Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Categorical CombinatoricsCombining the Concrete and the Conceptual
Tien Chih
Department of Science and MathematicsNewberry College, Newberry SC
June 15, 2016
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Outline of the Presentation
1 Introduction:
2 Graph Theory
3 Enumerative Combinatorics
4 Conclusion
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
The 20th Century
The 20th century saw major advances in mathematics:
Topology: dimension invariance, Betti numbers (homological invariance),Algebraic Topology, Knots in low dimension.
Algebra: Noether isomorphism theorems, Representation theory, K -theory,Modern ring, group and module theory.
Geometry: Algebraic Geometry, Varieties, Differential geometry.
Analysis: Functional Analysis, Hilbert and Banach Spaces, Fourier seriesand Harmonic Analysis.
Combinatorics: Basically all of it. (Graph Theory, Designs, Matroids etc.)
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Paul Erdos (1913-1996)
Highlights
Co-authored over 1,500 papers.
Solved problems in combinatorics, graph theory, number theory, classical analysis,approximation theory, set theory and probability theory.
Once solved a problem at a functional analysis seminar with no background, onlyasking for a few definitions.
Saw Mathematics as an oppurtunity to solve problems
“The purpose of life is to prove and to conjecture.”
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Alexander Grothendieck (1928-2014)
Highlights
Revolutionized Geometry by tying it fundamentally to Algebra and Topology.
Introduced the theory of Schemes (Zariski topologies glued together).
Belived all proofs could be expressed as tautologies, given sufficientdevelopment of theory.
“The first analogy that came to my mind is of immersing the nut in somesoftening liquid, and why not simply water? From time to time you rub so theliquid penetrates better, and otherwise you let time pass. The shell becomesmore flexible through weeks and months - when the time is ripe, hand pressure isenough, the shell opens like a perfectly ripened avocado!”
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Categories
Definition
A category C consists of:
A class of ob(C) objects.
A class hom(C) of morphisms or “arrows” denoted f : a → b or
a bf
satisfying:
Given a, b, c ∈ ob(C), ∃f : a → b, g : b → c, then ∃!h : a → c.
a b
c
f
g∃!h
Given a ∈ ob(C), ∃!ida : a → a:
a
ida
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Example of Categories
Common Examples of Categories are:
Sample Categories
Set, where the objects are the class of sets, and the morphisms arefunctions.
Grp, where the objects are groups, and the morphisms are grouphomomorphisms.
Top, where the objects are topological spaces, and the morphisms arecontinuous functions.
VectK , where the objects are K -vector spaces and the morphisms areK -linear transformations.
More Generally
Let ob(C) denote a class of sets with some structure, and hom(C) denote theclass of functions which preserve that structure.
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Graphs
Definition
A Graph is a pair of sets G = (V ,E), where V is a set of vertices, and E is acollection of unordered pairs of vertices called edges.
“single” graph simple graph loopless graph
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Morphisms
Definition
Given graphs G ,H, a graph morphism f : G → H is a mapf : V (G) ∪ E(G)→ V (H) ∪ E(H) such that incidence and adjacency arepreserved.
If ∀e ∈ E(G) it follows that f (e) ∈ E(H), then f is called a strict morphism.
1 2
3 4
f
morphism
3 4
1,2 1 2
3 4
g
strict morphism
1 2,3 4
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Coloring as a Morphism
Definition
A proper n-vertex coloring of a graph g is a function c : V (G)→ [n] such thatif vw ∈ E(G), c(v) 6= c(w). If such a c exists, we say G is n colorable. χ(G)the chromatic number is the minimum such n.
Theorem
G is k colorable iff there exists a strict morphism f : G → Kk .
1 2
3 4
f
1, 4 2
3
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Classifying Graphs with Morphisms
Definition
A Graph is Eularian if there exists a circuit which traverses all edges exactlyonce.
Theorem
Let G be a graph, then G is Eulerian iff ∃ f : C|E(G)| � G , where f is a strictepimorphism.
1 2
3
45
6f
1, 4 2
3, 56
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Definition
A Graph G is Hamiltonian if there exists a circuit which traverses each vertexexactly once.
Theorem
Let G be a graph, then G is Hamiltonian iff ∃ f : C|V (G)| ↪→ G , where f is a(strict) monomorphism.
1 2
3
45
6f
1 4
3
25
6
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Matchings with Morphisms
Definition
Given a graph G , a matching of G is a loop less subgraph M ′ where eachvertex is incident to exactly one edge. If V (M ′) = V (G) then we say M ′ is aperfect matching
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Theorem
Let G be a graph. Also let M :=∐n
i=1(K2)i , and let f be a (strict)monomorphism f : M ↪→ G . Then the image of M under G is a matching. IfV (f (M)) = V (G), f (M) is a perfect matching.
e1
e2
e3
e4
f
e1
e2
e3
e4
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Coverings with Morphisms
Definition
Given a graph G , a covering is an edge-free subgraph C ′ such that for eachedge e ∈ V (G), there is a c ∈ V (C ′) such that e is incident to c.
