Travels and Talks

John Baez

Here are some of my past travels and talks, in reverse chronological order, based on my lectures page.

• Thursday May 9 - Sunday May 12, 2013 - Lisa and I went to Montreal for a memorial service for her mother.

• Friday May 3 - Saturday May 5, 2013 - I gave a colloquium talk on Friday May 3 at the the Department of Logic and Philosophy of Science at UC Irvine. My contact person was James Owen Weatherall.

  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.

  • Thursday May 2 - From 4:10 to 5 I gave this talk at the UCR Mathematics Club in Room 284 of the Surge Building.

    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.

  • Friday April 26, 2013 - I spoke at the Climate Science Seminar at California State Northridge. I arrived for lunch a bit before 12:30 and gave my talk at 2:00 in Live Oak 1325.

    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.

  • Thursday April 25, 2013 - I went to Moreno Valley College and gave a talk at 4 pm on "Energy and the Environment: What Physicists Can Do." My contact was Dipen Bhattacharya.

  • April 16-20, 2013 - I visited the Perimeter Institute. On Wednesday the 17th I'll give their colloquium talk at 3 pm. My contact person is Daniel Gottesman, and I'll talk to Tobias Fritz about entropy.

    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.

  • Saturday March 23-Saturday March 30, 2013 - I went to the British Mathematics Colloquium at Sheffield University. I was invited by Eugenia Cheng.

    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:

    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.

    My host there was John Barrett.

    On Thursday the 28th the conference ended at 12:30, and I met David Tweed and later also Jim Stuttard and Glyn Adgie.

  • January 27, 2013 - At 9 am my time I gave my talk The Mathematics of Planet Earth virtually at the Mathematics Institute of the University of Warwick, using a pre-recorded video, and using Skype to answer questions. My contact person was Xinyu He.

  • Wednesday January 16, 2013 - At 4:10-5:30 pm I gave a seminar on "Network Theory" in 2206 Sproul Hall at the Econometrics Colloquium of the Economics Department of U.C. Riverside. My contact person was Gloria Gonzalez-Rivera.

  • January 10-12, 2013 - I drove to San Diego to pick up the Conant Prize at the annual Joint Mathematics Meeting. There was a prize session and reception on Thursday, January 10, 2013, from 4:25 to 7:00 pm. At 7 there was a Lecturers' Dinner. I stayed in the Marriott Marquis & Marina on 333 West Harbor Drive, San Diego, CA 92101. I rented a car until Monday the 14th.

  • December 5-6, 2012 - I went to Climate Modeling in a Transparent World and Integrated Test Beds I, organized by Steve M. Easterbrook, V. Balaji et al as part of the AGU (American Geophysical Union) Fall Meeting in San Francisco. I spoke on The Azimuth Project: an Open-Access Educational Resource on Thursday December 6, 2012 from 9:00 to 9:15 am. I spoke in 3010 (Moscone West), which is on 800 Howard Street.

  • November 15-18, 2012 - Lisa and I visited Berkeley. On Friday November 16 I gave the Serge Lang Lecture at U. C. Berkeley. My contact person was Kenneth Ribet.

    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.

  • Friday November 2, 2012 - Mike Stay visited Riverside, arriving 8:20 am and leaving 8:35 pm.

  • Tuesday October 30, 2012 - I gave a public lecture on The Mathematics of Planet Earth at Stellenbosch University in South Africa as part of the 55th annual meeting of the South African Mathematical Society. I did this via videoconferencing, and my host Bruce Bartlett created a video of the lecture and put it on YouTube.
    

    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.

  • Wednesday October 24, 2012 - I gave a colloquium talk on The Mathematics of Planet Earth at 3:30 at the mathematics department of the University of Southern California. My contact was Aaron Lauda and the colloquium schedule is here.

  • Friday September 21, 2012 - Lisa and I returned to California.

