Here are some of my past travels and talks, in reverse chronological order, based on my lectures page. Elsewhere you can find a list of transparencies and/or videos of talks I've given, but you can also reach those by clicking on some of the links here.
Probabilities versus Amplitudes
Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, the master equation for a chemical reaction network describes the interactions of molecules in a stochastic rather than quantum way. If we look at it from the perspective of quantum theory, this formalism turns out to involve creation and annihilation operators, coherent states and other well-known ideas — but with a few big differences.
I gave the opening talk at 9 am on Sunday June 28th:
I also gave a talk at 2:30 on Monday June 29th:Categories in Control
Control theory is the branch of engineering that studies dynamical systems with inputs and outputs, and seeks to stabilize these using feedback. Control theory uses "signal-flow diagrams" to describe processes where real-valued functions of time are added, multiplied by scalars, differentiated and integrated, duplicated and deleted. In fact, these are string diagrams for the symmetric monoidal category of finite-dimensional vector spaces, but where the monoidal structure is direct sum rather than the usual tensor product. Jason Erbele has given a presentation for this symmetric monoidal category, which amounts to saying that it is the PROP for bicommutative bimonoids with some extra structure.A broader class of signal-flow diagrams also includes 'caps' and 'cups' to model feedback. This amounts to working with a larger symmetric monoidal category where objects are still finite-dimensional vector spaces but the morphisms are linear relations. Erbele also found a presentation for this larger symmetric monoidal category. It is the PROP for a remarkable thing: roughly speaking, an object with two special commutative dagger-Frobenius structures, such that the multiplication and unit of either one and the comultiplication and counit of the other fit together to form a bimonoid.
Circuits, Categories and Rewrite Rules
We describe a category where a morphism is an electrical circuit made of resistors, inductors and capacitors, with marked input and output terminals. In this category we compose morphisms by attaching the outputs of one circuit to the inputs of another. There is a functor called the 'black box functor' that takes a circuit, forgets its internal structure, and remembers only its external behavior. Two circuits have the same external behavior if and only if they impose same relation between currents and potentials at their terminals. This is a linear relation, so the black box functor goes from the category of circuits to the category of finite-dimensional vector spaces and linear relations. Constructing this functor makes use of Brendan Fong's theory of 'decorated cospans' — and the question of whether two circuits map to the same relation has an interesting answer in terms of rewrite rules.
On Monday a little after 10:30 am, I gave this talk:
Network Theory
Nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, signal-flow graphs, Bayesian networks, Feynman diagrams and the like. Mathematically minded people know that in principle these diagrams fit into a common framework. But we are still far from a unified theory of networks. Here we propose compact categories as a general framework - or for a more detailed treatment, compact bicategories. We illustrate this with a number of key examples, and show how these examples are connected by functors.
Networks in climate science
The El Niño is a powerful but irregular climate cycle that has huge consequences for agriculture and perhaps global warming. Predicting its arrival more than 6 months ahead of time has been difficult. A recent paper by Ludescher et al caused a stir by using ideas from network theory to predict the start of an El Niño toward the end of 2014 with a 3-in-4 likelihood. After explaining the basics of El Niño and climate network theory, we critically analyze their method and address the question: are El Niños signaled by an increase in temperature correlations between regions of the Pacific within the El Niño basin and those outside it?
At 2:30 pm my time I gave a virtual public lecture at the University of York. This was 7:30 pm in the UK. The talk actually started half an hour later. My host is Stijn Hanson. I talked for about an hour and then answered some questions.The exceptional Jordan algebra and the Leech lattice
When Jordan, Wigner and von Neumann classified algebras of observables in their work on the foundations of quantum mechanics, they found 4 infinite series and one exception. This 'exceptional Jordan algebra' is 27-dimensional and consists of 3×3 self-adjoint octonionic matrices. The Leech lattice is another exceptional structure: the unique 24-dimensional even unimodular lattice with no vectors of length squared 2. I'll explain these entities and describe some work with Greg Egan where we made the Leech lattice into a 'Jordan subring' of the exceptional Jordan algebra.
8Different numbers have different personalities, and 8 is one of my favorites. The number 8 plays a special role in mathematics due to the "octonions", an 8-dimensional number system where one can add, multiply, subtract and divide, but multiplication is noncommutative and nonassociative. The octonions were discovered by Hamilton's friend John Graves in 1843 after Hamilton told him about the "quaternions". While much neglected, they stand at the crossroads of many interesting branches of mathematics and physics. For example, superstring theory works in 10 dimensions because 10 = 8+2, where 8 is the dimension of the octonions. Also, the densest known packing of spheres in 8 dimensions occurs when the spheres are centered at certain "integer octonions", which form the root lattice of the exceptional Lie group E8. The octonions also explain the curious way in which topology in dimension n resembles topology in dimension n+8.
After this talk I spoke with Aissa Wade at 317 McAllister.
Around 10:30 am I met Jason Morton at 219B McAllister.
At 12 I met Abhay Ashtekar at 316 Whitmore Laboratory and have lunch with him, Marc Geiller and Eugenio Bianchi
At 2:30 - 3:20 pm I gave this talk in the Geometry-Analysis-Physics Seminar at 106 McAllister:
Split octonions and the rolling ball
Understanding exceptional Lie groups as the symmetry groups of more familiar objects is a fascinating challenge. The compact form of the smallest exceptional Lie group, G_{2}, is the symmetry group of an 8-dimensional nonassociative algebra called the octonions. However, another form of this group arises as symmetries of a simple problem in classical mechanics! The space of configurations of a ball rolling on another ball without slipping or twisting defines a manifold where the tangent space of each point is equipped with a 2-dimensional subspace describing the allowed infinitesimal motions. Under certain special conditions, the split real form of G_{2} acts as symmetries. We can understand this using the quaternions together with an 8-dimensional algebra called the 'split octonions'. The rolling ball picture makes the geometry associated to G_{2} quite vivid. This is joint work with James Dolan and John Huerta, with animations created by Geoffrey Dixon.
At 6 pm there was a party at John Roe's house.
At 2:30 pm I gave a talk on Climate Networks in John Roe's class at 2:30 pm in 115 Osmond. It's a 50-minute class:
Math 033 - Mathematics for SustainabilityThe course meets Mondays, Wednesdays and Fridays at 2:30 in 115 Osmond. Class sessions last for 50 minutes and are divided roughly 60:40 between 'theory' classes — where the central theme is a mathematical idea — and 'case studies' — where the central theme is a particular example of sustainability practice and how it can be analyzed mathematically.
The aim of the class is to reach students who do not have an extensive mathematical background. High school algebra is the only prerequisite for the class. As the semester progresses we will introduce some more ideas related to 'measuring' (unit systems, scientific notation, specific discussion of energy); 'changing' (stocks, flows, equilibrium, dynamics, tipping points); 'risking' (probability, inference, decision-making under uncertainty); and 'networking' (networks, connectivity, strong and weak ties). Please bear this level of background knowledge in mind as you prepare your presentation. The students will be eager to hear how the ideas that they have been learning relate to the real-world experience that you can bring to the table.
Here is my talk:
Information and entropy in biological systems
Information and entropy are being used in biology in many different ways: for example, to study biological communication systems, the 'action-perception loop', the thermodynamic foundations of biology, the structure of ecosystems, measures of biodiversity, and evolution. Can we unify these? To do this, we must learn to talk to each other. This will be easier if we share some basic concepts which I'll sketch here.
I went to the NIPS Speaker's dinner at the Hyatt Regency Hotel Hotel on Tuesday, December 9th, 2014.
I gave this talk at 9 am on Wednesday December 10th:
Networks in climate science
The El Niño is a powerful but irregular climate cycle that has huge consequences for agriculture and perhaps global warming. Predicting its arrival more than 6 months ahead of time has been difficult. A recent paper by Ludescher et al caused a stir by using ideas from network theory to predict the start of an El Niño toward the end of 2014 with a 3-in-4 likelihood. After explaining the basics of El Niño and climate network theory, we critically analyze their method and address the question: are El Niños signaled by an increase in temperature correlations between regions of the Pacific within the El Niño basin and those outside it?
Biodiversity, entropy and thermodynamics
The most popular measures of biodiversity are formally identical to the measures of entropy developed by Shannon, Rényi and others. This fits into a larger analogy between thermodynamics and the mathematics of biodiversity. For example, in certain models of evolutionary game theory one can show that as population approaches an 'evolutionary optimum', the amount of information it has 'left to learn' is nonincreasing. This is mathematically analogous to the Second Law of Thermodynamics.
