Categories, Logic, Physics Higher Dimensional Birmingham Group - PowerPoint PPT Presentation

What is and what should be Categories, Logic, Physics ‘Higher Dimensional Birmingham Group Theory’? September 21, 2010 Ronnie Brown What it should be Dreams shattered! Back to basics Groupoids to the rescue Homotopy What is and what should be theory Commutative ‘Higher Dimensional Group Theory’? cubes Connections Rotations Ronnie Brown Historical context The end

What is and what should What should be higher be ‘Higher Dimensional dimensional group theory? Group Theory’? Ronnie Brown Optimistic answer: What it should be Real analysis ⊆ many variable analysis Dreams Group theory ⊆ higher dimensional group theory shattered! Back to basics What is 1-dimensional about group theory? Groupoids to We all use formulae on a line (more or less): the rescue Homotopy theory w = ab 2 a − 1 b 3 a − 17 c 5 Commutative cubes subject to the relations ab 2 c = 1, say. Connections Can we have 2-dimensional formulae? Rotations What might be the logic of 2-dimensional (or 17-dimensional) Historical formulae? context The end

What is and what should be ‘Higher The idea is that we may need to get away from ‘linear’ thinking Dimensional Group in order to express intuitions clearly. Theory’? Thus the equation Ronnie Brown What it 2 × (5 + 3) = 2 × 5 + 2 × 3 should be Dreams shattered! is more clearly shown by the figure Back to basics Groupoids to | | | | | | | | the rescue Homotopy | | | | | | | | theory Commutative But we seem to need a linear formula to express the general law cubes Connections Rotations a × ( b + c ) = a × b + a × c . Historical context The end

What is and what should be ‘Higher Dimensional Group Theory’? Ronnie Brown Published in 1884, What it should be available on the Dreams internet. shattered! Back to basics The linelanders Groupoids to the rescue had limited Homotopy interaction theory Commutative capabilities! cubes Connections Rotations Historical context The end

What is and what should be ‘Higher Consider the figures: Dimensional Group Theory’? Ronnie Brown What it should be Dreams shattered! Back to basics Groupoids to the rescue From left to right gives subdivision. Homotopy theory From right to left should give composition. Commutative What we need for local-to-global problems is: cubes Algebraic inverses to subdivision. Connections We know how to cut things up, but how to control Rotations Historical algebraically putting them together again? context The end

What is and what should be ‘Higher Dimensional Group Theory’? Ronnie Brown What it Look towards should be higher dimensional, Dreams shattered! noncommutative methods Back to basics for local-to-global problems Groupoids to and contributing to the unification of mathematics. the rescue Homotopy theory Commutative cubes Connections Rotations Historical context The end

What is and what should Higher dimensional group theory be ‘Higher Dimensional cannot exist (it seems)! Group Theory’? First try: A 2-dimensional group should be a set G with two Ronnie Brown group operations ◦ 1 , ◦ 2 each of which is a morphism What it should be G × G → G Dreams shattered! Back to basics for the other. Groupoids to Write the two group operations as: the rescue Homotopy theory Commutative x cubes y x Connections z Rotations Historical context x ◦ 1 z x ◦ 2 y The end

� What is and That each is a morphism for the other gives the what should be interchange law: ‘Higher Dimensional Group Theory’? ( x ◦ 2 y ) ◦ 1 ( z ◦ 2 w ) = ( x ◦ 1 z ) ◦ 2 ( y ◦ 1 w ) . Ronnie Brown What it should be This can be written in two dimensions as Dreams shattered! y x Back to basics Groupoids to the rescue z w Homotopy theory can be interpreted in only one way, and so may be written: Commutative cubes Connections � 2 � x � y Rotations z w 1 Historical context This is another indication that a ‘2-dimensional formula’ can be The end more comprehensible than a 1-dimensional formula!