  • June 19 to July 7, 2012 - I attended a research program on the Mathematics of Biodiversity at the Centre de Recerca Matemàtica in Barcelona, organized by Tom Leinster and others. There was a conference from the 2nd to the 6th.

    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.

    

  • June 9, 2012 - I went with Lisa to Paris, where we stayed in the Hotel Victoria at 2 Bis Cité Bergère, in the 9th arrondissement.

    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.

  • May 10 - 25, 2012 - trip to Hong Kong: from the 10th to the 12th I stayed with Lisa at Chinese University; from the 13th to the 25th I'll be going to the University of Hong Kong visiting Jiang-Hua Lu. Our flight back was on the 25th. I gave two talks at the University of Hong Kong:

    G₂ 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, G₂, 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.

  • March 25 - April 7, 2012 - John Huerta visited the CQT and we worked on the paper G₂ and the rolling ball.

  • Saturday February 18th - Sunday February 26th, 2012 - recess week at NUS. Visited Chiang Mai from February 17th to 24th.

  • Wednesday February 15th, 2012 - talk in the NUS math department Topology and Geometry Seminar on "The Beauty of Roots". My contact is Jon Berrick. The talk is Seminar Room 5, S17-05-11. It starts at 3:15 and lasts until 4:15, but there's tea first.

  • Monday February 13th, 2012 - I gave a talk at Google in the form of a robot:

    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?

  • Friday February 3rd - Thursday February 9, 2012 - visit to Macquarie University.

    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.

  • Saturday January 28 - Thursday February 2, 2012 - I spoke at the Coogee '12 Sydney Quantum Information Theory Workshop in Sydney, Australia. I stayed at the Crowne Plaza Coogee, 242 Arden Street Street, Coogee +61 (0)2 9315 9178.

    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?

  • January 8-15, 2012 - I gave talks on "Network theory" and also a public lecture as part of Expository Quantum Lecture Series 5 (EQuaLS5) at the Institute for Mathematical Research (INSPEM) at Universiti Putra Malaysia. This is near but not in Kuala Lumpur. My contact person is Hishamuddin Zainuddin. I will get picked up at the airport and driven to the Mines Wellness Hotel.

    The theme for EQuaLS5 was "Geometry, Topology and Physics 2012" and the speakers were:

    1. John Baez (NUS, Univ of California, Riverside) "Network Theory"
    2. Do Ngoc Diep (Inst of Math, Hanoi) "A Procedure for Quantization of Fields"
    3. Varghese Mathai (University of Adelaide) "Noncommutative Geometry and the Fractional Quantum Hall Effect"
    4. Fredrik Stroemberg (Technical Univ. of Darmstadt) "Arithmetic Quantum Chaos"
    5. S. Twareque Ali (Concordia University, Montreal) "Coherent States: Theory and Applications"

    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?

  • Monday January 9, 2012 - classes started at NUS.

  • Thursday January 5 - Friday January 6, 2012 - Yale-NUS College Workshop in Singapore, beginning at 8:30 a.m. on Thursday and adjourning after dinner on Friday.

  • Saturday December 24 - Friday December 30, 2011 - Lisa and I went to Luang Prabang in Laos.

  • Wednesday December 7, 2011 - I gave a colloquium as part of the CQT Annual Symposium.

    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?

  • Sunday December 4, 2011 - Sunday January 8th 2012 - five-week vacation at NUS.

  • November 9-22, 2011 - Lisa visited Shanghai and Beijing; Jason Morton visited from the 11th to the 16th.

  • September 17-25, 2011 - recess week at NUS; Lisa and I visited Lombok and Bali.

  • August 21-28, 2011 - My sister visited Singapore.

  • August 1, 2011 - Orientation period at NUS began. Classes started on Monday August 8th.

  • July 29-August 12, 2011 - Chenchang Zhu ran a conference Higher Structures in China II at Jilin University in the city of Changchun.

    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.

