two graphs: the same, or not the same? 可以区分两张图么？ · sci. springer, cham, 2015,...

Two graphs:The same, or not the same?

可以区分两张图么？Yaokun Wu (吴耀琨)

Nov. 29, 2020 (贰零贰零年壹拾壹月贰拾玖日)

乱花渐欲迷人眼图的移位等价火眼金睛 Graph Theory Done Wrong Linear Algebra Done Right

Overview

1 乱花渐欲迷人眼

2 图的移位等价

3 火眼金睛

4 Graph Theory Done Wrong

5 Linear Algebra Done Right

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走马观花

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欲做图像识别，首先得对什么叫一样的图给出合理的规定。

如果千里马的图片有太多种，即使可以按图索骥，也很难及时对眼前一匹白马做出准确判断。

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猪八戒三十六变

• 23 =1015 =

−210−315?

• [3 48 7

] [7 24 9

]=

[37 4284 79

]?

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孙悟空七十二变

Figure-Eight knot Haken’s unknothttp://geomschool2018.univ-mlv.fr/slides/hass1.pdf

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http://geomschool2018.univ-mlv.fr/slides/hass1.pdf


A common theme in all of mathematics is deciding when twoseemingly different objects are actually the same in some sense,and when two seemingly identical objects are actually different,and when we have no way to tell whether or not two objects arethe same.

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Probability theory is concerned only with those proper-ties of a stochastic process that are shared by all equivalentstochastic processes. – Edward Nelson1

This book treats isomorphism theory - that branchof ergodic theory dealing with the question of when twomeasure-preserving systems are, in a certain sense, es-sentially equivalent. – Steven Kalikow and Randall Mc-cutcheon2

1Edward Nelson. Radically Elementary Probability Theory. Vol. 117. Annalsof Mathematics Studies. Princeton University Press, Princeton, NJ, 1987,pp. x+98.

2Steven Kalikow and Randall McCutcheon. An Outline of Ergodic Theory.Vol. 122. Cambridge Studies in Advanced Mathematics. Cambridge UniversityPress, Cambridge, 2010, pp. viii+174.

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Decidable or Undecidable

• (Haken 1961) There is an algorithm to decides whether or nottwo given knots are equivalent.

• (Markov 1958) Given two manifolds of dimension n ≥ 4, thereis no algorithm to decide whether or not they arehomeomorphic.

• Undecidability everywherehttp://www-math.mit.edu/~poonen/slides/cantrell3.pdf

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http://www-math.mit.edu/~poonen/slides/cantrell3.pdf


When a topologist is asked, “How do you visualize afour-dimensional object?”the appropriate response is a So-cratic rejoinder:“How do you visualize a three-dimensionalobject?”We do not see in three spatial dimensions directly,but rather via sequences of planar projections integrated ina manner that is sensed if not comprehended. We spenda significant portion of our first year of life learning howto infer three-dimensional spatial data from paired planarprojections. Years of practice have tuned a remarkableability to extract global structure from representations ina strictly lower dimension. – Robert Ghrist3

But a simpler question may be: Do you really see and recognizeone-dimensional objects?

3Robert Ghrist. “Barcodes: the persistent topology of data”. In:Bull. Amer. Math. Soc. (N.S.) 45.1 (2008), pp. 61–75.

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Two Graphs

Figure: The same, or not the same?

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指鹿为马

1

2

3

4

5

6

7

8

1 2

34

5 6

78

Yes, they are isomorphic!

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Two Digraphs

12

3

45

6

7

8

Figure: The same, or not the same?

This is a question, to which we will come back later.12 / 67


Recognizing Cayley Digraphs

The one on the left is more symmetric4.

4Eric L. McDowell. “Recognizing Cayley Digraphs”. In: Math. Mag. 93.4(2020), pp. 261–270.

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Recognizing Cayley Digraphs

The one on the left is more symmetric4.