If you arrive before 4pm, then we could just meet at the theory chair, which is rather easier to find: we're in the so-called blue highrise, Martensstr. 3, which you cannot miss once you're at Technische Fakultät (it is, well, a blue highrise...), on floor 11 (the floors are small, so you'll find us). I myself will have to teach until 3:45 pm, but most of the other guys (some of them in the CC), including, I believe, our secretary Mrs. Schünberger (CC), will be there all afternoon, so you're welcome to drop by for coffee.Here is my talk:
Network Theory
We are entering the Anthropocene, a new geological epoch in which the biosphere is strongly affected by human activities. So, we can expect the science of this century to draw inspiration from biology, ecology and the environmental problems that confront us. What can category theorists, of all people, have to say about this? Researchers in many fields draw diagrams of networks: flow charts, Petri nets, Bayesian networks, electrical circuit diagrams, signal-flow graphs, Feynman diagrams and the like. In principle these diagrams fit into a common framework: category theory. But we are still far from a unified theory of networks. We give an overview of the theory as it stands now, with an emphasis on topics for future research, some involving higher categories.
Spans and the Categorified Heisenberg Algebra
Heisenberg reinvented matrices while discovering quantum mechanics, and the algebra generated by annihilation and creation operators obeying the canonical commutation relations was named after him. It turns out that matrices arise naturally from 'spans', where a span between two objects is just a third object with maps to both those two. In terms of spans, the canonical commutation relations have a simple combinatorial interpretation.More recently, Khovanov introduced a 'categorified' Heisenberg algebra, where the canonical commutation relations hold only up to isomorphism, and these isomorphisms obey new relations of their own. The categorified Heisenberg algebra naturally acts on the '2-Fock space' describing collections of particles in a 4-dimensional topological quantum field theory.
The meaning of the new relations in the categorified Heisenberg algebra was initially rather mysterious. However, Jeffrey Morton and Jamie Vicary have shown that they again have a nice interpretation in terms of spans. We can begin to formalize this using the work of Alex Hoffnung and Mike Stay, who have shown that spans of groupoids are morphisms in a symmetric monoidal bicategory.
Operads and the Tree of Life
Trees are not just combinatorial structures: they are also biological structures, both in the obvious way but also in the study of evolution. Starting from DNA samples from living species, biologists seek to reconstruct the most likely phylogenetic tree describing how these species evolved from earlier ones. In fact, phylogenetic trees are operations in an operad, the "phylogenetic operad", which plays a universal role in the study of branching Markov processes. To understand this operad, and more generally the relation between operads and trees, we use the fact that operads are themselves the algebras of a (typed) operad. This is joint work with Nina Otter and Todd Trimble.
Operads and the Tree of Life
Trees are not just combinatorial structures: they are also biological structures, both in the obvious way but also in the study of evolution. Starting from DNA samples from living species, biologists seek to reconstruct the most likely phylogenetic tree describing how these species evolved from earlier ones. In fact, phylogenetic trees are operations in an operad, the "phylogenetic operad", which plays a universal role in the study of branching Markov processes. To understand this operad, and more generally the relation between operads and trees, we use the fact that operads are themselves the algebras of a (typed) operad. This is joint work with Nina Otter and Todd Trimble.
On May 2, I gave this talk:
Network Theory
We are entering the Anthropocene, a new geological epoch in which the biosphere is strongly affected by human activities. Thus, just as 20th-century mathematics drew inspiration from physics, we can expect 21st-century mathematics to draw inspiration from biology, ecology and the environmental problems that confront us. What do category theorists, of all people, have to say about this? Researchers in many fields draw diagrams of networks: flow charts, electrical circuit diagrams, signal-flow graphs, Bayesian networks, Feynman diagrams and the like. In principle these diagrams fit into a common framework: category theory. But we are still far from a unified theory of networks. We give an overview of the theory as it stands now, with an emphasis on topics for future research, some involving higher categories.
Arrival and departure information:
Kindly check in with our reception staff when you arrive at Schloss Dagstuhl. Our reception office, located in the newer facility just left of the main building, is open from 3 p.m. to 7 p.m. on Sundays and holidays and from 8 a.m. to 4 p.m. on other days. If reception is closed when you arrive, please [....] A buffet dinner is available at the Schloss on Sundays from 6 p.m. until the following morning.
The Mathematics of Planet Earth
The global warming crisis is part of a bigger transformation in which humanity realizes that the Earth is a finite system and that our population, energy usage, and the like cannot continue to grow exponentially. If civilization survives this transformation, it will affect mathematics — and be affected by it — just as dramatically as the agricultural revolution and industrial revolution did. We cannot know for sure what the effect will be, but we can already make some guesses.
On Wednesday after my talk a bunch of people including students of Martin Hyland and Peter Johnstone had drinks at a pub, and then I went to dinner at Trinity with Bela Bollobas. On Thursday I had lunch at Trinity with Tim Gowers and Huw Price. On Thursday evening Lisa and I had dinner at the Gowers' house. On Friday we returned to Erlangen.
The Mathematics of Planet Earth
The global warming crisis is part of a bigger transformation in which humanity realizes that the Earth is a finite system and that our population, energy usage, and the like cannot continue to grow exponentially. If civilization survives this transformation, it will affect mathematics — and be affected by it — just as dramatically as the agricultural revolution and industrial revolution did. We cannot know for sure what the effect will be, but we can already make some guesses.
This took place in the Mathematical Institute building.Operads and the Tree of Life
Trees are not just combinatorial structures: they are also biological structures, both in the obvious way but also in the study of evolution. Starting from DNA samples from living species, biologists use increasingly sophisticated mathematical techniques to reconstruct the most likely 'phylogenetic tree' describing how these species evolved from earlier ones. In their work on this subject, they have encountered an interesting example of an operad, which is obtained by applying a variant of the Boardmann–Vogt 'W construction' to the operad for commutative monoids. The operations in this operad are labelled trees of a certain sort, and it plays a universal role in the study of stochastic processes that involve branching. It also shows up in tropical algebra. This talk is based on work in progress with Nina Otter.
Network Theory
Nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, signal-flow graphs, Bayesian networks, Feynman diagrams and the like. Mathematically minded people know that in principle these diagrams fit into a common framework: category theory. But we are still far from a unified theory of networks. After an overview, we will look at three portions of the jigsaw puzzle in three separate talks.
All these talks took place in Lecture Theatre B in the Computer Science Department:
The first talk was part of the OASIS series, meaning the "Oxford Advanced Seminar on Informatic Structures".
Also: Minhyong Kim organized an informal seminar on quantum gravity meeting 10:30-12+ε on the Fridays February 28 and March 7, meeting at Merton College, followed by lunch.
Categories in Control
Control theory is the branch of classical mechanics that deals with 'open systems': physical systems like machines, where the time evolution depends on parameters that can be changed in a time-dependent way by an external agent. We can take small open systems and glue them together to form larger ones. This means that category theory is relevant: we can treat open systems as 'morphisms', and glue them together by composing or tensoring them. Here we describe some small steps toward a category-theoretic approach to control theory.
The talk was on Tuesday at midday and it lasted for an hour, followed by 15 minutes of questions. They videotaped the talk and put it here. My contact was Adrian Brown.Life's Struggle to Survive
When pondering the number of extraterrestrial civilizations, it is worth noting that even after it got started, the success of life on Earth was not a foregone conclusion. We recount some thrilling episodes from the history of our planet, some well-documented but others merely theorized: our collision with the planet Theia, the oxygen catastrophe, the snowball Earth events, the Permian-Triassic mass extinction event, the asteroid that hit Chicxulub, and more, including the global warming episode we are causing now. All of these hold lessons for what may happen on other planets.
The Mathematics of Planet Earth
The International Mathematical Union has declared 2013 to be the year of The Mathematics of Planet Earth. The global warming crisis is part of a bigger transformation in which humanity realizes that the Earth is a finite system and that our population, energy usage, and the like cannot continue to grow exponentially. If civilization survives this transformation, it will affect mathematics — and be affected by it — just as dramatically as the agricultural revolution or industrial revolution. We cannot know for sure what the effect will be, but we can already make some guesses.
Petri nets, chemistry, and quantum theory
Chemists use "chemical reaction networks" to describe random interactions between things of different types. These are essentially the same as what computer scientists call "Petri nets", or mathematicians would call "free symmetric monoidal categories". The reachability problem for a Petri net asks which collections of things can turn into which other collections of things: it is decidable but hard. More relevant to chemistry is the master equation, a differential equation describing how the probability that some collection will turn into some other collection changes with time. This turns out to have a nice description using some math from quantum field theory, but with probabilities replacing amplitudes.
The invited scholars, each from a different disciplines, were asked to make two fairly short presentations on the basis of prepared position papers. On the first day each presenter will address the question "what is climate change?" On the second day each presenter addressed the question "what should we do about it?"
Each day consisted of three sessions for presentations followed by a lengthy roundtable where the commonalities and differences between the various disciplines can be teased out, with the aid of some BSIA and MCC graduate student facilitators. Presentations, and the papers they are based on, were not circulated prior to the workshop so that there is a certain surprise element during the event to focus attention on the details of both definition and response.
My talks are here: What is Climate Change? and What To Do About Climate Change?
Learning to Live on a Finite Planet
The global warming crisis is part of a bigger transformation in which humanity realizes that the Earth is a finite system and that our population, energy usage, and the like cannot continue to grow exponentially. The transformation is inevitable. The big question is, what can we do to make it more pleasant?