  • Monday July 11, 2011 - Lisa and I flew from Montreal to Frankfurt, arriving on the Tuesday the 12th, and then directly on to Singapore, arriving on the 13th.

  • July 3, 2011 - Lisa and I flew from Nuremberg to Frankfurt and then on to Montreal, where we visited her family. On Friday July 8th at 10 am I gave the following talk in the combinatorics seminar at the Université du Québec à Montréal. My host was André Joyal, and the seminar was run by Franco Saliola.

    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. We shall explain these ideas assuming a bare minimum of prerequisites.

  • June 25, 2011 - Lisa and I took a train to Erlangen, where I spoke with Derek Wise, and she attended a conference and then a meeting of her project.

  • June 13, 2011 - Lisa and I flew from Singapore to the conference Quantum Theory and Gravitation, to take place at ETH Zürich from June 14-24. The local organizers were Matthias Gaberdiel and Jürg Fröhlich. We arrived a day before and left a day after. We stayed in the Hotel Sunnehus at Sonneggstrasse 17.

    I gave this talk on the 17th:

    Higher gauge theory, division algebras and superstrings

    Classically, 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.

  • May 16 - 24, 2011 - I flew from Singapore to LAX on Monday May 16 to attend these thesis defenses:
    • John Huerta: Wednesday 5/18 from 1 pm to 2:30 pm.
    • Chris Rogers: Thursday 5/19 from 3:40 pm to 6:40 pm.
    • Christopher Walker: Friday 5/20 from 2:30 pm to 4:00 pm.

    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.

  • Sunday May 8 - Sunday July 31, 2011 - vacation at NUS for 12 weeks.

  • April 19 - 29, 2011 - Lisa took a trip to California.

  • Saturday April 16 to Saturday May 7, 2011 - reading week followed by two weeks of exams at NUS.

  • March 14 - June 10, 2011, Probability and Discrete Mathematics in Mathematical Biology, at the Institute for Mathematical Sciences at NUS.

  • Tuesday March 22 - Sunday March 28 - Visit to the mathematics department at Hong Kong University, sponsored by Jiang-Hua Lu. On Wednesday at 4 pm I gave a colloquium talk:

    Energy, the environment, and what mathematicians 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 mathematicians do to help?

    On 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:

    Higher gauge theory, division algebras and superstrings

    Classically, 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.

  • January 8-14, 2011 - 14th Workshop on Quantum Information Processing (QIP2011), Capella Hotel, Sentosa, Singapore.

  • December 25, 2010 - January 9, 2011 - Lisa and I went first to Hanoi, then Hue from the 27th to the 1st, and then back to Hanoi.

  • Monday, November 1, 2010 - Lisa, Walter Blackstock and I went to hear the Prime Minister of Singapore, Lee Hsien Loong, give the Singapore Energy Lecture at 9:30 am in the Suntec Singapore International Convention and Exhibition Centre.

  • Saturday September 18th to Sunday September 26th, 2010 - recess week at NUS; Lisa and I took part of this time to travel to Bali.

  • Friday July 9th, 2010 - Lisa and I left to catch a plane for Singapore departing from LAX very early in the morning of July 10th. We will spend two years in Singapore on leave from UCR, during which I will work at the Centre for Quantum Technologies in Singapore. During this time I hope to visit various people and places including Jiang-Hua Lu at the University of Hong Kong.

  • A trip to CUNY and Oxford:
    • Wednesday May 19, 2010 - Talk at the Einstein Chair Mathematics Seminar at CUNY, run by Dennis Sullivan. My contact person has been Aron Fischer. I'll fly to New York on Tuesday May 18th and stay at Sullivan's apartment, then leave for London on Saturday May 22nd.

      Electrical Circuits

      While 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.