4McDowell, “Recognizing Cayley Digraphs”.13 / 67


Petersen Graph: the smallest vertex-transitive graph that is not aCayley graph

https://blogs.ams.org/visualinsight/2015/07/01/petersen-graph/

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https://blogs.ams.org/visualinsight/2015/07/01/petersen-graph/https://blogs.ams.org/visualinsight/2015/07/01/petersen-graph/https://blogs.ams.org/visualinsight/2015/07/01/petersen-graph/


A Cubical Complex

5

5Louis J. Billera, Susan P. Holmes, and Karen Vogtmann. “Geometry of thespace of phylogenetic trees”. In: Adv. in Appl. Math. 27.4 (2001),pp. 733–767.

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And its Link

6

6R. Ghrist. Elementary Applied Topology. ed. 1.0. Createspace, 2014.url: https://www2.math.upenn.edu/~ghrist/notes.html.

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https://www2.math.upenn.edu/~ghrist/notes.html


The Space of Phylogenetic Trees on Five Taxa

B. Sturmfels, Can biology lead tonew theorems?

http://groups.claymath.org/library/annual_report/ar2005/05report_featurearticle.pdf

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http://groups.claymath.org/library/annual_report/ar2005/05report_featurearticle.pdfhttp://groups.claymath.org/library/annual_report/ar2005/05report_featurearticle.pdfhttp://groups.claymath.org/library/annual_report/ar2005/05report_featurearticle.pdfhttp://groups.claymath.org/library/annual_report/ar2005/05report_featurearticle.pdf


Figure: Jack Edmonds and his stonehttp://www.maths.qmul.ac.uk/~fink/Edmonds_NPstone.jpg

I spent from 1961 to 1965 on the subject NP versus P,and in 1966 made the conjecture NP ∩ coNP = P ,NP.– Jack Edmonds

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http://www.maths.qmul.ac.uk/~fink/Edmonds_NPstone.jpg


The Graph Isomorphism Problem

Babai7 showed that the graph isomorphism (GI) problem for twon-vertex graphs can be solved in quasipolynomial time, that isnp(log n) for some polynomial p.

It is not known whether GI belongs to coNP8.

GI and Weisfeiler-Leman algortihm: https://www.famnit.upr.si/sl/resources/files/seminars/koper151.pdf

7László Babai. “Group, graphs, algorithms: the graph isomorphismproblem”. In:Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. IV. Invited lectures.World Sci. Publ., Hackensack, NJ, 2018, pp. 3319–3336.

8Martin Grohe and Pascal Schweitzer. “The Graph Isomorphism Problem”.In: Commun. ACM 63.11 (2020), pp. 128–134.

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https://www.famnit.upr.si/sl/resources/files/seminars/koper151.pdfhttps://www.famnit.upr.si/sl/resources/files/seminars/koper151.pdf


Fractional Isomorphism

For any graph G, we use AGto denote its adjacencymatrix. Two graphs G and Hare fractionally isomorphicprovided there exists a doublystochastic matrix S such that

SAG = AHS. Figure: Two fractionally isomorphic graphsa.

aM.V. Ramana, E.R. Scheinerman, and D. Ullman. “Fractional isomorphismof graphs”. In: Discrete Math. 132.1-3 (1994), pp. 247–265.

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Theorem• 9 Two graphs are fractionally isomorphic if and only if they

cannot be distinguished by color refinement (1-dimensional WLalgorithm). Henceforth, it is polynomial-time solvable to deter-mine if two given finite graphs are fractionally isomorphic.

• 10 Suppose that F and G are fractionally isomorphic graphsand that F is a forest. Then F is isomorphic to G.

9Ramana, Scheinerman, and Ullman, “Fractional isomorphism of graphs”.10V. Arvind et al. “On the power of color refinement”. In:

Fundamentals of computation theory. Vol. 9210. Lecture Notes in Comput.Sci. Springer, Cham, 2015, pp. 339–350; Ramana, Scheinerman, and Ullman,“Fractional isomorphism of graphs”.

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Quantum Isomorphism

The quantum isomorphism problem is known to be undecidable11.

A planar graph has a 4-coloring, namely has a homomorphism toK4, if and only if it has a homomorphism to the Cayley graph Hon the symmetric group S4 whose connection set consists of allelements of order two12. The graph H has chromatic number 5.