Spans and the Categorified Heisenberg Algebra
Heisenberg reinvented matrices while discovering quantum mechanics, and the algebra generated by annihilation and creation operators obeying the canonical commutation relations was named after him. It turns out that matrices arise naturally from 'spans', where a span between two objects is just a third object with maps to both those two. In terms of spans, the canonical commutation relations have a simple combinatorial interpretation. More recently, Khovanov introduced a 'categorified' Heisenberg algebra, where the canonical commutation relations hold only up to isomorphism, and these isomorphisms obey new relations of their own. The meaning of these new relations was initially rather mysterious. However, Jeffrey Morton and Jamie Vicary have shown that these, too, have a nice interpretation in terms of spans. Moreover, the categorified Heisenberg algebra naturally acts on the '2-Fock space' describing collections of particles in a 4-dimensional topological quantum field theory.
Learning to Live on a Finite Planet
The global warming crisis is part of a bigger transformation in which humanity realizes that the Earth is a finite system and that our population, energy usage, and the like cannot continue to grow exponentially. The transformation is inevitable. The big question is, what can we do to make it more pleasant?
Spans and the Categorified Heisenberg Algebra
Heisenberg reinvented matrices while discovering quantum mechanics, and the algebra generated by annihilation and creation operators obeying the canonical commutation relations was named after him. It turns out that matrices arise naturally from 'spans', where a span between two objects is just a third object with maps to both those two. In terms of spans, the canonical commutation relations have a simple combinatorial interpretation. More recently, Khovanov introduced a 'categorified' Heisenberg algebra, where the canonical commutation relations hold only up to isomorphism, and these isomorphisms obey new relations of their own. The meaning of these new relations was initially rather mysterious. However, Jeffrey Morton and Jamie Vicary have shown that these, too, have a nice interpretation in terms of spans. We can begin to formalize this using the work of Alex Hoffnung and Mike Stay, who have shown that spans of groupoids are morphisms in a symmetric monoidal bicategory.
Key Developments in Category Theory
From the invention of categories in 1945 to the rise of homotopy type theory in recent years, category theory has been exceptionally rich in philosophically interesting ideas. I will try to outline some of the key developments with a minimum of technical detail.
Then I went to their Category-Theoretic Foundations of Mathematics Workshop that weekend, where I gave the following talk at Sunday May 5th at 9 am:
The Foundations of Applied Mathematics
Suppose we take "applied mathematics" in an extremely broad sense that includes math developed for use in electrical engineering, population biology, epidemiology, chemistry, and many other fields. Suppose we look for mathematical structures that repeatedly appear in these diverse contexts — especially structures that aren't familiar to pure mathematicians. What do we find? The answers may give us some clues about the concepts that underlie the most applicable kinds of mathematics. We should not be surprised to find some category theory here.
5
Different numbers have different personalities. The number 5 is quirky and intriguing, thanks in large part to its relation with the golden ratio, the "most irrational" of irrational numbers. It is impossible to tile the plane with regular pentagons, or form a crystal with perfect 5-fold symmetry... but trying leads to some beautiful things.
Milankovitch Cycles and the Earth's Climate
In the last few million years the Earth's climate has been dominated by dramatic swings in temperature between cold 'glacials' and warm 'interglacials'. The most widely accepted theory says that these are triggered by Milankovitch cycles: periodic changes in the Earth's orbital parameters. However, the details are far from fully understood, leaving many puzzles for us to study.
Energy and the Environment - What Physicists Can Do
The global warming crisis is part of a bigger transformation in which humanity realizes that the Earth is a finite system and that our population, energy usage, and the like cannot continue to grow exponentially. While politics and economics pose the biggest challenges, physicists are in a good position to help make this transition a bit easier. After a quick review of the problems, we discuss a few ways physicists can help.
I left on Saturday, I arrived at Heathrow at about 2:50 on Sunday the 24th, and then I took a train to Sheffield, arriving at a pub called Brown's at about 9:00 pm, where a bunch of people had gathered, including Eugenia Cheng, Tom Leinster, Simon Willerton and Nick Gurski.
On Monday the 25th from 2 to 2:25 pm I gave a talk on "Bicategories and Tricategories of Spans" at a satellite meeting of the British Mathematics Colloquium, 94th Peripatetic Seminar on Sheaves and Logic.
Later that day, from 18:30 to 19:30, I gave this public lecture:
The Mathematics of Planet Earth
The International Mathematical Union has declared 2013 to be the year of The Mathematics of Planet Earth. The global warming crisis is part of a bigger transformation in which humanity realizes that the Earth is a finite system and that our population, energy usage, and the like cannot continue to grow exponentially. If civilization survives this transformation, it will affect mathematics — and be affected by it — just as dramatically as the agricultural revolution or industrial revolution. We cannot know for sure what the effect will be, but we can already make some guesses.
After this a bunch of us went to an Indian restaurant called Aagrah, where Eugenia Cheng booked a large table.
On Wednesday the 27th, I spoke on a panel on open access from 11:30 to 12:45. Later that day I went to Nottingham to give a talk at 17:00:
My host there was John Barrett.Spans and the Categorified Heisenberg Algebra
Heisenberg reinvented matrices while discovering quantum mechanics, and the algebra generated by annihilation and creation operators obeying the canonical commutation relations was named after him. It turns out that matrices arise naturally from 'spans', where a span between two objects is just a third object with maps to both those two. In terms of spans, the canonical commutation relations have a simple combinatorial interpretation. More recently, Khovanov introduced a 'categorified' Heisenberg algebra, where the canonical commutation relations hold only up to isomorphism, and these isomorphisms obey new relations of their own. The meaning of these new relations was initially rather mysterious. However, Jeffery Morton and Jamie Vicary have shown that these, too, have a nice interpretation in terms of spans.
On Thursday the 28th the conference ended at 12:30, and I met David Tweed and later also Jim Stuttard and Glyn Adgie.
The Mathematics of Planet Earth
The International Mathematical Union has declared 2013 to be the year of The Mathematics of Planet Earth. The global warming crisis is part of a bigger transformation in which humanity realizes that the Earth is a finite system and that our population, energy usage, and the like cannot continue to grow exponentially. If civilization survives this transformation, it will affect mathematics — and be affected by it — just as dramatically as the agricultural revolution or industrial revolution. We cannot know for sure what the effect will be, but we can already make some guesses.
The Mathematics of Planet Earth
The International Mathematical Union has declared 2013 to be the year of The Mathematics of Planet Earth. The global warming crisis is part of a bigger transformation in which humanity realizes that the Earth is a finite system and that our population, energy usage, and the like cannot continue to grow exponentially. If civilization survives this transformation, it will affect mathematics — and be affected by it — just as dramatically as the agricultural revolution or industrial revolution. We cannot know for sure what the effect will be, but we can already make some guesses.
On July 5, I gave this talk:
Diversity, Entropy and Thermodynamics
As is well known, some popular measures of biodiversity are formally identical to measures of entropy developed by Shannon, Rényi and others. This fact is part of a larger analogy between thermodynamics and the mathematics of biodiversity, which we explore here. Any probability distribution can be extended to a 1-parameter family of probability distributions where the parameter has the physical meaning of 'temperature'. This allows us to introduce thermodynamic concepts such as energy, entropy, free energy and the partition function in any situation where a probability distribution is present . for example, the probability distribution describing the relative abundances of different species in an ecosystem. The Rényi entropy of this probability distribution is closely related to the change in free energy with temperature. We give one application of thermodynamic ideas to population dynamics, coming from the work of Marc Harper: as a population approaches an 'evolutionary optimum', the amount of Shannon information it has 'left to learn' is nonincreasing. This fact is closely related to the Second Law of Thermodynamics.
On June 14 I gave a talk at the Preuves, Programmes et Systèmes group at Université Paris 7:
Stochastic Petri Nets and Chemical Reactions
Chemists use "chemical reaction networks" to describe random interactions between things of different types. These are essentially the same as what computer scientists call "stochastic Petri nets". From such a structure, we can obtain a differential equation describing time evolution. A simple criterion implies that this equation has stationary solutions. This criterion involves a generalization of Euler characteristic to stochastic Petri nets.
G_{2} and the Rolling Ball
Understanding the exceptional Lie groups as the symmetry groups of simpler objects is a longstanding challenge. Here we describe how the smallest exceptional Lie group, G_{2}, shows up as symmetries of a simple physics problem: a ball rolling on a larger ball without slipping or twisting. G2 acts as symmetries of this problem, but only when we treat the smaller ball as a 'spinor', which returns to its orientation not after one full turn but only after two — and only when the larger ball is 3 times as big as the smaller one. We show how to understand this special ratio, describe the geometry of the rolling ball system in terms of imaginary split octonions, and show how geometric quantization applied to this system lets us recover the imaginary split octonions together with their cross product.Teleparallel Gravity as a Higher Gauge Theory
Higher gauge theory uses '2-connections' to describe parallel transport not only along curves, but also over surfaces. Just as gauge theory uses Lie groups, higher gauge theory uses Lie 2-groups. We show that general relativity can be viewed as a higher gauge theory. On any semi-Riemannian manifold M, we construct a principal 2-bundle with the the 'teleparallel 2-group' as its structure 2-group. Any flat metric-preserving connection on M gives a flat 2-connection on this 2-bundle, and the key ingredient of this 2-connection is the torsion. Taking advantage of Einstein and Cartan's formulation of general relativity in which a flat connection and its torsion are are key ingredients, this lets us rewrite general relativity as a theory with a 2-connection for the teleparallel 2-group as its only field.