� � What is and what should be ‘Higher Dimensional Group Theory’? Ronnie Brown Theorem Let X be a set with two binary operations ◦ 1 , ◦ 2 , each with identities e 1 , e 2 , and satisfying the interchange law. What it should be Then the two binary operations coincide, and are commutative Dreams and associative. shattered! Back to basics Proof � e 1 � Groupoids to e 2 � e 1 � e 2 the rescue 2 e 1 = = e 2 . e 2 e 1 e 2 e 1 Homotopy 1 theory We write then e for e 1 and e 2 . Commutative cubes Connections Rotations Historical context The end

What is and what should be ‘Higher Dimensional Group Theory’? Ronnie Brown � x � e What it x ◦ 1 w = x ◦ 2 w . e w should be So we write ◦ for each of ◦ 1 , ◦ 2 . Dreams shattered! � e � y Back to basics y ◦ z = z ◦ y . z e Groupoids to We leave the proof of associativity to you. This completes the the rescue proof. Homotopy theory Commutative cubes Connections Rotations Historical context The end

What is and what should be ‘Higher Dimensional Group Theory’? Ronnie Brown Dreams shattered! What it Back to basics! should be How does group theory work in mathematics? Dreams shattered! Symmetry Back to basics An abstract algebraic structure, e.g. in number theory, Groupoids to the rescue geometry. Homotopy Paths in a space: fundamental group theory Commutative cubes Connections Rotations Historical context The end

What is and what should Algebra structuring space be ‘Higher Dimensional F.W. Lawvere: The notion of space is associated with Group Theory’? representing motion. Ronnie Brown How can algebra structure space? What it Dirac String Trick should be The space of rotations in 3-dimensions Dreams shattered! The group theory equation is: Back to basics x 2 = 1 Groupoids to the rescue Homotopy theory Commutative cubes Connections Rotations Historical context The end

What is and what should The space around a knot be ‘Higher Dimensional Group Theory’? Ronnie Brown What it should be Dreams shattered! Back to basics Groupoids to the rescue Homotopy theory Commutative cubes Connections Rotations Historical context The end

What is and what should be ‘Higher Dimensional Group Theory’? Ronnie Brown What it should be Local and global issue. Dreams shattered! Use rewriting of relations. Back to basics Classify the ways of pulling the loop off the knot! Groupoids to the rescue Homotopy theory Commutative cubes Connections Rotations Historical context The end

� What is and what should Groupoids to the rescue be ‘Higher Dimensional Group Theory’? Groupoid: underlying geometric structure is a graph Ronnie Brown What it s i should be � G � G 0 G 0 Dreams t shattered! Back to basics such that si = ti = 1. Write a : sa → ta . Groupoids to Multiplication ( a , b ) �→ ab defined if and only if ta = sb ; the rescue so it is a partial multiplication, assumed associative. Homotopy theory ix is an identity for the multiplication: ( isa ) a = a = a ( ita ) Commutative So G is a small category, and we assume all a ∈ G are cubes invertible. Connections (groups) ⊆ (groupoids) Rotations Historical context The end

What is and The notion of groupoid first arose in number theory, what should be generalising work of Gauss from binary to quaternary quadratic ‘Higher Dimensional forms. Group Theory’? Groupoids clearly arise in the notion of composition of paths, Ronnie Brown giving a geography to the intermediate steps. What it should be Dreams shattered! Back to basics Groupoids to the rescue The objects of a groupoid add a spatial component to group Homotopy theory. theory Groupoids have a partial multiplication, and this opens the Commutative cubes door into the world of partial algebraic structures. Connections Higher dimensional algebra: algebra structures with partial Rotations operations defined under geometric conditions. Historical Allows new combinations of algebra and geometry, new kinds context of mathematical structures, and new ways of describing their The end inter-relations.

What is and what should be ‘Higher Theorem Let G be a set with two groupoid compositions Dimensional Group satisfying the interchange law (a double groupoid). Then G Theory’? contains a family of abelian groups. Ronnie Brown Double groupoids are more nonabelian than groups. What it n -fold groupoids are even more nonabelian! should be Masses of algebraic and geometric examples, linking with Dreams shattered! classical themes, particularly crossed modules. Rich algebraic Back to basics structures! Groupoids to Are there applications in geometry? in physics? in the rescue neuroscience? Homotopy theory Credo: Commutative Any simply defined and intuitive mathematical structure is cubes Connections bound to have useful applications, eventually! Rotations Search on the internet for “higher dimensional algebra”. 51,000 Historical hits recently context The end