In 1976, Appel and Haken used 1200 hours of computer time towork through the details of the first proof of 4CT.

11Albert Atserias et al. “Quantum and non-signalling graph isomorphisms”.In: J. Combin. Theory Ser. B 136 (2019), pp. 289–328.

12Laura Mančinska and David E. Roberson. “Quantum isomorphism isequivalent to equality of homomorphism counts from planar graphs”. In:(2019). arXiv: 1910.06958v2.

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https://arxiv.org/abs/1910.06958v2


太上老君八十一变

Figure: Photo by Gerrard Gethings

Terry Gannon (in Moonshine Beyond the Monster)calls ADE a“meta-pattern”in mathematics, i.e., a struc-ture that shows up more often than we would expect.Vladimir Arnold (in Symplectization, Complexification andMathematical Trinities) describes ADE as “a kind of re-ligion rather than mathematics.”– Drew Armstrong https://www.math.miami.edu/~armstrong/Talks/What_is_ADE.pdf 23 / 67

https://www.math.miami.edu/~armstrong/Talks/What_is_ADE.pdfhttps://www.math.miami.edu/~armstrong/Talks/What_is_ADE.pdf


Walks in a Digraph 凌波微步

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Symbolic Dynamics

• Dynamical system is“something”that“evolves”with time.• Symbolic dynamics can be viewed, in a sense, as dynamical

systems in which space and time have been discretised.• Representation theory studies how groups act on linear spaces;

Symbolic dynamics studies how the shift maps act ontwo-sided (one-sided) topological Markov chains, or simpy agood set of infinite words.

• Main open problem of this theory includes the isomorphismproblem for subshifts of finite type (SFT), namelyclassification of (finite) digraphs in terms of strong shiftequivalence.

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Let A be an alphabet of size n. For x = (xi) ∈ AZ, the shift mapσ sends x to σ(x) = y such that yi = xi+1 for all i ∈ Z. The pair(AZ, σ) is known as a full shift and denoted by Xn.

A closed shift-invariant subset X of a full shift is called a subshift.We often use σX for the restriction of σ on X.

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Sliding Block Code 邯郸学步

Curtis-Lyndon-Hedlund: A continuous shift-commuting map from(X,σX) to (Y, σY ) must be a sliding block code.

i−1y . ... ..

. .. . ... ..

Φ

iy i+1y

Figure: Sliding block code (滑块码)

If there is a sliding block code f from (X,σX) to (Y, σY ) and asliding block code g from (Y, σY ) to (X,σX) such that f ◦ g = IdYand g ◦ f = IdX , then we say that (X,σX) and (Y, σY ) areconjugate and we view them as the same dynamical system.

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Digraph and Matrix

[2 33 0

] [1 25 1

]

Walks in digraphs correspond to matrix multiplications.

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SFT and Graph Theory

For each digraph G, XG denotes the subshift consisting of allbi-infinite walks of G – a walk is a sequence of arcs such that thehead of an arc is the tail of its successive arc.We say that two digraphs are strong shift equivalent if their shiftspaces are conjugate to each other. A shift of finite type (SFT) isa subshift which is conjugate to the shift of some finite digraph.If we allow the arcs to have not necessarily distinct lables and weread the labels of the arcs when walking across an arc-labelleddigraph, we get the sofic shift of a digraph. In general, a sofic shiftis a shift space which is conjugate to a sofic shift of a labelleddigraph.

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相由心生

Many basic objects in mathematics should be viewed as morphismsin various senses13:

Figure: Morphisms: Loops in a topological space; Lines in an Euclideanspace; Elements in a set.

原相？现相？13Tom Leinster. “Rethinking set theory”. In: Amer. Math. Monthly 121.5

(2014), pp. 403–415.30 / 67


The Question

The study of SFT (sofic shift) is partly the study of walks in a(labelled) digraph14. By a walk, here we mean a morphism to thedigraph from the Cayley digraph of the monoid of non-negativeintegers or from the Cayley digraph of the additive group ofintegers.

QuestionHow to decide whether or not two (labelled) digraphs specifyconjugate SFT (sofic shift)? Is this a decidable problem?