Energy, the Environment and What We Can Do
Our heavy reliance on fossil fuels is causing two serious problems: global warming, and the decline of cheaply available oil reserves. Unfortunately the second problem will not cancel out the first. Each one individually seems extremely hard to solve, and taken together they demand a major worldwide effort starting now. After an overview of these problems, we turn to the question: what can we do about them?
I gave the Mathematics Colloquium at 1 pm Monday 6 February 2012, in E7B T2:
Probabilities versus Amplitudes
Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, there is a widely used and successful theory of "chemical reaction networks", which describes the interactions of molecules in a stochastic rather than quantum way. If we look at it from the perspective of quantum theory, this turns out to involve creation and annihilation operators, coherent states and other well-known ideas - but with a few big differences. The stochastic analogue of quantum field theory is also used in population biology, and here the connection is well-known. But what does it mean to treat wolves as identical bosons?
Tuesday the 7th at 1 pm in E7B T2 I gave this talk:
Energy, the Environment and What We Can Do
Our heavy reliance on fossil fuels is causing two serious problems: global warming, and the decline of cheaply available oil reserves. Unfortunately the second problem will not cancel out the first. Each one individually seems extremely hard to solve, and taken together they demand a major worldwide effort starting now. After an overview of these problems, we turn to the question: what can we do about them?
Wednesday the 8th at 2 pm in E7A 333, after lunch with the category theorists, I gave this talk in the Australian Category Seminar:
Symmetric Monoidal Categories in Chemistry and Biology
Chemists use "chemical reaction networks" to describe interactions between things of different types. In population biology and the study of infectious diseases, "stochastic Petri nets" are sometimes used for the same purpose. In fact chemical reaction networks and stochastic Petri nets are essentially the same thing. The theory of symmetric monoidal categories can help us understand what this thing is, and how to work with it.
Talks started at 10 am on Monday the 30th, finishing 5 pm on Thursday the 2nd. There was be a workshop banquet on Tuesday 31 January, and a free afternoon on Wednesday 1 February. My contact people were Gavin Brennen and Stephen Bartlett. My talk lasted 45 minutes, starting 9 am on Wednesday:
Probabilities versus Amplitudes
Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, there is a widely used and successful theory of "chemical reaction networks", which describes the interactions of molecules in a stochastic rather than quantum way. If we look at it from the perspective of quantum theory, this turns out to involve creation and annihilation operators, coherent states and other well-known ideas - but with a few big differences. The stochastic analogue of quantum field theory is also used in population biology, and here the connection is well-known. But what does it mean to treat wolves as identical bosons?
The theme for EQuaLS5 was "Geometry, Topology and Physics 2012" and the speakers were:
My talks on "Network Theory" were at 10:00 on Monday the 9th, 15:00 on Tuesday, 9:00 on Thursday and 9:00 on Friday. My public talk was at 15:00 on Friday, with the following topic:
Energy, the Environment and What We Can Do
Our heavy reliance on fossil fuels is causing two serious problems: global warming, and the decline of cheaply available oil reserves. Unfortunately the second problem will not cancel out the first. Each one individually seems extremely hard to solve, and taken together they demand a major worldwide effort starting now. After an overview of these problems, we turn to the question: what can we do about them?
Probabilities versus Amplitudes
Some ideas from quantum theory are just beginning to percolate back to classical probability theory. For example, there is a widely used and successful theory of "chemical reaction networks", which describes the interactions of molecules in a stochastic rather than quantum way. If we look at it from the perspective of quantum theory, this turns out to involve creation and annihilation operators, coherent states and other well-known ideas - but with a few big differences. The stochastic analogue of quantum field theory is also used in population biology, and here the connection is well-known. But what does it mean to treat wolves as fermions or bosons?
Lisa and I flew from Singapore to Beijing on Air China flight CA976, leaving at 9:30 on July 29 and arriving at 15:30 at Terminal 3.
We took a train to Changchun on August 2. The train ride took 6 hours and 20 minutes.
From the 3rd to 6th we went on an excursion to Baekdu Mountain, tallest of the Changbai Mountains.
On Sunday the 7th we returned to Changchun and Lisa flew back to Beijing and thence to Singapore.
August 8th-10th I gave three one-hour talks on Higher gauge theory, division algebras and superstrings.
Then I took a train or plane back to Beijing. I left Beijing from Terminal 3 on Air China flight CA969, at 15:35 on August 12.
Yunhe Sheng of the department of mathematics at Jilin University wrote:
Probably, you will arrive at Beijing first and spend several days there. We can also arrange your lodging in Beijing. If so, you are also invited to give a talk at Capital Normal University in Beijing. It is very convenient to take a train to go to Changchun from Beijing, which will take about 6 hours and 20 minutes. I will arrange students to buy train tickets for you and pick you up at Changchun Railway station.
Here's a more detailed description of the trip to the Changbai Mountains:
... we can travel 4 days, 3-6 August. On August 3rd we start after breakfast; hopefully we will arrive at Lanjing (a hotel inside the park, Landscape of English) around 3-4 pm and we can have a walk up Changbai Mountain, maybe. On August 4th we will go to see Tianchi, Pubu, Dixiasenlin, and etc. (points to see in the park). We will go to another hotel for Wenquan (hot spring) that night. On August 5th we will go to Senjiaolongwan. We do not need to start very early that day since we have enough time. We will stay around Sanjiaolongwan. Then on the morning August 6th we will come back, and arrive at Changchun before dinner time.
Operads and the Tree of LifeTrees are not just combinatorial structures: they are also biological structures, both in the obvious way but also in the study of evolution. Starting from DNA samples from living species, biologists use increasingly sophisticated mathematical techniques to reconstruct the most likely "phylogenetic tree" describing how these species evolved from earlier ones. In their work on this subject, they have encountered an interesting example of an operad, which is obtained by applying a variant of the Boardmann-Vogt "W construction" to the operad for commutative monoids. The operations in this operad are labelled trees of a certain sort, and it plays a universal role in the study of stochastic processes that involve branching. We shall explain these ideas assuming a bare minimum of prerequisites.
I gave this talk on the 17th:
Higher gauge theory, division algebras and superstringsClassically, superstrings make sense when spacetime has dimension 3, 4, 6, or 10. It is no coincidence that these numbers are two more than 1, 2, 4, and 8, which are the dimensions of the normed division algebras: the real numbers, complex numbers, quaternions and octonions. We sketch an explanation of this already known fact and its relation to "higher gauge theory". Just as gauge theory describes the parallel transport of supersymmetric particles using Lie supergroups, higher gauge theory describes the parallel transport of superstrings using "Lie 2-supergroups". Recently John Huerta has shown that we can use normed division algebras to construct a Lie 2-supergroup extending the Poincaré supergroup when spacetime has dimension 3, 4, 6 and 10.
On Sunday May 22nd I will leave Riverside and visit Christopher Lee in Los Angeles. I returned to Singapore on Tuesday May 24, Lisa was in Erlangen May 18 - 27, 2011.
Energy, the environment, and what mathematicians can doOn Thursday Jiang-Hua Lu and I went to the Chinese University of Hong Kong to talk to Conan Leung, and at 11 am I gave a talk at the Institute of Mathematical Sciences on the number workshop on geometry and Lie groups, where Peter Bouwknegdt, Varghese Mathai, and David Vogan also spoke:Our heavy reliance on fossil fuels is causing two serious problems: global warming, and the decline of cheaply available oil reserves. Unfortunately the second problem will not cancel out the first. Each one individually seems extremely hard to solve, and taken together they demand a major worldwide effort starting now. After an overview of these problems, we turn to the question: what can mathematicians do to help?
Higher gauge theory, division algebras and superstringsClassically, superstrings make sense when spacetime has dimension 3, 4, 6, or 10. It is no coincidence that these numbers are two more than 1, 2, 4, and 8, which are the dimensions of the normed division algebras: the real numbers, complex numbers, quaternions and octonions. We sketch an explanation of this already known fact and its relation to "higher gauge theory". Just as gauge theory describes the parallel transport of supersymmetric particles using Lie supergroups, higher gauge theory describes the parallel transport of superstrings using "Lie 2-supergroups". Recently John Huerta has shown that we can use normed division algebras to construct a Lie 2-supergroup extending the Poincaré supergroup when spacetime has dimension 3, 4, 6 and 10.