    • May 22nd, 2010 - I took a flight from JFK to Heathrow leaving at 10:30 pm and arriving at 10:30 am Sunday May 23rd. I took the Airline (a one-hour bus ride) to Oxford and went to the Cotswold Lodge, 66a Banbury Road. Then from May 24th to 28th I attended the school on Foundational Structures in Quantum Computation and Information, run by Bob Coecke and Ross Duncan. I talked to Tim Palmer over lunch on Tuesday the 24th, Thomas Fischbacher on Thursday the 27th starting at 12:00, and also Dan Ghica.

    • Saturday/Sunday May 29-30, 2010 - I gave a talk at the Quantum Physics and Logic workshop at Oxford, organized by Bob Coecke, Prakash Panangaden, and Peter Selinger. Then I caught a flight from Heathrow to Los Angeles at 3:45 pm on Monday May 31st.

      Duality in Logic and Physics

      Duality 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.

  • Friday April 9th, 2010 - I gave a colloquium talk at Cal State Fresno. My contact people were Douglas Singleton in the physics department and Carmen Caprau in math. I drove up Thursday afternoon and drove back down on Sunday after spending Saturday night in Sequoia National Park. On Friday, I began by giving a math talk from 10 to 11 am:

    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. 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.

    Then I gave my colloquium talk at 3:

    Physics, Topology, Logic and Computation: a Rosetta Stone

    In 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.

  • Monday March 22nd, 2010 - colloquium talk at Fullerton College. My contact person is Dana Clahane in the math department.
    

    8

    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 E₈. The octonions also explain the curious way in which topology in dimension n resembles topology in dimension n+8.

  • Saturday January 23rd, 2010 - A talk on quantum gravity, 3-5 pm at the LPS seminar room (777 Social Science Tower), for the Southern California Reading Group in the Philosophy of Physics. This is a group that meets about three times per quarter at UC Irvine to for discussions of recent articles in the foundations of physics and talks on the philosophy of physics. Contact person: Christian Wüthrich.

  • November 7-9, 2009 - 2009 Fall Western Section Meeting of the AMS, on Saturday and Sunday. Julie Bergner and I are organizing a special session on Homotopy Theory and Higher Algebraic Structures. On Saturday November 7 from 3:00 pm to 3:20 pm I'll give a talk at the session on the History and Philosophy of Mathematics organized by Jim Tattersall and Shawnee McMurran:

    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.

  • September 13-20, 2009 - I gave 5 lectures on Categorification in Mathematical Physics at the 2nd School and Workshop on Quantum Gravity and Quantum Geometry at the Corfu Summer Institute, along with Abhay Ashtekar (Loop Quantum Gravity), John Barrett (Spin Networks and Quantum Gravity), Vincent Rivasseau (Renormalization in Fundamental Physics), and Carlo Rovelli (Covariant Loop Quantum Gravity and its Low-Energy Limit). The main organizer is George Zoupanos, but I was also in contact with Harald Grosse.

    Categorification in Fundamental Physics

    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.

    1. Connections on abelian gerbes
    2. Lie n-groups and Lie n-algebras
    3. Multisymplectic geometry and classical field theory
    4. Higher gauge theory and the string 2-group
    5. Higher gauge theory, strings and branes

    Lisa and I left on September 11th and flew back on the 22nd.

  • September 8-10, 2009 - I gave several talks as the Cecil & Ida Green Honors Chair at the mathematics department of Texas Christian University (TCU) in Forth Worth, Texas. My contact was Bob Doran, chair of the math department.

    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.

  • August 12-14, 2009 - I gave a talk at the 24th Annual IEEE Symposium on Logic in Computer Science (LICS 2009) at UCLA at 8:30 am on August 13th. I was supposed to talk about "work going on at the frontiers between programming language semantics, algebra, logic and mathematical physics".

    Computation and the Periodic Table

    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.

    On 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.

  • June 12th, 2009 - Lisa gave a talk in Paris. I have a final exam on June 9th, so we left on the evening of June 10th and arrived on the 11th. I spent two months visiting Paul-André Melliès and we came back to California on August 11th.