14Terrence Bisson and Aristide Tsemo. “Symbolic dynamics and the categoryof graphs”. In: Theory Appl. Categ. 25 (2011), No. 22, 614–640.

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Multilingual dictionary15:transformation minimal presentation

symbolic dynamics sliding block code Fischerconvolutional code encoder Forney

automata transducer Hopcroft-Ullmansystem input-state-output Williams

15Brian Marcus. “Symbolic dynamics and connections to coding theory,automata theory and system theory”. In:Different aspects of coding theory (San Francisco, CA, 1995). Vol. 50. Proc.Sympos. Appl. Math. Amer. Math. Soc., Providence, RI, 1995, pp. 95–108;Dave Forney et al. “Multilingual dictionary: system theory, coding theory,symbolic dynamics, and automata theory”. In:Different aspects of coding theory (San Francisco, CA, 1995). Vol. 50. Proc.Sympos. Appl. Math. Amer. Math. Soc., Providence, RI, 1995, pp. 109–138.

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Constrained Code

In contrast to error correction coding, the main philosophy ofconstrained coding is to avoid patterns that are more prone toerror rather than to correct error patterns.

Figure: Encoding and decoding

B. Marcus, R. Roth, P. Siegel, An Introduction to Coding forConstrained Systems,https://ronny.cswp.cs.technion.ac.il/wp-content/uploads/sites/54/2016/05/chapters1-9.pdf

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https://ronny.cswp.cs.technion.ac.il/wp-content/uploads/sites/54/2016/05/chapters1-9.pdf


From Torus to Disk 环面与硬盘

Theorem (Finite-State Coding Theorem16)There is a finite-state (X,n)-code if and only if the sofic shift Xhas entropy at least logn.

Adler17 told us the story about how the state-splitting algorithmhidden behind the above theorem, which won the authors the IEEEInformation Theory Group 1985 Best Paper prize, came as anunexpected practical applications of pure mathematics research.

16Roy L. Adler, Don Coppersmith, and Martin Hassner. “Algorithms forsliding block codes. An application of symbolic dynamics to informationtheory”. In: IEEE Trans. Inform. Theory 29.1 (1983), pp. 5–22.

17R. L. Adler. “The torus and the disk”. In: IBM J. Res. Develop. 31.2(1987), pp. 224–234.

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I

(a) (b)

I1

I2

a

b

c

d

e

d1

d2

f 1f 2f

J

K

J1

K1

K2

a1

b1

c1

c2

e1

Figure: An out-splitting (out-amalgamation)18

18Douglas Lind and Brian Marcus.An Introduction to Symbolic Dynamics and Coding. Cambridge UniversityPress, Cambridge, 1995, pp. xvi+495.

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Some Facts

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A digraph is strongly shift equivalent to another if and only if itcan be transformed to the other by a sequence of splitting andamalgamation operations. Equivalently, two nonnegative integermatrices are strong shift equivalent if and only if they areconnected by a series of elementary strong shift equivalence:19

Figure: Elementary strong shift equivalence20

19R. F. Williams. “Classification of subshifts of finite type”. In:Ann. of Math. (2) 98 (1973), 120–153; errata, ibid. (2) 99 (1974), 380–381.

20K. H. Kim, F. W. Roush, and J. B. Wagoner. “The shift equivalenceproblem”. In: Math. Intelligencer 21.4 (1999), pp. 18–29.

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Shift Equivalence and Eventual Conjugacy

A and B are eventually conjugate if An and Bn are conjugate forall sufficiently large n.

Figure: Shift equivalence21

21Kim, Roush, and Wagoner, “The shift equivalence problem”.38 / 67


Shift equivalence and strong shift equivalence can be defined overany semiring, which turns out to be useful even if we aim tounderstand it over the semiring of nonnegative integers.

Algebraic K-theory is the branch of algebra dealingwith linear algebra (especially in the limiting case of largematrices) over a general ring R instead of over a field. –Jonathan Rosenberg22

K-theory was first considered from the appropriate point of viewby Atiyah in the last 1950s and 1960s as the most importantgeneralization of cohomology theory.