Electrical CircuitsWhile category theory has many sophisticated applications to theoretical physics — especially quantum fields and strings — it also has interesting applications to a seemingly more pedestrian topic: electrical circuits. The pictorial resemblance between circuit diagrams and Feynman diagrams is an obvious clue, but what is the underlying mathematics? This question quickly leads us to an interesting combination of category theory, symplectic geometry, complex analysis and graph theory. Moreover, electrical circuits are just one example of 'open systems': physical systems that that interact with their environment. While textbooks on classical mechanics usually focus on closed systems, open systems are more important in engineering, and their mathematics is arguably deeper and more interesting.
Duality in Logic and PhysicsDuality has many manifestations in logic and physics. In classical logic, propositions form a partially ordered set and negation is an order-reversing involution which switches "true" and "false". The same holds in quantum logic, with propositions corresponding to closed subspaces of a Hilbert space. But the full structure of quantum physics involves more: at the very least, the category of Hilbert spaces and bounded linear operators. This category has another kind of duality, a contravariant involution that switches "preparation" and "observation". Other closely related dualities in quantum physics include "charge conjugation" (switching matter and antimatter), "parity" (switching left and right), and "time reversal" (switching future and past). The quest to find a unified mathematical framework for dualities in logic and physics leads to a fascinating variety of structures: star-autonomous categories, n-categories with duals, and more. We give a tour of these, with an effort to focus on conceptual rather than technical issues.
5Then I gave my colloquium talk at 3:Different numbers have different personalities. The number 5 is quirky and intriguing, thanks in large part to its relation with the golden ratio, the "most irrational" of irrational numbers. The plane cannot be tiled with regular pentagons, but there exist quasiperiodic planar patterns with pentagonal symmetry of a statistical nature, first discovered by Islamic artists in the 1600s, later rediscovered by the mathematician Roger Penrose in the 1970s, and found in nature in 1984.
The Greek fascination with the golden ratio is probably tied to the dodecahedron. Much later, the symmetry group of the dodecahedron was found to give rise to a 4-dimensional regular polytope, the 120-cell, which in turn gives rise to the Poincaré homology sphere and the root system of the exceptional Lie group E_{8}. So, a wealth of exceptional objects arise from the quirky nature of 5-fold symmetry.
Physics, Topology, Logic and Computation: a Rosetta StoneIn particle physics, Feynman diagrams are used to reason about quantum processes. Similar diagrams can be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of interest in quantum cryptography and quantum computation, the explanation became clear: there is extensive network of analogies between physics, topology, logic and computation. In this introductory talk I, will make some of these analogies precise using a wonderfully general branch of mathematics called category theory.
8The number 8 plays a special role in mathematics due to the "octonions", an 8-dimensional number system where one can add, multiply, subtract and divide, but where the commutative and associative laws for multiplication — ab = ba and (ab)c = a(bc) — fail to hold. The octonions were discovered by Hamilton's friend John Graves in 1843 after Hamilton told him about the "quaternions". While much neglected, they stand at the crossroads of many interesting branches of mathematics and physics. For example, superstring theory works in 10 dimensions because 10 = 8+2: the 2-dimensional worldsheet of a string has 8 extra dimensions in which to wiggle around, and the theory crucially uses the fact that these 8 dimensions can be identified with the octonions. Or: the densest known packing of spheres in 8 dimensions arises when the spheres are centered at certain "integer octonions", which form the root lattice of the exceptional Lie group E_{8}. The octonions also explain the curious way in which topology in dimension n resembles topology in dimension n+8.
Who Discovered the Icosahedron?It has been suggested that the regular icosahedron, not being found in nature, is the first example of a geometrical object that is the free creation of human thought. Regardless of the truth of this, it is interesting to try to track down the origin of the icosahedron. A scholium in Book XIII of Euclid's "Elements" speaks of "the five so-called Platonic figures which, however, do not belong to Plato, three of the five being due to the Pythagoreans, namely the cube, the pyramid, and the dodecahedron, while the octahedron and the icosahedron are due to Theaetetus." More recently, Atiyah and Sutcliffe have claimed that a regular icosahedron appears among a collection of stone balls in the Ashmolean Museum - balls that were unearthed in Scotland and may date back to 2000 BC. However, Lieven le Bruyn has argued that these authors are the victims of a hoax. We examine the evidence with a critical eye.
Categorification in Fundamental PhysicsLisa and I left on September 11th and flew back on the 22nd.Categorification is the process of replacing set-based mathematics with analogous mathematics based on categories or n-categories. In physics, categorification enters naturally as we pass from the mechanics of particles to higher-dimensional field theories. For example, higher gauge theory is a generalization of gauge theory that describes the parallel transport not just of particles, but also strings or higher-dimensional branes. To handle strings, we must categorify familiar notions from gauge theory and consider connections on "principal 2-bundles" with a given "structure 2-group". One of the simplest 2-groups is the shifted version of U(1). U(1) gerbes are really principal 2-bundles with this structure 2-group, and the B field in string theory can be seen as a connection on this sort of 2-bundle. The relation between U(1) bundles and symplectic manifolds, so important in the geometric quantization, extends to a relation between U(1) gerbes and "2-plectic manifolds", which arise naturally as phase spaces for 2-dimensional field theories, such as the theory of a classical string. More interesting 2-groups include the "string 2-group" associated to a compact simple Lie group G. This is built using the central extension of the loop group of G. A closely related 3-group plays an important role in Chern-Simons theory, and it appears that n-groups for higher n are important in the study of higher-dimensional membranes.
- Connections on abelian gerbes
- Lie n-groups and Lie n-algebras
- Multisymplectic geometry and classical field theory
- Higher gauge theory and the string 2-group
- Higher gauge theory, strings and branes
I took a flight from Ontario Airport to Dallas-Forth Worth on Monday September 7th, leaving at 3:20 pm and arriving at 8:10 pm. I came back on Thursday the 10th at 9:15 pm.
At 1 pm on Tuesday the 8th I spoke about the number 5.
At 7 pm on Tuesday the 8th I spoke about Zooming Out in Time.
At 1 pm on Wednesday the 9th I spoke about the number 8.
At 4 pm on Wednesday the 9th I spoke about Fundamental Physics: Where We Stand Today.
At 1 pm on Thursday the 10th I spoke about the number 24.
Computation and the Periodic TableOn Wednesday August 12th I drove into Los Angeles and spent a night in a dorm in Sunset Village, in the northwest part of the campus. I spent another night there on the 13th, then went home.In physics, Feynman diagrams are used to reason about quantum processes. Similar diagrams can also be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of topological quantum field theory and quantum computation, it became clear that diagrammatic reasoning takes advantage of an extensive network of interlocking analogies between physics, topology, logic and computation. These analogies can be made precise using the formalism of symmetric monoidal closed categories. But symmetric monoidal categories are just the n = 1, k = 3 entry of a hypothesized "periodic table" of k-tuply monoidal n-categories. This raises the question of how these analogies extend. An important clue comes from the way symmetric monoidal closed 2-categories describe rewrite rules in the lambda calculus and multiplicative intuitionistic linear logic. This talk is based on work in progress with Paul-André Melliès and Mike Stay.
Why Smooth Spaces?The category of smooth manifolds and smooth maps is often taken as the default context for research on differential geometry. However, in many applications it is convenient or even necessary to work in a more general framework. In this introductory talk we explain various reasons for this. We also describe the advantages of various alternative frameworks, including Banach manifolds and other types of infinite-dimensional manifolds, Chen spaces and diffeological spaces, orbifolds and differentiable stacks, and synthetic differential geometry.
Categorification and TopologyThe relation between n-categories and topology is clarified by a collection of hypotheses, some of which have already been made precise and proved. The "homotopy hypothesis" says that homotopy n-types are the same as n-groupoids. The "stabilization hypothesis" says that each column in the periodic table of n-categories stabilizes at a certain precise point. The "cobordism hypothesis" gives an n-categorical description of cobordisms, while the "tangle hypothesis" does the same for tangles and their higher-dimensional relatives. We shall sketch these ideas, describe recent work by Lurie and Hopkins on the cobordism and tangle hypotheses, and, time permitting, say a bit about how these ideas are related to other lines of work on categorification.
Lectures on Higher Gauge TheoryGauge theory describes the parallel transport of point particles using the formalism of connections on bundles. In both string theory and loop quantum gravity, point particles are replaced by 1-dimensional extended objects: paths or loops in space. This suggests that we seek some sort of "higher gauge theory" that describes parallel transport as we move a path through space, tracing out a surface. To find the right mathematical language for this, we must "categorify" concepts from topology and geometry, replacing smooth manifolds by smooth categories, Lie groups by Lie 2-groups, Lie algebras by Lie 2-algebras, bundles by 2-bundles, sheaves by stacks or gerbes, and so on. This overview of higher gauge theory will emphasize its relation to homotopy theory and the cohomology of groups and Lie algebras.