  • May 1-3, 2009 - Fields Institute workshop on Smooth Structures in Logic, Physics and Category Theory organized by Rick Blute and the Logic and Foundations of Computing group at the University of Ottawa. There will be talks by Kristine Bauer, Thomas Ehrhard, Anders Kock, Andrew Stacey and me. I will fly there on Thursday April 30th and fly back on Monday May 4th. I'll be staying at a B&B called the Gasthaus Switzerland.

    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.

  • April 18-19, 2009 - Keynote speaker at Graduate Student Topology Conference at the University of Wisconsin, invited by Nicolas Addington. I gave my talk on Categorification and Topology, in two 1-hour sessions, one in the morning and one in the afternoon. I'll show up on Friday the 17th and leave on Monday the 20th.

  • April 12-17, 2009 - Talk at a workshop on Categorification and Geometrisation from Representation Theory in Glasgow. I arrived on Sunday April 12th and left early, Friday April 17th, so I could go to Wisconsin! I stayed at The Glasgow Pond Hotel, Great Western Road, West End. My talk was on Wednesday April 15th at 11 am:

    Categorification and Topology

    The 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.

  • February 5-6, 2009 - Three 1-hour talks at Higher Structures in Topology and Geometry II at the Mathematisches Institut, Georg-August-Universität Göttingen. This is organized by Chenchang Zhu and Giorgio Trentinaglia of the CRCG. I left the night of Tuesday the 3rd and returned Sunday the 8th. I went with Chris Rogers; we missed our connection, had to spent a night in Frankfurt, and showed up late.

    Lectures on 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. This overview of higher gauge theory will emphasize its relation to homotopy theory and the cohomology of groups and Lie algebras.

    • Lecture 1 - Lie 2-groups and Lie 2-algebras. 2-3 pm, Thursday February 5th.
    • Lecture 2 - 2-bundles and classifying spaces for 2-groups. 9-10 am, Friday February 6th.
    • Lecture 3 - 2-connections on 2-bundles. 3-4 pm, Friday February 6th.

  • January 3-10, 2009 - Trip to the Joint Mathematics Meeting in Washington DC. I flew from LAX to Dulles on Saturday January 3rd and came back on Saturday January 10th.

    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-Groups

    Categorifying 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, Stevenson and myself 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|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:

    Groupoidification

    There 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.

  • Saturday and Sunday, November 22-23, 2008 - the Groupoidfest at U. C. Riverside. I spoke from 12 to 12:50 on Sunday.

    Groupoidification

    There 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.

  • Thursday, October 23rd, 2008 - Two talks at the University of Illinois at Urbana-Champaign, invited by Matthew Ando. I went there on the 22nd and returned on the 25th. The first talk was the department colloquium:

    8

    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.

    The second was some sort of math/physics seminar:

    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, 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.

  • Friday, October 17th, 2008 - UCOLASC meeting in Oakland. I left from the Ontario airport around 6:30 pm on Thursday and returned around 9 pm on Friday.

  • September 14-21, 2008 - The 2008 Robert Rankin Lectures at the University of Glasgow, sponsored by the Glasgow Mathematical Journal Trust. I will arrive Sunday the 14th, give lectures on Monday, Wednesday and Friday, and leave Sunday the 21st.

    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 E₈. 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 E₈. 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 + ... = -1/12

    which 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

    1² + 2² + 3² + ... + 24²

    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.

  • Sunday July 6 - Friday August 15, 2008 - Visited Paris, working with Paul-André Melliès at the Preuves, Programmes et Systemes group at Université Paris 7. From July 6th to August 4th, Lisa and I stayed in an apartment on Rue Sufflot between the Pantheon and Jardin du Luxembourg. Then we moved to a short-term apartment at 6 Rue Victor Cousin behind the Hotel Sorbonne from the 5th to the 11th. Finally, from the 11th to the 115th, we stayed in the Hotel des Trois Colleges at the corner of Rue Victor Cousin and Rue Cujas.
    • July 7-11, 2008 - Algebraic Topological Methods in Computer Science 2008 at University of Paris Diderot (Paris 7). I gave this one-hour plenary talk at 2:30-3:30 pm on Monday July 7th:

      Computation and the Periodic Table

      By 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.