22Jonathan Rosenberg. Algebraic K-Theory and its Applications. Vol. 147.Graduate Texts in Mathematics. Springer-Verlag, New York, 1994, pp. x+392.

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Finite Equivalence and Almost Conjugacy

Two sofic shifts X and Y are finitely equivalent provided there isan SFT Z together with finite-to-one factor codes from it to bothX and Y . Two irreducible sofic shifts are finitely equivalent if andonly if they have the same entropy23.

Two sofic shifts X and Y are almost conjugate provided there is anSFT Z which possess almost invertible factor codes to both X andY . Let X and Y be irreducible sofic shifts with minimal rightresolving presentations (G,L) and (H,M). Then X and Y arealmost conjugate if and only if X and Y have the same entropywhile G and H have the same period24.

23William Parry. “A finitary classification of topological Markov chains andsofic systems”. In: Bull. London Math. Soc. 9.1 (1977), pp. 86–92.

24Roy L. Adler and Brian Marcus. “Topological entropy and equivalence ofdynamical systems”. In: Mem. Amer. Math. Soc. 20.219 (1979), pp. iv+84.

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Proper Embedding

For a shift space X, let qn(X) denote the number of points ofleast period n in X.

The Embedding Theorem of Krieger25 claims the following:Let X and Y be two irreducible SFTs. Then there is a properembedding of X into Y if and only if h(X) < h(Y ) andqn(X) ≤ qn(Y ) for all positive integer n.

25Wolfgang Krieger. “On the subsystems of topological Markov chains”. In:Ergodic Theory Dynam. Systems 2.2 (1982), pp. 195–202.

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Weak Equivalence

The following two are equivalent for two SFTs X and Y 26:• There exists a homomorphism ϕ from Xn to Xm where Xn

contains Y and Xm contains Z such that ϕ−1(Z) = Y , andthere exists a homomorphism ψ from Xk to Xℓ where Xkcontains Z and Xℓ contains Y such that ψ−1(Y ) = Z.

• There exists a homomorphism from Y to Z and there exists ahomomorphism from Z to Y .

26Joseph Barth and Andrew Dykstra. “Weak equivalence for shifts of finitetype”. In: Indag. Math. (N.S.) 18.4 (2007), pp. 495–506.

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Shear 剪切变换

For two different numbers i, j ∈ {1, . . . , n}, let Un[i, j] denote theunit shear (or unit transvection matrix) which is obtained from then by n identity matrix by adding an extra 1 in the (i, j) position.For two n× n nonnegative integer matrices A and B, add an arcfrom A to B if there exist i , j such that A(i, j)B(i, j) > 0 andUn[i, j]AUn[i, j]−1 = B.

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In a shear digraph, any two matrices in the same component arestrongly shift equivalent.27

Figure: Shear graph

27Kirby A. Baker. “Strong shift equivalence and shear adjacency ofnonnegative square integer matrices”. In: Linear Algebra Appl. 93 (1987),pp. 131–147, Theorem 3.2.

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年年岁岁花相似，岁岁年年人不同

Riedel28 showed that the following two are strongly shift equivalentfor every positive integer k:[

k − 1 11 k + 1

] [k 21 k

]

28Norbert Riedel. “An example on strong shift equivalence of positiveintegral matrices”. In: Monatsh. Math. 95.1 (1983), pp. 45–55.

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Cocyclic Subshifts

Cocyclic subshift is a generalization of sofic shifts.

Theorem29 The question of equality of two cocyclic subshifts isalgorithmically undecidable.

29David Buhanan and Jaroslaw Kwapisz. “Cocyclic subshifts fromDiophantine equations”. In: Dyn. Syst. 29.1 (2014), pp. 56–66.

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简易图论

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Flower Shift

A renewal system, first defined by R. Adler, is a subshift which isgenerated by infinite bilateral free concatenations of a finite set ofwords30. It is also known as a flower shift31.

30Susan Williams. “Notes on renewal systems”. In: Proc. Amer. Math. Soc.110.3 (1990), pp. 851–853.