I gave a talk at the special session on Homotopy Theory and Higher Categories run by Tom Fiore, Mark Johnson, Jim Turner, Steve Wilson and Donald Yau. This was on Wednesday January 7th, 1-1:20 pm in Virginia Suite C, Lobby Level, Marriott:
Classifying Spaces for Topological 2-GroupsCategorifying the concept of topological group, one obtains the notion of a topological 2-group. This in turn allows a theory of "principal 2-bundles" generalizing the usual theory of principal bundles. It is well-known that under mild conditions on a topological group G and a space M, principal G-bundles over M are classified by either the Cech cohomology H^{1}(M,G) or the set of homotopy classes [M,BG], where BG is the classifying space of G. Here we review work by Bartels, Jurco, Baas-Bökstedt-Kro, Stevenson and myself generalizing this result to topological 2-groups. We explain various viewpoints on topological 2-groups and the Cech cohomology H^{1}(M,G) with coefficients in a topological 2-group G, also known as "nonabelian cohomology". Then we sketch a proof that under mild conditions on M and G there is a bijection between H^{1}(M,G) and [M,B|NG|], where B|NG| is the classifying space of the geometric realization of the nerve of G.
I also gave a talk at the special session on Categorification and Link Homology, run by Aaron Lauda and Mikhail Khovanov. This was on Wednesday January 7th, 5:10 - 5:30 pm, in the Harding Room, Mezzanine Level, Marriott:
GroupoidificationThere is a systematic process that turns groupoids into vector spaces and spans of groupoids into linear operators. "Groupoidification" is the attempt to reverse this process, taking familiar structures from linear algebra and enhancing them to obtain structures involving groupoids. Like quantization, groupoidification is not entirely systematic. However, examples show that it is a good thing to try! For example, groupoidifying the quantum harmonic oscillator yields combinatorial structures associated to the groupoid of finite sets, while groupoidifying the q-deformed oscillator yields structures associated to finite-dimensional vector spaces over the field with q elements. Starting with flag varieties defined over the field with q elements, we can also groupoidify Hecke and Hall algebras.
GroupoidificationThere is a systematic process that turns groupoids into vector spaces and spans of groupoids into linear operators. "Groupoidification" is the attempt to reverse this process, taking familiar structures from linear algebra and enhancing them to obtain structures involving groupoids. Like quantization, groupoidification is not entirely systematic. However, examples show that it is a good thing to try! For example, groupoidifying the quantum harmonic oscillator yields combinatorial structures associated to the groupoid of finite sets. We can also groupoidify mathematics related to quantum groups - for example, Hecke algebras and Hall algebras. It turns out that we obtain structures related to algebraic groups defined over finite fields. After reviewing the basic idea of groupoidification, we shall describe as many examples as time permits.
8The second was some sort of math/physics seminar:Different numbers have different personalities, and 8 is one of my favorites. The number 8 plays a special role in mathematics due to the "octonions", an 8-dimensional number system where one can add, multiply, subtract and divide, but multiplication is noncommutative and nonassociative. The octonions were discovered by Hamilton's friend John Graves in 1843 after Hamilton told him about the "quaternions". While much neglected, they stand at the crossroads of many interesting branches of mathematics and physics. For example, superstring theory works in 10 dimensions because 10 = 8+2, where 8 is the dimension of the octonions. Also, the densest known packing of spheres in 8 dimensions occurs when the spheres are centered at certain "integer octonions", which form the root lattice of the exceptional Lie group E8. The octonions also explain the curious way in which topology in dimension n resembles topology in dimension n+8.
Higher Gauge Theory and the String GroupHigher gauge theory is a generalization of gauge theory that describes the parallel transport not just of particles, but also strings or higher-dimensional branes. To handle strings, one can categorify familiar notions from gauge theory and consider "principal 2-bundles" with a given "structure 2-group". These are a slight generalization of nonabelian gerbes. We focus on examples related to the 2-group String_{k}(G) associated to any compact simple Lie group G. We describe how this 2-group is built using the level-k central extension of the loop group of G, and how it is related to the "string group". Finally, we discuss 2-bundles with String_{k}(G) as structure 2-group, and characteristic classes for these 2-bundles.
The overall title will be My Favorite Numbers and
individual titles "5", "8", and "24". These talks will be written up and
published with some help from the Trust.
5 (Monday September 15th, 4 pm)Different numbers have different personalities. The number 5 is quirky and intriguing, thanks in large part to its relation with the golden ratio, the "most irrational" of irrational numbers. The plane cannot be tiled with regular pentagons, but there exist quasiperiodic planar patterns with pentagonal symmetry of a statistical nature, first discovered by Islamic artists in the 1600s, later rediscovered by the mathematician Roger Penrose in the 1970s, and found in nature in 1984. The Greek fascination with the golden ratio is probably tied to the dodecahedron. Much later, the symmetry group of the dodecahedron was found to give rise to a 4-dimensional regular polytope, the 120-cell, which in turn gives rise to the Poincaré homology sphere and the root system of the exceptional Lie group E_{8}. So, a wealth of exceptional objects arise from the quirky nature of 5-fold symmetry.
8 (Wednesday September 17th, 4 pm)The number 8 plays a special role in mathematics due to the "octonions", an 8-dimensional number system where one can add, multiply, subtract and divide, but where the commutative and associative laws for multiplication — ab = ba and (ab)c = a(bc) — fail to hold. The octonions were discovered by Hamilton's friend John Graves in 1843 after Hamilton told him about the "quaternions". While much neglected, they stand at the crossroads of many interesting branches of mathematics and physics. For example, superstring theory works in 10 dimensions because 10 = 8+2: the 2-dimensional worldsheet of a string has 8 extra dimensions in which to wiggle around, and the theory crucially uses the fact that these 8 dimensions can be identified with the octonions. Or: the densest known packing of spheres in 8 dimensions arises when the spheres are centered at certain "integer octonions", which form the root lattice of the exceptional Lie group E_{8}. The octonions also explain the curious way in which topology in dimension n resembles topology in dimension n+8.
24 (Friday September 19th, 4 pm)The numbers 12 and 24 play a central role in mathematics thanks to a series of "coincidences" that is just beginning to be understood. One of the first hints of this fact was Euler's bizarre "proof" that
1 + 2 + 3 + 4 + ... = -1/12which he obtained before Abel declared that "divergent series are the invention of the devil". Euler's formula can now be understood rigorously in terms of the Riemann zeta function, and in physics it explains why bosonic strings work best in 26=24+2 dimensions. The fact that
1^{2} + 2^{2} + 3^{2} + ... + 24^{2}is a perfect square then sets up a curious link between string theory, the Leech lattice (the densest known way of packing spheres in 24 dimensions) and a group called the Monster. A better-known but closely related fact is the period-12 phenomenon in the theory of "modular forms". We shall do our best to demystify some of these deep mysteries.
Computation and the Periodic TableBy now there is an extensive network of interlocking analogies between physics, topology, logic and computer science, which can be seen most easily by comparing the roles that symmetric monoidal closed categories play in each subject. However, symmetric monoidal categories are just the n = 1, k = 3 entry of a hypothesized "periodic table" of k-tuply monoidal n-categories. This raises the question of how these analogies extend. We present some thoughts on this question, focusing on how symmetric monoidal closed 2-categories might let us understand the lambda calculus more deeply.
I stayed at the SERHS Campus Hotel on the campus of the Universitat Autonoma de Barcelona (UAB) campus in Bellaterra, 18 km north of Barcelona. This campus is where the CRM is located.
I gave the first talk, at 9:30 am on Monday June 30th:
GroupoidificationThere is a systematic process that turns groupoids into vector spaces and spans of groupoids into linear operators. "Groupoidification" is the attempt to reverse this process, taking familiar structures from linear algebra and enhancing them to obtain structures involving groupoids. Like quantization, groupoidification is not entirely systematic. However, examples show that it is a good thing to try! For example, groupoidifying the quantum harmonic oscillator yields combinatorial structures associated to the groupoid of finite sets. Groupoidifying the q-deformed oscillator yields combinatorial structures associated to finite-dimensional vector spaces over the field with q elements. We can also groupoidify some mathematics related to quantum groups and representations of finite groups. We first describe the basic ideas, and then as many examples as time permits.
An overall map is here. I stayed at the residence Carmen de la Victoria, 9 Cuesta del Chapiz, from June 21st to 29th. On Sunday the 29th my flight left for Barcelona at 8:45 am.
Here's an overall map.
I stayed at the Residència d'Investigadors, Carrer de l'Hospital, 64 08001 Barcelona.
The IMUB is here, inside the Facultat de Mathemàtiques in the historic building of the Universitàt de Barcelona, at Gran Via 585. The morning lectures took place in Aula B6 (that's in the north-most corner of the building), while the afternoon lectures took place in the "Aula Aribau" (that's in the south-most corner, in fact in the modern building just next to the historical building).
I gave a talk on Monday June 16th, starting at 10 am: first 1 hour of background, then 45 minutes of actual talk, then 45 minutes for discussion.