    • Sunday July 13 - Wednesday July 16, 2008 - Visit to Lisbon, Portugal for the agregãço of Alexsandar Mikovic, which will be from 2-4 pm on Monday the 14th and Tuesday the 15th, with some bureacratic stuff lasting until 5 pm on Tuesday. I'm giving a talk on Classifying Spaces for Topological 2-Groups on Monday morning. Invited by Roger Picken.

    • Thursday July 24 - Talk on "Groupoidification" at the Preuves, Programmes et Systemes group at Université Paris 7.

  • June 30-July 5, 2008 - Conference on Homotopy Theory and Higher Categories at the Centre de Recerca Matemàtica (CRM) in Barcelona, organized by Carles Casacuberta, André Joyal, Joachim Kock, Amnon Neeman and Frank Neumann.

    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:

    Groupoidification

    There 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.

  • June 21-29, 2008 - Visit to Granada, sponsored by Pilar Carrasco. Two talks on "Lie 2-Algebras" on Tuesday the 24th and Thursday the 26th.

    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.

  • June 16-20, 2008 - Workshop on Categorical Groups at the Institut de Matemàtica de la Universitat de Barcelona (or IMUB), organized by Pilar Carrasco.

    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 Spaces

    Categorifying 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.

  • May 19-23, 2007 - trip to northern California.
    • Tuesday May 20, 2008 - Talk at the Groupoids in Analysis and Geometry seminar at U. C. Berkeley - visit organized by Alan Weinstein.

      Groupoidification

      There 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.

    • Thursday May 22nd, 2008 - Talk on "5" at Google headquarters in Mountain View, hosted by Mike Stay.

  • May 6-9, 2008 - Visit to the math department at George Washington University, organized by Bill Schmitt. I visited my parents from Friday the 9th to Wednesday the 14th, after giving this talk at GWU at 2-3 pm on Wednesday May 7th:

    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. 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.

  • April 8-26, 2008 - Visit to the CAS-MPG Partner Institute for Computational Biology in Shanghai, invited by Andreas Dress. Leaving the night of April 7th for a flight departing 1:45 am on April 8th, arriving April 9th.

  • March 23-27, 2008 - Visit to the math and physics departments at the National University of Singapore. Flight left LAX late March 21st and arrived back here March 27th.

    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.

  • November 1-2, 2007 - public lecture, talk for physics majors and lunch talk at the Department of Physics and Astronomy of James Madison University in Harrisonburg, Virginia. Invited by Sean T. Scully as part of JMU's visiting scholars program:
    • Thursday the 1st, 6 pm: public talk:

      Zooming Out in Time: A Long Term History of Climate Change

      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.

    • Friday the 2nd, around noon: lunchtime chalk talk on mathematical physics.
    • Friday the 2nd, 3:45-4:45: departmental colloquium:

      Fundamental Physics: Where We Stand Today

      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.

  • October 3-4, 2007 - keynote speaker at Deep Beauty: Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World. This conference at Princeton University, organized by Hans Halvorson of the Philosophy Department, will celebrate the 75th anniversary of von Neumann's The Mathematical Foundations of Quantum Mechanics. I went there on Tuesday October 2nd and returned Friday October 5th.

    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.

  • July 28 - September 17, 2007 - visit to Greenwich, punctuated by side trips:
    • August 5-10, 2007 - the fourth Abel Symposium in Oslo, Norway. This symposium was organized by Eric Friedlander, Stefan Schwede and Graeme Segal together with the local representatives Nils Baas, Bjørn Ian Dundas, Bjørn Jahren and John Rognes. It covered the following aspects of modern algebraic topology:

      • algebraic K-theory and motivic homotopy theory
      • structured ring spectra and homotopical algebraic geometry
      • elliptic objects and quantum field theory

      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.