∗
∗
∗ ∗
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Theorem
Let G be a graph, and let C be an empty edge graph, and f : C ↪→ G be a(strict) monomorphism. Also let ρ : K2 � K1 be the unique morphism from K2
to K1. If given any (strict) monomorphism ι : K2 → G , it follows thatρ(ι−1(f (C))) is non-empty, then f (C) is a covering of G .
c1
c2
c3
c4
f
∗c1
∗c2
∗c3 ∗c4ι
K2
ρ
K1
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Independent Sets and Morphisms
Definition
Given a graph G , an independent set is a subgraph I where given i1, i2 ∈ V (I ),it follows that i1i2 6∈ E(G).
i1
i2
i3
i4
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Theorem
Let T ′ be the graph with V (T ′) = {v1, v2} and E(T ′) = {v1v2, v2v2}. Then letf : G → T ′ be a strict morphism. It follows that f −1(v1) is an independent setof G . In fact there is a one-to-one correspondence between all possible choicesof f and the independent sets of G .
fv2
v1i1, i2, i3, i4
i1
i2
i3
i4
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
A General Perspective
In general, we would like to answer the question:
Problem
Given a graph G , and a graph H, does there exist f : G → H
G Hf
where f has [Some Property]?
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Species
Definition
For a finite set U, a species is a rule F that produces the set of F structures ofU. Additionally for any bijection σ : U → V , F [σ] is a bijection
F [σ] : F [U]→ F [V ]
called the transport of F-strcutures along σ. F must satisfy the following:
Given σ : U → V , τ : V →W ,
F [σ ◦ τ ] = F [σ] ◦ F [τ ].
F [idU ] = idF [U].
In other words: F is a functor F : B→ B where B is the category of finite setsand bijections.
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Examples of Species
G the species of labeled simple graphs
Given a finite set U let G[U] be the set of all labeled graphs on U.
G[{1, 2, 3}] =1 2
3
, 1 2
3
1 2
3
, 1 2
3
, 1 2
3
, 1 2
3
, 1 2
3
, 1 2
3
Additionally σ : {1, 2, 3} → {a, b, c} produces similar graphs labeled a, b or cand there is a natural bijection
G[σ] : G[{1, 2, 3}]→ G[{a, b, c}]
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Utility of Species
Properties of Species
One can define the Sum, Product, Composition and Differentiation ofspecies.
One can define a “power series” type expansion for a species F :
F (x) :=∑
n≥0,|Un|=n
|F [Un]|xn
n!.
Given F [U], once can consider equivalence classes under the symmetricgroup SU , and construct the isomorphism classes of F [U].
1 2
3
, 1 2
3
1 2
3
, 1 2
3
, 1 2
3
, 1 2
3
, 1 2
3
, 1 2
3
→
, , ,
The theory of species allows us to enumerate collections of objects thatare difficult to count with direct counting methods.
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Current & Future Work
Current Work
Studying several problems of graph isomorphisms between limits andco-limits of graphs (with Dr. Demitri Plessas.)
Identifying the relationship between graph morphisms and minors ofgraphs (with D.P.).
Working with an undergraduate to write a program which can computelimits and co-limits in the category of Grphs.
Future Work
Understand species a lot better!
Consider species of combinatorial structures (matroids, t-designs etc.) andtry to find functorial relations between them.
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
Conclusion
In Summary: The level of abstraction that category theory provided to thefields of Algebra, Topology and Geometry allowed great advances to be madein those areas, and exposed the connections between the three. The samecould be done for Combinatorics as well, advancing the theory and connectingit to other branches of Mathematics.
Acknowledgements:
I’d like to thank the MAA for giving me the opportunity to present this talk. Iwould like to thank the math faculty of UNC Wilmington and the organizingcommittee for all their work hosting this event.Finally, I’d like to thank all in attendance for taking the time to listen to thispresentation.
Questions?
Tien Chih Categorical Combinatorics
Introduction: Graph Theory Enumerative Combinatorics Conclusion References
References
Bibliography
[1] Michael Aatiyah, Mathematics in the 20th Century, The MAA Monthly (2001).
[2] Francois Bergeron, Gilbert Labelle, and Pierre Leroux, Introduction to the Theory of Species,Universite du Quebec a Montreal, 2008.
[3] Adrian Bondy and U.S.R. Murty, Graph Theory, Springer, 2008.
[4] Edward Frenkel, The Lives of Alexander Grothendieck, a Mathematical Visionary, 2014.
[5] Andrew Gainer-Dewar, Γ-Species and the Enumeration of k-trees, The Electronic Journal ofCombinatorics 19 (2012), no. 4.
[6] Pavol Hell and Jaroslav Nesetril, Graphs and Homomorphisms, Oxford University Press, 2004.
[7] Paul Hoffman, The Man Who Only Loved Numbers: The Story of Paul Erdos and the Searchfor Mathematical Truth, Hachette Books, 1999.
[8] Saunders Macclane, Categories for the Working Mathematician, Springer-Verlag, 1998.
[9] Demitri Plessas, The Categories of Graphs, Ph.D. Thesis, 2011.
Tien Chih Categorical Combinatorics
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