Doing physics with math ...and vice versa Alexander Tabler Thessaloniki, December 19, 2019 Thessaloniki, December 19, 2019 0 / 14 Outline 1 Physics vs. Mathematics 2 Case study: a question about shapes 3 Enter physics & quantum field theory 4 Summary Thessaloniki, December 19, 2019 1 / 14 1. Physics vs. Mathematics Thessaloniki, December 19, 2019 1 / 14 Common origins, divergent futures At some point, Mathematics and Physics were the same discipline: Natural philosophy. Calculus: mathematics or physics? \Isaac Newton" - Godfrey Kneller - commons.curid=37337 Thessaloniki, December 19, 2019 2 / 14 Today, Math=Physics6 Mathematics: - Definition. Theorem. Proof. (Bourbaki-style mathematics) - Logical rigor, \waterproof" - \Proof"-driven (Theoretical) Physics: - Less rigor - Computation-oriented (\shut up and calculate") - Experiment-driven ...in principle Thessaloniki, December 19, 2019 3 / 14 An early 20th century divorce Freeman Dyson, 1972: \The marriage between mathematics and physics has recently ended in divorce"1 F. Dyson - by ioerror - commons.curid=71877726 1 Dyson - \Missed Opportunities", Bull. Amer. Math. Soc., Volume 78, Number 5 (1972), 635-652. Thessaloniki, December 19, 2019 4 / 14 A fruitful relation However, strong bonds still remain: - Modern theoretical physics uses more abstract mathematics - There is a deep relationship between advances in both fields - This relationship is mutually beneficial https://www.ias.edu/ideas/power-mirror-symmetry Thessaloniki, December 19, 2019 5 / 14 2. Case study: a question about shapes Thessaloniki, December 19, 2019 5 / 14 Can we classify shapes? A typical question in mathematics. - What does it mean to classify? - What is a shape? Field of topology Thessaloniki, December 19, 2019 6 / 14 A weak notion of classification: Earth∼=billiard ball! Examples Rules for classification: Stretching 3 Tearing 7 Squeezing 3 Gluing 7 Bending 3 Puncturing 7 Thessaloniki, December 19, 2019 7 / 14 Examples Rules for classification: Stretching 3 Tearing 7 Squeezing 3 Gluing 7 Bending 3 Puncturing 7 A weak notion of classification: Earth∼=billiard ball! Thessaloniki, December 19, 2019 7 / 14 Examples | are these pairs (X1;X2) the \same"? One line & two lines A sphere & a torus A line & a circle A coffee mug & a torus Thessaloniki, December 19, 2019 7 / 14 Examples | are these pairs (X1;X2) the \same"? One line & two lines A sphere & a torus A line & a circle A coffee mug & a torus Thessaloniki, December 19, 2019 7 / 14 Topological invariants questions that possibly distinguish X1 and X2: # of connected components? Remove a point from each. Are they the different now? Are there non-trivial loops? Are there \holes" in the space? Topological (manifold) invariants t(X) Given a space X, a topological invariant t(X) is the answer to a question t about X Answer must remain invariant under stretching, squeezing and bending of X Remain agnostic about tearing, gluing and puncturing. Thessaloniki, December 19, 2019 8 / 14 Topological invariants Here's how to use them: ∼ if X1 = X2; then t(X1) = t(X2) , if t(X1) 6= t(X2); then X1 X2 Thessaloniki, December 19, 2019 8 / 14 Trouble with invariants Invariants usually a long way from classification. t's are hard to find. Even if t's are found, they are hard to answer! Examples were in dimension 1&2. Problem is harder in higher dimensions! Thessaloniki, December 19, 2019 9 / 14 3. Enter physics & quantum field theory Thessaloniki, December 19, 2019 9 / 14 Meanwhile in 20th century Physics... + = Quantum field theory Forces are described by fields. Fields are created by charges. Charges move due to the fields. Equations describe this complicated interplay. Answers in the form of probabilities (not deterministic). Thessaloniki, December 19, 2019 10 / 14 ...unless Lots of exact results! Caveats of QFT Mathematically ill-defined Usually impossible to solve Thessaloniki, December 19, 2019 11 / 14 Caveats of QFT Mathematically ill-defined Usually impossible to solve ...unless Lots of exact results! Thessaloniki, December 19, 2019 11 / 14 A new spark Questions answered in QFT, string theory and quantum gravity renewed interest between ...ex-spouses Cover of Deligne et al. - \Quantum fields and strings; A course for Mathematicians". Thessaloniki, December 19, 2019 12 / 14 Back-and-forth Math Realization in `80's (Donaldson, Freedman): there are field theories whose solutions compute topological invariants (a.k.a. new t's!) Physics Late `80's (Witten): Birth of topological quantum field theory Math TFTs & higher categories Physics Seiberg-Witten theory Math Surprising results in dim = 4! Thessaloniki, December 19, 2019 13 / 14 Back-and-forth Math Physics Top. invariants, Donaldson invariants YM Instantons TFTs, Higher categories Seiberg-Witten theory Mirror symmetry, String theory, quantum cohomology, GLSMs Gromov-Witten invariants D-branes ?? More invariants? ??? Thessaloniki, December 19, 2019 12 / 14 Greg Moore's \Physical Mathematics" On YouTube: \The Shapes of Spaces and the Nuclear Force " \Physical Mathematics and the Future" Thessaloniki, December 19, 2019 13 / 14 Greg Moore's \Physical Mathematics" Physical mathematics: Fusion of mathematical and physical ideas with aim to 1 elucidate the laws of nature 2 discover deep mathematical truths Deviates from strict notions of both Physics and Mathematics Rich source of profound questions and (my) excitement Thessaloniki, December 19, 2019 13 / 14 4. Summary & some propaganda Thessaloniki, December 19, 2019 13 / 14 My advice: Learn as much (modern) mathematics as possible. There is much to learn, and much more to do in the intersection of mathematics and physics! Summary & propaganda Math and physics are different disciplines, although they didn't use to be. The new blossomed relationship is not one-sided, but fruitful and mutually beneficial: example of manifold invariants computed using QFT Thessaloniki, December 19, 2019 14 / 14 Summary & propaganda Math and physics are different disciplines, although they didn't use to be. The new blossomed relationship is not one-sided, but fruitful and mutually beneficial: example of manifold invariants computed using QFT My advice: Learn as much (modern) mathematics as possible. There is much to learn, and much more to do in the intersection of mathematics and physics! Thessaloniki, December 19, 2019 14 / 14 Thank you! Thessaloniki, December 19, 2019 14 / 14.