    • August 18-26, 2007 - visit to the Erwin Schrödinger Institut in Vienna, to attend a Workshop on Poisson Geometry and Sigma Models organized by Anton Alekseev, Henrique Bursztyn and Thomas Strobl, which lasts from August 20th to 24th. I have a reservation at the Pension ANI, Kinderspitalgasse 1, 9th district - arriving on Saturday the 18th and leaving on Sunday the 26th.

      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.

    • September 4, 2007 - talk at the London Analysis and Probability Seminar, University College, London:

      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.

    • September 10-12, 2007 - 22nd British Topology Meeting at Sheffield University. I gave my talk on 4 pm on Monday September 10th:

      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.

  • July 24-28, 2007 - my wife attended a meeting of the Center for Hellenic Studies in Olympia; I went along as a tourist.

  • July 20-23, 2007 - talk at a meeting of Thales and Friends in Delphi, organized by Apostolos Doxiadis, to discuss contributions for a book on Mathematics and Narrative. (Arrive on the 19th and depart on the 24th.)

    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.

  • July 1-19, 2007 - visit to Paul-André Melliès and the PPS (Preuves, Programmes et Systèmes) group at CNRS, Université Paris 7. (Leaving US June 30, arriving Paris July 1st.)
    • Tuesday July 10th - talk at the AstroParticule et Cosmologie (APC) group at Université Paris 7, invited by Marc Lachièze-Rey. The talk will be at 2 pm.

      Cartan Geometry and MacDowell–Mansouri Gravity: the Work of Derek Wise

      MacDowell and Mansouri invented a clever formulation of general relativity in which the Lorentz connection and coframe field are combined into a single connection with the DeSitter group SO(4,1) or anti-DeSitter group SO(3,2) as gauge group, depending on the sign of the cosmological constant. While this formulation may seem like a 'trick', it actually has a deep geometrical meaning. This is best understood in terms of Cartan's approach to connections — an approach which was somewhat forgotten after his student Ehresmann developed the simpler approach that eventually became standard. Witten's formulation of 3d gravity as a Chern-Simons theory is also clarified using Cartan geometry. However, in 3 dimensions the relevant Cartan connection is flat and gravity is a topological field theory, while in 4 dimensions this is true only in a certain limit. In this limit, point particles and certain string-like excitations can be nicely described as topological defects. This talk is an exposition of the work of Derek Wise.

  • April 22-29 (Sunday-Sunday), 2007 - 40-minute talk at the conference Philosophical and Formal Foundations of Modern Physics. This was being held at the estate of Les Treilles in the Var near Draguignan in southern France, and it was organized by Michel Bitbol and Alexei Grinbaum.

    Quantum Quandaries: a Category-Theoretic Perspective

    Category 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.

  • March 11-15, 2007 - Eugenia Cheng visited UCR from Sunday to Thursday.

  • January 9-13, 2007 - workshop on Higher Categories and Their Applications, organized by Eugenia Cheng, Peter May and myself. This was part of the Fields Institute program Geometric Applications of Homotopy Theory, organized by Rick Jardine, Gunnar Carlsson and Dan Christensen. You can see photos, abstracts and talks from this workshop.

    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.

  • Thursday December 7, 2006 - mathematics colloquium at Stanford University:

    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.

  • Friday November 17th, 2006 - joint math/physics talk at Louisiana State University, invited by Jorge Pullin:

    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.

  • Friday November 10th, 2006 - I gave the Reese Prosser Memorial Lecture at Dartmouth College:

    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.

  • Friday October 13th, 2006 - I gave a talk in San Francisco as part of the Long Now Foundation's Seminars About Long-Term Thinking, organized by Stewart Brand.

    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.