31Wit Foryś, Piotr Oprocha, and Slawomir Bakalarski. “Symbolic Dynamics,Flower Automata and Infinite Traces”. In:Implementation and Application of Automata. Ed. by Michael Domaratzki andKai Salomaa. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011,pp. 135–142.

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看花容易绣花难

QuestionAre they conjugate to any renewal system?32 Moreover, is it truethat every irreducible SFT is conjugate to a renewal system?

32Aimee S. A. Johnson and Kathleen Madden. “Renewal systems,sharp-eyed snakes, and shifts of finite type”. In: Amer. Math. Monthly 109.3(2002), pp. 258–272; Rune Johansen. “Is every irreducible shift of finite typeflow equivalent to a renewal system?” In: Operator algebra and dynamics.Vol. 58. Springer Proc. Math. Stat. Springer, Heidelberg, 2013, pp. 187–209. 49 / 67


Two Families

Ak =[

1 kk − 1 1

]Bk =

[1 k(k − 1)1 1

]

When k ≤ 3, it is known that Ak and Bk are strong shiftequivalent. What about k ≥ 4?

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Ashley’s Eight by Eight 相逢何必曾相识

12

3

45

6

7

8

Figure: It is an unsolved question whether or not they are strong shiftequivalent. It is only easy to see that they are shift equivalent.

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The Same or Not? 笑问客从何处来

It is a shame that we have no idea if the above two digraphs arestrong shift equivalent33.

33Baker, “Strong shift equivalence and shear adjacency of nonnegative squareinteger matrices”.

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Open Problems in Symbolic Dynamics

http://www.math.umd.edu/~mboyle/open/

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http://www.math.umd.edu/~mboyle/open/


简约但不简单

• Digraphs are just nonnegative integer square matrices.• Number theory is about digraphs of one vertex!• We may need to consider weighted digraphs where an arc of a

digraph can carry a weight. This is seen from the fact thatnumber theorists are not really just playing with thosenumbers which you learned in elementary school.

• Group: one object category with all morphisms beingisomorphisms

• Monoid: one object category• Preorder: A category in which there exists at most one map

from any object to any other object54 / 67


实战矩阵论

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Integral matrices, or matrices over rings, is a huge sub-ject which extends into many different areas of mathemat-ics. For example, the entire theory of finite groups couldbe placed in this category. – Morris Newman, Preface of34

34Morris Newman. Integral Matrices. Pure and Applied Mathematics, Vol.45. Academic Press, New York-London, 1972, pp. xvii+224.

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Bowen-Franks Group

https://bowen.pims.math.ca/

Exercise35 Let R be a commutative ring. Let p(x) = a0 + a1x+ · · · + adxdbe a polynomial in x whose coefficients are from R. Let A be ann× n matrix over R. If a0 is invertible in the ring R, then the map

A∗ : α+ p(A)Rn → Aα+ p(A)Rn, ∀α ∈ Rn,

induces an isomorphism of the additive group Rn/p(A)Rn.35Rufus Bowen and John Franks. “Homology for zero-dimensional

nonwandering sets”. In: Ann. of Math. (2) 106.1 (1977), pp. 73–92. 57 / 67

https://bowen.pims.math.ca/


Spectral Conjecture

ExerciseLet A be an n×n nonnegative real matrix with largest nonnegativeeigenvalue 1. Let λ > 1. Then 0 < det(λI −A) ≤ λn − 1.

Boyle-Handelman conjecture36

36Mike Boyle and David Handelman. “The spectra of nonnegative matricesvia symbolic dynamics”. In: Ann. of Math. (2) 133.2 (1991), pp. 249–316;Mike Boyle and Scott Schmieding. “Strong shift equivalence and theGeneralized Spectral Conjecture for nonnegative matrices”. In:Linear Algebra and its Applications 498 (2016), pp. 231–243.

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Eventual Conjugacy and Shift Equivalence

ExerciseLet n be a positive integer and let C and D be two nonsingularcomplex matrices with Cn = Dn. If C , D, then we can find aneigenvalue λ of C and an eigenvalue µ of D such that λ , µ butλn = µn.

