nankai)

ML Mathematical Structures

EAST, Jul, 202019 / 34

Summary and Outlook

PHYSICS

Use AI (Neural Networks, SVMs, Regressor . . . ) as
1. Classier deep-learn and categorize landscape data
2. Predictor estimate results beyond computational power

MATHS

how is AI doing maths w/o knowing any maths? (Alg Geo/C,
combinatorics, RT = integer matrices, NT ??)1. Predictor form new conjectures/formulae
2. Classier stochastically do NP-hard problems
Hierarchy of Diculty ML struggles with:numerical < algebraic geometry over C <combinatorics/algebra < number theory
YANG-HUI HE (London/Oxford/Nankai)

ML Mathematical Structures

EAST, Jul, 2020

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THANK YOU

Boris Zilber [Merton Professor of Logic,
Oxford]: you've managed syntax
without semantics. . . 

Alpha Go→

Alpha ZeroML→

Voevodsky's Dream;
Automated Thm Pf

cf. Renner et al., PRL/Nature News,
2019: ML (SciNet, autoencoder) nds

Sophia (Hanson Robotics, HK)

heliocentrism from Mars positions alone.
cf. Lample-Charton, 2019: ML Symolic
manipulations in mathematics

YANG-HUI HE (London/Oxford/Nankai)

ML Mathematical Structures

1st non-human citizen (2017, Saudi Arabia)
1st non-human with UN title (2017)
1st String Data Conference (2017)

EAST, Jul, 202021 / 34

Digressions

YANG-HUI HE (London/Oxford/Nankai)

ML Mathematical Structures

EAST, Jul, 202022 / 34χ(Σ) = 2

χ(Σ) = 0

χ(Σ) < 0

Spherical

Ricci-Flat

Hyperbolic+ curvature 0 curvature− curvature

Fano

Calabi-Yau

General Type

Euler, Gauss, Riemann, Bourbaki, Atiyah-Singer . . . ; generalize1χ(Σ) = 2 − 2g(Σ) = [c1 (Σ)] · [Σ] =
2π

Z
R=
Σ

2
X

(−1)i hi (Σ)

i=0

CONJECTURE [E. Calabi, 1954, 1957] / Thm [ST. Yau, 1977-8] M compact
Kähler manifold (g, ω) and ([R] = [c1 (M )])H 1,1 (M ) . Then ∃!(g̃, ω̃) such that([ω] = [ω̃])H 2 (M ;R) and Ricci(ω̃) = R.
Strominger & Yau were neighbours at IAS in 1985: CHSW named Ricci-Flat
Kähler as Calabi-Yau

YANG-HUI HE (London/Oxford/Nankai)

Back

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EAST, Jul, 202023 / 34

16 Reexive Polygons Back to Reexives

classify

convexlatticepolytopes with single interior point and all faces
are distance 1 therefrom(up to SL(n; Z))

Kreuzer-Skarke: 4319 reexive polyhedra, 473,800,776 reexive 4-polytopes,
Skarke: next number is at least 185,269,499,015.

YANG-HUI HE (London/Oxford/Nankai)

ML Mathematical Structures

EAST, Jul, 202024 / 34

Heterotic Comp: Recent Development

E6 GUTs a toy, SU (5) and SO(10) GUTs and SM: general embedding

Instead of T X , use (poly-)stable holomorphic vector bundle V
Gauge group(V ) = G = SU (n), n = 3, 4, 5, gives H = Commutant(G, E8 ):Breaking Pattern

E8 → G × H

MSSM: HSU (3) × E6

248→

(1, 78) ⊕ (3, 27) ⊕ (3, 27) ⊕ (8, 1)

SU (4) × SO(10)248→(1, 45) ⊕ (4, 16) ⊕ (4, 16) ⊕ (6, 10) ⊕ (15, 1)

SU (5) × SU (5)248→(1, 24) ⊕ (5, 10) ⊕ (5, 10) ⊕ (10, 5) ⊕ (10, 5̄) ⊕ (24, 1)