Topological 2-Groups and Their Classifying SpacesCategorifying the concept of topological group, one obtains the notion of a topological 2-group. This in turn allows a theory of "principal 2-bundles" generalizing the usual theory of principal bundles. It is well-known that under mild conditions on a topological group G and a space M, principal G-bundles over M are classified by either the Cech cohomology H^{1}(M,G) or the set of homotopy classes [M,BG], where BG is the classifying space of G. Here we review work by Bartels, Jurco, Baas-Bökstedt-Kro, and others generalizing this result to topological 2-groups. We explain various viewpoints on topological 2-groups and the Cech cohomology H^{1}(M,G) with coefficients in a topological 2-group G, also known as "nonabelian cohomology". Then we sketch a proof that under mild conditions on M and G there is a bijection between H^{1}(M,G) and [M,B|G|], where B|G| is the classifying space of the geometric realization of the nerve of G.
GroupoidificationThere is a systematic process that turns groupoids into vector spaces and spans of groupoids into linear operators. "Groupoidification" is the attempt to reverse this process, taking familiar structures from linear algebra and enhancing them to obtain structures involving groupoids. Like quantization, groupoidification is not entirely systematic. However, examples show that it is a good thing to try! For example, groupoidifying the quantum harmonic oscillator yields combinatorial structures associated to the groupoid of finite sets. Groupoidifying the q-deformed oscillator yields combinatorial structures associated to finite-dimensional vector spaces over the field with q elements. We can also groupoidify some mathematics related to quantum groups and representations of finite groups. We first describe the basic ideas, and then as many examples as time permits.
5Different numbers have different personalities. The number 5 is quirky and intriguing, thanks in large part to its relation with the golden ratio, the "most irrational" of irrational numbers. The plane cannot be tiled with regular pentagons, but there exist quasiperiodic planar patterns with pentagonal symmetry of a statistical nature, first discovered by Islamic artists in the 1600s, later rediscovered by the mathematician Roger Penrose in the 1970s, and found in nature in 1984. The Greek fascination with the golden ratio is probably tied to the dodecahedron. Much later, the symmetry group of the dodecahedron was found to give rise to a 4-dimensional regular polytope, the 120-cell, which in turn gives rise to the Poincaré homology sphere and the root system of the exceptional Lie group E_{8}. So, a wealth of exceptional objects arise from the quirky nature of 5-fold symmetry.
Higher Gauge Theory
Higher gauge theory is a generalization of gauge theory that describes the parallel transport not just of particles, but also strings or higher-dimensional branes. To handle strings, we generalize familiar notions from gauge theory and consider connections on "2-bundles" with a given "structure 2-group". After an introduction to these ideas, we discuss one of the most interesting 2-groups: the "string 2-group" associated to any compact simple Lie group G. This 2-group is built using a central extension of the loop group of G. We describe the theory of characteristic classes for 2-bundles with this structure 2-group.
How can we detect and understand oncoming crises in time to avert them? Sometimes we must "zoom out": expand our perspective and find similar situations in the distant past. A good example is climate change. What can a few degrees of warming do? To answer this, we need to know some history: how the Earth's climate has changed over the last 65 million years.
Since the discovery of the W and Z particles over twenty years ago, few truly novel predictions of fundamental theoretical physics have been confirmed by experiment. On the other hand, observations in astronomy have revealed shocking facts that our theories do not really explain: most of our universe consists of "dark matter" and "dark energy". Where does fundamental physics stand today, and why has theory become divorced from experiment?
I went on Wednesday October 31st and came back on Monday November 5th after spending a weekend in Great Falls visiting my parents.
Spans in Quantum Theory
Many features of quantum theory — quantum teleportation, violations of Bell's inequality, the no-cloning theorem and so on — become less puzzling when we realize that quantum processes more closely resemble pieces of spacetime than functions between sets. In the language of category theory, the reason is that Set is a "cartesian" category, while the category of finite-dimensional Hilbert spaces, like a category of cobordisms describing pieces of spacetime, is "dagger compact". Here we discuss a possible explanation for this curious fact. We recall the concept of a "span", and show how categories of spans are a generalization of Heisenberg's matrix mechanics. We explain how the category of Hilbert spaces and linear operators resembles a category of spans, and how cobordisms can also be seen as spans. Finally, we sketch a proof that whenever C is a cartesian category with pullbacks, the category of spans in C is dagger compact.
My talk was on Wednesday the 8th at 11 am:
Higher Gauge Theory and Elliptic Cohomology
The concept of elliptic object suggests a relation between elliptic cohomology and "higher gauge theory", a generalization of gauge theory describing the parallel transport of strings. In higher gauge theory, we categorify familiar notions from gauge theory and consider "principal 2-bundles" with a given "structure 2-group". These are a slight generalization of nonabelian gerbes. After a quick introduction to these ideas, we focus on the 2-groups String_{k}(G) associated to any compact simple Lie group G. We describe how these 2-groups are built using central extensions of the loop group ΩG, and how the classifying space for String_{k}(G)-2-bundles is related to the "string group" familiar in elliptic cohomology. If there is time, we shall also describe a vector 2-bundle canonically associated to any principal 2-bundle, and how this relates to the von Neumann algebra construction of Stolz and Teichner.
I gave my talk at 9 am on Wednesday August 22nd:
Higher Gauge Theory and the String Group
Higher gauge theory is a generalization of gauge theory that describes the parallel transport not just of particles, but also strings or higher-dimensional branes. To handle strings, we categorify familiar notions from gauge theory and consider "principal 2-bundles" with a given "structure 2-group". These are a slight generalization of nonabelian gerbes. We focus on examples related to the 2-group String_{k}(G) associated to any compact simple Lie group G. We describe how this 2-group is built using the level-k central extension of the loop group of G, and how it is related to the "string group". Finally, we discuss 2-bundles with String_{k}(G) as structure 2-group, and pose the problem of computing characteristic classes for such 2-bundles in terms of connections.
2-Hilbert Spaces
In work inspired by topological quantum field theory, Kapranov and Voevodsky found it useful to "categorify" the concept of a finite-dimensional vector space and invent the concept of "2-vector space". In simple terms, this amounts to replacing vectors - viewed as lists of numbers - by "2-vectors", which are lists of vector spaces. For purposes of analysis it is important to go further and find a good notion of "2-Hilbert space". For example, just as the category of finite-dimensional representations of a finite group is a 2-vector space, the category of unitary representations of a Lie group should be a 2-Hilbert space. We sketch some attempts to define 2-Hilbert spaces using measurable fields of Hilbert spaces.
Higher Gauge Theory and the String Group
Higher gauge theory is a generalization of gauge theory that describes the parallel transport not just of particles, but also strings or higher-dimensional branes. To handle strings, we categorify familiar notions from gauge theory and consider "principal 2-bundles" with a given "structure 2-group". These are a slight generalization of nonabelian gerbes. We focus on examples related to the 2-group String_{k}(G) associated to any compact simple Lie group G. We describe how this 2-group is built using the level-k central extension of the loop group of G, and how it is related to the "string group". Finally, we discuss 2-bundles with String_{k}(G) as structure 2-group, and pose the problem of computing characteristic classes for such 2-bundles in terms of connections.
Why Mathematics is Boring
Storytellers have many strategies for luring in their audience and keeping them interested. These include standardized narrative structures, vivid characters, breaking down long stories into episodes, and subtle methods of reminding the readers of facts they may have forgotten. The typical style of writing mathematics systematically avoids these strategies, since the explicit goal is "proving a fact" rather than "telling a story". Readers are left to provide their own narrative framework, which they do privately, in conversations, or in colloquium talks. As a result, even expert mathematicians find papers - especially those outside their own field - boring and difficult to understand. This impedes the development of mathematics. In my attempts at mathematics exposition I have tried to tackle this problem by using some strategies from storytelling, which I illustrate here.
Quantum Quandaries: a Category-Theoretic PerspectiveCategory theory is a general language for describing things and processes - called "objects" and "morphisms". In this language, the counterintuitive features of quantum theory turn out to be properties that the category of Hilbert spaces shares with the category of cobordisms - in which objects are choices of "space", and morphisms are choices of "spacetime". In particular, both these categories - but not the category of sets and functions - are noncartesian monoidal categories with duals. We show how this accounts for many of the famously puzzling features of quantum theory: the failure of local realism, the impossibility of duplicating quantum information, and so on. We argue that these features only seem puzzling when we try to treat the category of Hilbert spaces as analogous to the category of sets rather than the category of cobordisms, so that quantum theory will make more sense when regarded as part of a theory of spacetime. To find such a theory, it may be helpful to study categories of "spans" and "cospans".
The actual conference was from Tuesday April 24nd to Friday April 27th, but I flew there on Sunday April 22nd, arriving at Nice airport on Monday April 23rd. I left the conference on the morning of Saturday April 28th, spent a day in Nice visiting Eugenia Cheng, and departed from Nice airport on Sunday morning.