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Balanced Loading

Exercise (J. Friedman 1984 37)Let A be a nonnegative integer matrix with spectral radius λ.Assume that n ≤ λ < n+ 1 for some nonnegative integer n. Thenthere exists a nonnegative integer matrix B with each row/columnsum being either n or n+ 1 such that A and B are strong shiftequivalent.

37He published this result when he was still an undergraduate student atHarvard. It is the output of his visit during summer 1983 to the coding theorygroup at IBM San Jose led by Brian Marcus.

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1975: The Furstenberg (Ergodic) Magic Year!

Figure: https://mathshistory.st-andrews.ac.uk/Biographies/Furstenberg/pictdisplay/

I experienced the love of mathematics blended withhuman-kindness, an experience I can only wish I couldreplicate for others. – H. Furstenberg38

38Harry Furstenberg. “Ergodic behavior of diagonal measures and a theoremof Szemerédi on arithmetic progressions”. In: J. Analyse Math. 31 (1977),pp. 204–256; Parry, “A finitary classification of topological Markov chains andsofic systems”.

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https://mathshistory.st-andrews.ac.uk/Biographies/Furstenberg/pictdisplay/


Finite Equivalence

Exercise (H. Furstenberg, 1975)Suppose A and B are square, nonnegative integer irreduciblematrices. Then there exists a positive integer matrix F such thatAF = FB if and only if A and B have the same spectral radius.

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Diophantine Equation

ExerciseShow that

[19 45 1

]and

[19 54 1

]are not shift equivalent.

Hilbert’s 10th problem asked if an algorithm exists for determiningwhether an arbitrary Diophantine equation has a solution. YuriMatiyasevich39 demonstrated the impossibility of obtaining ageneral solution in 1970.

39Ju. V. Matijasevič. “The Diophantineness of enumerable sets”. In:Dokl. Akad. Nauk SSSR 191 (1970), pp. 279–282.

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8-line Proof of the Quillen-Suslin Theorem by Vaserstein

• Nil Garcés de Marcilla Escubedo, The Quillen-Suslin Theorem,http://diposit.ub.edu/dspace/bitstream/2445/125803/2/memoria.pdf, 2018.

• http://math.sjtu.edu.cn/faculty/ykwu/data/TeachingMaterial/limit.pdf

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http://diposit.ub.edu/dspace/bitstream/2445/125803/2/memoria.pdfhttp://diposit.ub.edu/dspace/bitstream/2445/125803/2/memoria.pdfhttp://math.sjtu.edu.cn/faculty/ykwu/data/TeachingMaterial/limit.pdfhttp://math.sjtu.edu.cn/faculty/ykwu/data/TeachingMaterial/limit.pdf


Let R be a commutative ring with identity. Let Mr,s(R) denotethe set of r × s matrices over R and let GLs(R) denote the set ofs× s invertible matrices over R 40. An elementary matrix is eitherthe identity mtrix or obtained from the identity matrix bymodifying exactly one off-diagonal entry. The group generated byall elementary matrices in GLn(R) is denoted En(R).

Exercise (L.N. Vaserstein)Let α ∈ Mr,s(R) and β ∈ Ms,r(R). Assume thatIr + αβ ∈ GLr(R). Show that[

(Ir + αβ)−1 00 Is + βα

]∈ Er+s(R).

40those matrices with both left inverse and right inverse.65 / 67


Every point above the average is a waste of effort. https://www.zhihu.com/question/36093280

I wasn’t getting very far. Most of the time I was strug-gling to keep my job. I’d see other people my age, such asSimon Donaldson (1986 Fields Medallist), being consider-ably more successful, and I thought I’m obviously not allthat good. There were times when I thought of droppingout. https://mathshistory.st-andrews.ac.uk/Biographies/Borcherds/– Richard Ewen Borcherds (b. 29 November 1959)

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https://www.zhihu.com/question/36093280https://www.zhihu.com/question/36093280https://mathshistory.st-andrews.ac.uk/Biographies/Borcherds/


Thank you!

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乱花渐欲迷人眼图的移位等价火眼金睛Graph Theory Done WrongLinear Algebra Done Right

two graphs: the same, or not the same? 可以区分两张图么？ · sci. springer, cham, 2015,...

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