  • July 3 - September 20, 2006 - I stayed in Shanghai, with two side trips:
    • August 18-21, 2006 - Suzhou.
    • September 14-17, 2006 - Hangzhou.

  • May 15 - June 20, 2006 - I visited Laurent Freidel and other people at the Perimeter Institute, and Dan Christensen at the University of Western Ontario.

    • Wednesday, May 31, 2006 - I gave this week's colloquium at the Perimeter Institute at 2 pm:

      Higher-Dimensional Algebra: A Language for Quantum Spacetime

      Category 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.

    • Thursday, June 1, 2006 - I gave a colloquium in the mathematics department at the University of Western Ontario at 3:30-4:30 pm, after coffee at 3.

      Fundamental Physics: Where We Stand Today

      Since 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?

    • I visited Cambridge Massachusetts on June 2-6 (Friday-Tuesday), reading grant proposals.

    • I visited my parents in DC on June 6-11 (Tuesday-Sunday).

    • Back at Waterloo June 11th-20th (Sunday-Tuesday). Talk with Howard Burton at 11 am on Wednesday the 14th.

    • Tuesday June 20th: back to Riverside.

  • April 1 - May 15, 2006 - I was on sabbatical visiting Peter May, Eugenia Cheng, and others at the University of Chicago.
    • April 7-11, 2006 - I attended the Mac Lane Memorial Conference at the University of Chicago. In conjunction with this, I gave the 24th annual Unni Namboodiri Lectures - a series of three lectures at 4 pm on the 7th (Friday), 4 pm on the 10th (Monday), and 4:30 pm on the 11th (Tuesday):

      Higher Gauge Theory, Higher Categories

      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".

    • Starting Wednesday April 19th, I gave a series of informal talks to the category theorists at the University of Chicago, about topics that may be covered in my book:
      • On Wednesday April 19th, at 2:30 pm, room E203: The Power of Negative Thinking
      • On Friday April 21st, at 2:30 pm, room E312: Cohomology: The Layer-Cake Philosophy
      • On Monday April 24th, at 3:00 pm, room E203: An Example: Classifying 2-Groupoids

      Michael Shulman took notes on these talks and wrote an extensive appendix on topos theory.

    • April 29-May 1, 2006 - three 1-hour talks at the 2006 Barrett Lectures, a conference on geometric topology at the University of Tennessee in Knoxville. I left on Friday April 28th, and returned on Wednesday 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.

      On May 2nd there was an excursion to the Smoky Mountains. I flew there on April 28th and flew back on May 3rd.

  • Tuesday, March 21, 2-5 pm - in Surge 284, Toby Bartels successfully defended defended his thesis on Higher Gauge Theory: 2-Bundles.

  • Monday March 13, 11 am - I gave a talk on loop quantum gravity at the Physics Department of Cal State Long Beach, invited by Zvonko Hlousek.

    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.

  • Friday March 10, 2:30-4:30 pm - oral exam for Richard W. McHard in EBU II Room 203.

  • January 30 - March 3, 2006 - I attended a workshop on the Geometry of Computation at CIRM (Centre Internationale de Recontres Mathématiques). This workshop was run by Thomas Ehrhard and Laurent Regnier of the Institut de Mathematiques de Luminy in Marseille. On the first week I gave nine lectures on the following topic:

    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 Today

    Since 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.

  • Monday January 21, 2006 - from 1:10 to 2 pm I gave a talk to UCR math department grad students on How to Teach Stuff.

  • January 9 - 13, 2005 - Dan Christensen visited UCR and we worked on a model category of smooth spaces.

  • December 25th-31st - Lisa and I drove around Arizona with some friends.

  • Sunday December 4, 2005 - I gave a talk from 9:30 to 10:30 am at the Union College Mathematics Conference on category theory, homotopy theory and commutative algebra. I arrived at this conference on Friday December 3rd and come back on Monday the 5th.

    Higher Gauge Theory: 2-Connections

    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. 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.

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