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Topological quantum field theory from path integrals to higher categories

Topological quantum field theory from path integrals to higher categories. Bruce Bartlett, Phd student, Sheffield University January 2008, Stellenbosch. Outline. A topological quantum field theory (TQFT) is a gadget which assigns invariants to geometric structures.

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Topological quantum field theory from path integrals to higher categories

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  1. Topological quantum field theory from path integrals to higher categories Bruce Bartlett, Phd student, Sheffield University January 2008, Stellenbosch

  2. Outline • A topological quantum field theory (TQFT) is a gadget • which assigns invariants to geometric structures. • They came out of physics in the late 1980's, but were soon • defined formally by mathematicians. • We will look at Chern-Simons theory, a famous example • of a TQFT. Much remains to be understood. • Amongst other things, this theory is about finding • invariants of knots in three-dimensional space. • Originally, Jones obtained these invariants • from studying von Neumann algebras. But • Witten explained them geometrically in terms of • a TQFT.

  3. Fields Medallists 1982-2006 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov

  4. Fields Medallists 1982-2006 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov

  5. Fields Medallists 1982-2006 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Used ideas from inter alia quantum field theory to "bridge probability, representation theory and algebraic geometry". 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov

  6. Fields Medallists 1982-2006 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Proved the Geometrization conjecture for 3-manifolds by essentially employing, and making rigorous, the "renormalization group flow" from string theory. 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov

  7. Grisha Perelman.

  8. Fields Medallists 1982-2006 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov Proved the "moonshine conjecture" by rigourously establishing a quantum field theory with the "Monster group" as its symmetry group.

  9. Fields Medallists 1982-2006 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov His work is about geometric aspects of mathematical physics : quantization, knot theory, mirror symmetry, and topological quantum field theory.

  10. Fields Medallists 1982-2006 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Quantum groups and their relation to knot theory. 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov

  11. Fields Medallists 1982-2006 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Used Von Neumann algebras and ideas from statistical mechanics to derive a new invariant of knots - the Jones polynomial. 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov

  12. Fields Medallists 1982-2006 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale Used quantum field theory to explain the Jones polynomial in terms of a topological quantum field theory known as Chern-Simons theory. Geometrically one is "summing over all connections". Many other things. 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998 Richard Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov In all this large and exciting field, which involves many of the leading physicists and mathematicians in the world, Edward Witten clearly stands out as the most influential and dominating figure. Michael Atiyah, Fields medal citation, 1990.

  13. Fields Medallists 1982-2006 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov Used gauge theory to find new invariants of smooth 4-dimensional manifolds. Corollary : there are infinitely many smooth structures onR4! This is another example of a topological quantum field theory.

  14. Fields Medallists 1982-2006 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov Revolutionized this subject in 1994 with the Seiberg-Witten equations.

  15. Fields Medallists 1982-2006 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov Classification theorems for von Neumann algebras. The "father" of noncommutative geometry... which nowadays he uses to try and explain the Standard Model of particle physics.

  16. Alain Connes at the British Mathematics Colloquium, Swansea, 2007.

  17. Fields Medallists 1982-2006 1990 Vladimir Drinfeld Vaughan Jones Shigefumi Mori Edward Witten 1986 Simon Donaldson Gerd Faltings Michael Freedman 1982 Alain Connes William Thurston Shing-Tung Yau ... 1966 Michael Atiyah Alexander Grothendieck Paul Cohen Stephen Smale 2006 Andrei Okounkov Grigori Perelman (declined) Terence Tao Wendelin Warner 2002 Laruent Lafforgue Vladimir Voevodsky 1998 Richard Borcherds Timothy Gowers Maxim Kontsevich Curtis McMullen Silver Plate : Andrew Wiles 1994 Jean Bourgain Pierre-Louis Lions Jean-Cristophe Yoccoz Efim Zelmanov K-Theory and the Atiyah-Singer index theorem. Met Witten at Harvard in 1977... revolutionizing their future research. Formalized the notion of a topological quantum field theory.

  18. John Baez Mathematician Urs Schreiber Mathematical Physicist David Corfield Philosopher

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