Wilson Line−→SU (3) × SU (2) × U (1)Issues in low-energy physics ∼ Precise questions in Alg Geo of (X, V )Particle Content ∼ (tensor powers) V Bundle Cohomology on X
LE SUSY ∼ Hermitian Yang-Mills connection ∼ Bundle Stability
Yukawa ∼ Trilinear (Yoneda) composition
Doublet-Triplet splitting ∼ representation of fundamental group of X
Back to LandscapeYANG-HUI HE (London/Oxford/Nankai)

ML Mathematical Structures

EAST, Jul, 202025 / 34

Various Databases

Kreuzer-Skarke:

http://hep.itp.tuwien.ac.at/~kreuzer/CY/

new PALP: Braun-Walliser: ArXiv 1106.4529
Triang: Altmann-YHH-Jejjala-Nelson: http://www.rossealtman.com/
CICYs:

resurrected Anderson-Gray-YHH-Lukas, http://www-thphys.

physics.ox.ac.uk/projects/CalabiYau/cicylist/index.html
q.v. other databases of interesting to the math/physics community:
Graded Rings/Varieties:
Finite Groups/Rings:
Modular Forms:

Brown, Kasprzyk, et al. http://www.grdb.co.uk/

GAP https://www.gap-system.org/

Sutherland, Cremona et al. https://www.lmfdb.org/

Knots & Invariants:

KnotAtlas http://katlas.org/

Return...

YANG-HUI HE (London/Oxford/Nankai)

ML Mathematical Structures

EAST, Jul, 2020

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Progress in String Theory Back to ML/Maths

Major International Annual Conference Series
1986- First Strings Conference
2002- First StringPheno Conference
2006 - 2010 String Vacuum Project (NSF)
2011- First String-Math Conference
2014- First String/Theoretical Physics Session in SIAM Conference
2017- First String-Data Conference

YANG-HUI HE (London/Oxford/Nankai)

ML Mathematical Structures

EAST, Jul, 202027 / 34

CICYs



n1

q11

q12

...

q1K

 n2

M=  .
 ..

q21

q22

...

q2K

.
.
.

.
.
.

.

.

.
.
.

1
qm

2
qm

...

K
qm

nm

.







−

Complete Intersection Calabi-Yau (CICY) 3-folds−

K eqns of multi-degree qji ∈ Z≥0nn
embedded in P 1 × . . . × P m−

KP

c1 (X) = 0 ;

j=1

m×K

−

qrj = nr + 1

M T also CICY

1,101

The Quintic Q = [4|5]−200 (or simply [5]);CICYs Central to string pheno in the 1st decade [Distler, Greene, Ross, et al.]

E6 GUTS unfavoured; Many exotics: e.g. 6 entire anti-generations
Back to CICYs

AdS/CFT as a Quiver Rep/Moduli Variety Corr.
a 20-year prog. joint with A. Hanany, S. Franco, B. Feng, et al.
Local CY3U(N)N D3−Branes

World−Volume =Generic

Quiver Gauge Theory.C3

D-Brane Gauge Theory

Toric

AbelianOrbifolds

Orbifolds(SCFT encoded as quiver)Singularities.C←→

CY3 Conedel Pezzo

Vacuum Space as ane Variety

Sasaki−Einstein 5−fold!

X(N = 4 SYM)

, W = Tr([x, y], z)Z

Y←→ C3 = Cone(S 5 ) [Maldacena]

THM [(P) Feng, Franco, Hanany, YHH, Kennaway, Martelli, Mekareeya, Seong, Sparks, Vafa, Vegh, Yamazaki,Zaaroni . . . (M) R. Böckland, N. Broomhead, A. Craw, A. King, G. Musiker, K. Ueda . . . ] (coherent
component of ) representation variety of a quiver is toric CY3 i quiver +superpotential graph dual to a bipartite graph on T

2

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combinatorial data/lattice polytopes ←→ gauge thy data as quivers/graphs
YANG-HUI HE (London/Oxford/Nankai)