I arrived January 6th and left January 15th. On Wednesday January 10th I gave this talk:
The Homotopy Hypothesis
Crudely speaking, the Homotopy Hypothesis says that n-groupoids are the same as homotopy n-types - nice spaces whose homotopy groups above the nth vanish for every basepoint. We summarize the evidence for this hypothesis. Naively, one might imagine this hypothesis allows us to reduce the problem of computing homotopy groups to a purely algebraic problem. While true in principle, in practice information flows the other way: established techniques of homotopy theory can be used to study coherence laws for n-groupoids, and a bit more speculatively, n-categories in general.
Higher Gauge Theory
Gauge theory describes the parallel transport of point particles using the formalism of connections on bundles. In both string theory and loop quantum gravity, point particles are replaced by 1-dimensional extended objects: paths or loops in space. This suggests that we seek some sort of "higher gauge theory" that describes parallel transport as we move a path through space, tracing out a surface. To find the right mathematical language for this, we must "categorify" concepts from topology and geometry, replacing Lie groups by Lie 2-groups, bundles by 2-bundles, and so on. Some interesting examples of these concepts show up in the mathematics of topological quantum field theory, string theory and 11-dimensional supergravity.
Higher Gauge Theory
Gauge theory describes the parallel transport of point particles using the formalism of connections on bundles. In both string theory and loop quantum gravity, point particles are replaced by 1-dimensional extended objects: paths or loops in space. This suggests that we seek some sort of "higher gauge theory" that describes parallel transport as we move a path through space, tracing out a surface. To find the right mathematical language for this, we must "categorify" concepts from topology and geometry, replacing Lie groups by Lie 2-groups, bundles by 2-bundles, and so on. Some interesting examples of these concepts show up in the mathematics of topological quantum field theory, string theory and 11-dimensional supergravity.
Tales of the Dodecahedron: from Pythagoras through Plato to Poincaré
The dodecahedron is a beautiful shape made of 12 regular pentagons. It doesn't occur in nature; it was invented by the Pythagoreans, and we first read of it in a text written by Plato. We shall see some of its many amazing properties: its relation to the Golden Ratio, its rotational symmetries - and best of all, how to use it to create a regular solid in 4 dimensions! Poincaré exploited this to invent a 3-dimensional space that disproved a conjecture he made. This led him to an improved version of his conjecture, which was recently proved by the reclusive Russian mathematician Grigori Perelman - who now stands to win a million dollars.
Zooming Out in Time
How can we detect and understand oncoming crises in time to avert them? Sometimes we must "zoom out": expand our perspective and find similar situations in the distant past. A good example is climate change. What can a few degrees of warming do? To answer this, we need to know some history: how the Earth's climate has changed over the last 65 million years.
Higher-Dimensional Algebra: A Language for Quantum SpacetimeCategory theory is a general language for describing things and processes - called "objects" and "morphisms". In this language, the counterintuitive features of quantum theory turn out to be properties that the category of Hilbert spaces shares with the category of cobordisms - in which objects are choices of "space", and morphisms are choices of "spacetime". The striking similarities between these categories suggests that "n-categories with duals" are a promising framework for a quantum theory of spacetime. We sketch the historical development of these ideas from Feynman diagrams, to string theory, topological quantum field theory, spin networks and spin foams, and especially recent work on open-closed string theory, quantum gravity coupled to point particles, and 4d BF theory coupled to strings.
Fundamental Physics: Where We Stand TodaySince the discovery of the W and Z particles over twenty years ago, no really novel prediction of fundamental theoretical physics has been confirmed by experiment, except perhaps Guth's inflationary cosmology. On the other hand, observations in astronomy have revealed shocking new facts which our theories do not really explain: most of our universe consists of "dark matter" and "dark energy". Where does fundamental physics stand today, and why has theory become divorced from experiment?
The work of Eilenberg and Mac Lane marks the beginning of a trend in which mathematics based on sets is generalized to mathematics based on categories and then higher categories. We illustrate this trend towards "categorification" by a detailed introduction to "higher gauge theory".Gauge theory describes the parallel transport of point particles using the formalism of connections on bundles. In both string theory and loop quantum gravity, point particles are replaced by 1-dimensional extended objects: paths or loops in space. This suggests that we seek some kind of "higher gauge theory" that describes the parallel transport as we move a path through space, tracing out a surface. Surprisingly, this requires that we "categorify" concepts from differential geometry, replacing smooth manifolds by smooth categories, Lie groups by Lie 2-groups, Lie algebras by Lie 2-algebras, bundles by 2-bundles, sheaves by stacks or gerbes, and so on.
To explain how higher gauge theory fits into mathematics as a whole, we start with a lecture reviewing the basic principle of Galois theory and its relation to Klein's Erlangen program, covering spaces and the fundamental group, Eilenberg-Mac Lane spaces, and Grothendieck's ideas on fibrations.
The second lecture treats connections on trivial bundles and 2-connections on trivial 2-bundles, explaining how they can be described either in terms of their holonomies or in terms of Lie-algebra-valued differential forms. For a clean treatment of these concepts, we recall Chen's theory of "smooth spaces", which generalize smooth finite-dimensional manifolds.
The third lecture explains connections on general bundles and 2-connections on general 2-bundles, explaining how they can be described either in terms of holonomies or local data involving differential forms. We also explain how 2-bundles are described using nonabelian Cech 2-cocycles, and how the theory of 2-connections relates to Breen and Messing's theory of "connections on nonabelian gerbes".
Michael Shulman took notes on these talks and wrote an extensive appendix on topos theory.
On May 2nd there was an excursion to the Smoky Mountains. I flew there on April 28th and flew back on May 3rd.Higher Gauge Theory
Gauge theory describes the parallel transport of point particles using the formalism of connections on bundles. In both string theory and loop quantum gravity, point particles are replaced by 1-dimensional extended objects: paths or loops in space. This suggests that we seek some sort of "higher gauge theory" that describes parallel transport as we move a path through space, tracing out a surface. To find the right mathematical language for this, we must "categorify" concepts from topology and geometry, replacing smooth manifolds by smooth categories, Lie groups by Lie 2-groups, Lie algebras by Lie 2-algebras, bundles by 2-bundles, sheaves by stacks or gerbes, and so on. We give an overview of higher gauge theory, with an emphasis on its relation to homotopy theory and the cohomology of groups and Lie algebras.
Loop Quantum Gravity
One of the great challenges facing physics today is to reconcile quantum theory and general relativity. Loop quantum gravity is an approach to this challenge that incorporates quantum theory into our description of spacetime from the very start. Quantum states of the geometry of space are described by "spin networks" - graphs with certain labellings of their edges and vertices. The theory predicts that geometrical quantities such as area and volume take on a discrete spectrum of possible values, and it explains the entropy of black holes by associating information to each point at which a spin network edge punctures the event horizon. This talk will be a nontechnical overview of the basic ideas behind loop quantum gravity.
Universal Algebra and Diagrammatic Reasoning
Since the introduction of category theory, the old subject of "universal algebra" has diversified into a large collection of frameworks for describing algebraic structures. These include "monads" (formerly known as "triples"), the "algebraic theories" of Lawvere, and the "PROPs" of Adams, Mac Lane, Boardman and Vogt. We give an overview of these different frameworks, which are closely related, and explain how one can reason diagrammatically about algebraic structures defined using them. Our treatment of monads focuses on the abstract "bar construction". Our treatment of algebraic theories and PROPs explains how the latter are related to Feynman diagrams, and leads up to an adjunction between algebraic theories and PROPs which is analogous to the relation between classical and quantum physics. We conclude with some reflections on how features of our physical universe have influenced our notions of universal algebra.
On Monday February 27th I gave a talk for a large scientific audience at the Faculté des Sciences de Luminy:
Fundamental Physics: Where We Stand TodaySince the discovery of the W and Z particles over twenty years ago, no really novel prediction of fundamental theoretical physics has been confirmed by experiment, except perhaps Guth's inflationary cosmology. On the other hand, observations in astronomy have revealed shocking new facts which our theories do not really explain: most of our universe consists of "dark matter" and "dark energy". Where does fundamental physics stand today, and why has theory become divorced from experiment?
During this time I also visited Carlo Rovelli and Alejandro Perez at the Centre de Physique Theorique de Luminy. Kirill Krasnov arrived on Sunday, February 26th around 1 pm, and left on Wednesday, March 1st early in the morning. We talked about his ideas on "2-Feynman diagrams" and their relation to my n-categorial approach to spin foams.
Higher Gauge Theory: 2-ConnectionsGauge theory describes the parallel transport of point particles using the formalism of connections on bundles. In both string theory and loop quantum gravity, point particles are replaced by 1-dimensional extended objects: paths or loops in space. This suggests that we seek some kind of "higher gauge theory" that describes the parallel transport as we move a path through space, tracing out a surface. To find the right mathematical language for this, it seems we must "categorify" concepts from differential geometry, replacing smooth manifolds by smooth categories, Lie groups by Lie 2-groups, Lie algebras by Lie 2-algebras, bundles by 2-bundles, sheaves by stacks or gerbes, and so on. We give an overview of higher gauge theory, with an emphasis on the concept of "2-connection" for a principal 2-bundle.