Here are some of my past travels and talks, in reverse chronological order. This page is to help me remember when I did what. It also gives me something to do with stuff on my page listing forthcoming lectures after I've given them.
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, focussing 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 H1(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 H1(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 H1(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 E8. 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 Stringk(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 Stringk(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 Stringk(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 Stringk(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 Stringk(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 Stringk(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.