ML Mathematical Structures

EAST, Jul, 202029 / 34

A Single Neuron: The Perceptron

began in 1957 (!!) in early AI experiments (using CdS photo-cells)
DEF: Imitates a neuron: activates upon certain inputs, so dene

Activation Function f (zi ) for input tensor zi for some multi-index i;
consider: f (wi zi + b) with wi weights and b bias/o-set;typically, f (z) is sigmoid, Tanh, etc.
(j)Given training data: D = {(xi, d(j) } with input xi and known output d(j) ,minimize

!2
SD =

X

X
(j)
f(
wi xi + b) − d(j)jito nd optimal wi and b ; learning, then check against Validation Data
Essentially (non-linear) regression

YANG-HUI HE (London/Oxford/Nankai)

ML Mathematical Structures

EAST, Jul, 202030 / 34

The Neural Network: network of neurons ; the brainDEF: a connected graph, each node is a perceptron (Implemented on

Mathematica 11.1 + / TensorFlow-Keras on Python)1

adjustable weights/bias;

2

distinguished nodes: 1 set for input and 1 for output;

3

iterated training rounds.

Simple case: forward directed only,
called multilayer perceptron

others: e.g., decision trees, support-vector machines (SVM), etc
Essentially how brain learns complex tasks; apply to our Landscape Data

Back to LandscapeYANG-HUI HE (London/Oxford/Nankai)

ML Mathematical Structures

EAST, Jul, 202031 / 34edComputing Hodge Numbers O(eRecall Hodge decomposition H?
H 1,1 (X) = H 1 (X, TX
),

p,q

)

(X) ' H q (X, ∧p T ? X) ;?
H 2,1 (X) ' H 1,2 = H 2 (X, TX) ' H 1 (X, TX )Euler Sequence for subvariety X ⊂ A is short exact:

0 → TX → TM |X → NX → 0
Induces long exact sequence in cohomology:

0

→

:0

0 
H
(X, TX ) → H 0 (X, TA |X ) →d
H 1 (X, TX ) → H 1 (X, TA |X ) →→H 2 (X, TX )→→H 0 (X, NX ) →H 1 (X, NX ) →...
i

Need to compute Rk(d), cohomology and H (X, TA |X ) (Cf. Hübsch)

Back to ML
YANG-HUI HE (London/Oxford/Nankai)

ML Mathematical Structures

EAST, Jul, 202032 / 34

ArXiv Word-Clouds

lattice scattering nonabelian scalar-field relativistic
massive d-brane
particleconformal
cosmologicaldimension instanton

dark-energy dynamical

spectrumpotential string-theory brane
newfermionyang-mills-theory
symmetryquantum-gravity
effect approachcosmology action
generalized
equationstringtheoryqcd
black-hole
algebragravitational qft

operator

local

note

backgroundenergy

state

gauge
amplitude
spin

quantumsystem
-

model

dynamics

entropychiralmatrix-model groupgeometry
matrix

field
space

cft

sugra

n=2 superstring ii

problem

limit
term

structureanomalyquantization

solutionsusy
field-theory gravityholographic

generalcorrection

solitoninflation gauge-theory topologicalcoupling
spacetime representation
boundary function
vacuumexact

classical duality

noncommutative interaction
universe holography invariant
constraint mass formalism matter
scalarrenormalization

new-physics structure
parameterchiral collisioncosmologyinteraction coupling
potential photon
fermiongravity spectrum
high-energy scale
string symmetry
analysis energy

largepion physics

newcurvatureanisotropic
gravitational cosmological-constantneutron-star

quantum-gravitysymmetry constantrelativistic

scalarsolutionmodelinflation branecosmicholographic

analysisradiation

fluid
general-relativitygeneralized

generalgravitational-wave charged

entropy datatime space
limit

state

spacetime

approachparticle

black-holetheoryspin

wave

energy iii

cmb

systemcollapsequantization

braneworldpropertyds -gravityequation

tensor

local

star

stringnew

background problem

parameter

cosmologydynamics
dark-energy
quantumgeometry
cosmological effect
field
universescalar-field mass
perturbation
spectrumcoupling
physics

modified-gravity

staticcosmological-model

singularity
dynamical

equation

sm

lhc
model
decay susyquark

cpv heavycorrelation
polarized->

vacuum processes

gamma

b

light

theorymeson
approachcorrection

spin

study

qcd

systemquantum

amplitude

gluon
scattering
limit
flavorbaryonmass neutrino nucleon
data

probing
matter inflation

constraint dynamics

proton heavy-quark phenomenology lattice-qcd thermal
property
lattice neutrino-mass cosmological
nuclear effective search determination mssm phase

scalarhep-phwormhole nonlinear correction

structure evolution constraint stability potential

methodeffectproduction particle

dark-matter
mixinghiggs

transition

hep-thinteraction classicalmodified

deformation

bound function evolution relativistic form-factor state
implications dynamical

field hadronicelectroweak

hadronthermodynamics

vacuum rotating dark-matter
einsteinmassive horizon conformal matter metricgr-qc

YANG-HUI HE (London/Oxford/Nankai)interaction relativistic stability

metricdiscrete hamiltonian dimension classicalintegrable representation
spectrum stochastic
fractional
potential functionlattice
waverandom symmetry
application form
point structureclass topologicalassociatedproblems

periodic particle

systemgeometry

algebraicquantization entropy

equationquantumspaceapproach

integraltensortype

matrixcomplexspin

asymptotic

group

energysolitongeometricii

propertyi

model

problemspacetime

analysis-

relation scattering

field-theory

gravityqft

spectruml

singular generalpolynomials

theorygeneralizednew susy
flow
limitalgebra
operator dynamics
noncommutative
field

graphsasymptotics

exact
quantum-mechanics

surfacesolutionnonlinear

method

time

local

linear boundary schrodinger-operator invariant finite
criticalconformal theorem manifold

state estimate
effectmath-phcritical interaction finite-volume scattering correction
simulation potential wilson-fermion
methodexact two-dimensional form-factor landau-gauge

resultmass

domain-wall-fermion

scalarstructuregauge-theory chiral analysis
action operator
heavy-quark chiral-perturbation-theory perturbative
approachfinite-temperature monopole flavor

quark-mass

nonperturbativedecay
systemvacuumising-model
su(3)

gluon

field

physics spin
improvedbproperty

abeliansymmetrylatticedensitylattice-qcd
quark qcd theorymodel- string

scaling

su(2)

pion

study

2d

large yang-mills-theory

phase
gauge

susy

Back to Word2Vec

lightbaryondynamics
new spectrum matter
quantum stateorder
dynamical lattice-gauge-theory

meson

coupling phase-transition
algorithm
confinement effectmagnetic chiral-symmetry
renormalization chemical-potential strong-coupling function

fermioneffective nucleon simulations
hadronic quencheddeterminationtransition expansion

topological monte-carlo numerical

hep-lat

ML Mathematical Structures

EAST, Jul, 202033 / 34

Classifying Titles

Compare, + non-physics sections, non-science (Times), pseudo-science (viXra)```
Actual```Word2Vec + SVM
```
```

1

2

3

4

5

1

40.2

6.5

8.7

24.0

20.6

2

7.8

65.8

12.9

9.1

4.4

3

7.5

11.3

72.4

1.5

7.4

4

12.4

4.4

1.0

72.1

10.2

5

10.9

2.2

4.0

7.8

75.1

1

2

3

4

5

6

7

viXra-hep

11.5

47.4

6.8

13.

11.

4.5

viXra-qgst

13.3

14.5

1.5

54.

8.4

1.8

PP

NN
PP
PP
P
P



 1


 2

:

hep-th:hep-ph3:

hep-lat




4



:

gr-qc

:

math-ph

8

9

10

0.2

0.3

2.2

3.1

0.1

1.1

2.8

3.

5

Actual

6: cond-mat,

7: q-n, 8: stat, 9: q-bio,

YANG-HUI HE (London/Oxford/Nankai)

10: Times of India

ML Mathematical Structures

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EAST, Jul, 202034 / 34