Yang{Mills for probabilists Sourav Chatterjee Sourav Chatterjee Yang{Mills for probabilists I The electromagnetic force. Interaction of light and matter. I The weak force. Interactions between sub-atomic particles. I The strong force. Force that holds together quarks that form protons and neutrons. I Gravity. I Physical theories: I Electromagnetic force: Quantum electrodynamics (QED). Dirac, Feynman, Tomonaga, Schwinger, Dyson, ... I Weak force: Electroweak theory. Glashow, Salam, Weinberg, ... I Strong force: Quantum chromodynamics (QCD). Yang, Mills, Gross, Politzer, Wilczek, Wilson, ... I Gravity: General relativity (GR). Einstein. Fundamental forces I The four fundamental forces: Sourav Chatterjee Yang{Mills for probabilists I The weak force. Interactions between sub-atomic particles. I The strong force. Force that holds together quarks that form protons and neutrons. I Gravity. I Physical theories: I Electromagnetic force: Quantum electrodynamics (QED). Dirac, Feynman, Tomonaga, Schwinger, Dyson, ... I Weak force: Electroweak theory. Glashow, Salam, Weinberg, ... I Strong force: Quantum chromodynamics (QCD). Yang, Mills, Gross, Politzer, Wilczek, Wilson, ... I Gravity: General relativity (GR). Einstein. Fundamental forces I The four fundamental forces: I The electromagnetic force. Interaction of light and matter. Sourav Chatterjee Yang{Mills for probabilists I The strong force. Force that holds together quarks that form protons and neutrons. I Gravity. I Physical theories: I Electromagnetic force: Quantum electrodynamics (QED). Dirac, Feynman, Tomonaga, Schwinger, Dyson, ... I Weak force: Electroweak theory. Glashow, Salam, Weinberg, ... I Strong force: Quantum chromodynamics (QCD). Yang, Mills, Gross, Politzer, Wilczek, Wilson, ... I Gravity: General relativity (GR). Einstein. Fundamental forces I The four fundamental forces: I The electromagnetic force. Interaction of light and matter. I The weak force. Interactions between sub-atomic particles. Sourav Chatterjee Yang{Mills for probabilists I Gravity. I Physical theories: I Electromagnetic force: Quantum electrodynamics (QED). Dirac, Feynman, Tomonaga, Schwinger, Dyson, ... I Weak force: Electroweak theory. Glashow, Salam, Weinberg, ... I Strong force: Quantum chromodynamics (QCD). Yang, Mills, Gross, Politzer, Wilczek, Wilson, ... I Gravity: General relativity (GR). Einstein. Fundamental forces I The four fundamental forces: I The electromagnetic force. Interaction of light and matter. I The weak force. Interactions between sub-atomic particles. I The strong force. Force that holds together quarks that form protons and neutrons. Sourav Chatterjee Yang{Mills for probabilists I Physical theories: I Electromagnetic force: Quantum electrodynamics (QED). Dirac, Feynman, Tomonaga, Schwinger, Dyson, ... I Weak force: Electroweak theory. Glashow, Salam, Weinberg, ... I Strong force: Quantum chromodynamics (QCD). Yang, Mills, Gross, Politzer, Wilczek, Wilson, ... I Gravity: General relativity (GR). Einstein. Fundamental forces I The four fundamental forces: I The electromagnetic force. Interaction of light and matter. I The weak force. Interactions between sub-atomic particles. I The strong force. Force that holds together quarks that form protons and neutrons. I Gravity. Sourav Chatterjee Yang{Mills for probabilists I Electromagnetic force: Quantum electrodynamics (QED). Dirac, Feynman, Tomonaga, Schwinger, Dyson, ... I Weak force: Electroweak theory. Glashow, Salam, Weinberg, ... I Strong force: Quantum chromodynamics (QCD). Yang, Mills, Gross, Politzer, Wilczek, Wilson, ... I Gravity: General relativity (GR). Einstein. Fundamental forces I The four fundamental forces: I The electromagnetic force. Interaction of light and matter. I The weak force. Interactions between sub-atomic particles. I The strong force. Force that holds together quarks that form protons and neutrons. I Gravity. I Physical theories: Sourav Chatterjee Yang{Mills for probabilists I Weak force: Electroweak theory. Glashow, Salam, Weinberg, ... I Strong force: Quantum chromodynamics (QCD). Yang, Mills, Gross, Politzer, Wilczek, Wilson, ... I Gravity: General relativity (GR). Einstein. Fundamental forces I The four fundamental forces: I The electromagnetic force. Interaction of light and matter. I The weak force. Interactions between sub-atomic particles. I The strong force. Force that holds together quarks that form protons and neutrons. I Gravity. I Physical theories: I Electromagnetic force: Quantum electrodynamics (QED). Dirac, Feynman, Tomonaga, Schwinger, Dyson, ... Sourav Chatterjee Yang{Mills for probabilists I Strong force: Quantum chromodynamics (QCD). Yang, Mills, Gross, Politzer, Wilczek, Wilson, ... I Gravity: General relativity (GR). Einstein. Fundamental forces I The four fundamental forces: I The electromagnetic force. Interaction of light and matter. I The weak force. Interactions between sub-atomic particles. I The strong force. Force that holds together quarks that form protons and neutrons. I Gravity. I Physical theories: I Electromagnetic force: Quantum electrodynamics (QED). Dirac, Feynman, Tomonaga, Schwinger, Dyson, ... I Weak force: Electroweak theory. Glashow, Salam, Weinberg, ... Sourav Chatterjee Yang{Mills for probabilists I Gravity: General relativity (GR). Einstein. Fundamental forces I The four fundamental forces: I The electromagnetic force. Interaction of light and matter. I The weak force. Interactions between sub-atomic particles. I The strong force. Force that holds together quarks that form protons and neutrons. I Gravity. I Physical theories: I Electromagnetic force: Quantum electrodynamics (QED). Dirac, Feynman, Tomonaga, Schwinger, Dyson, ... I Weak force: Electroweak theory. Glashow, Salam, Weinberg, ... I Strong force: Quantum chromodynamics (QCD). Yang, Mills, Gross, Politzer, Wilczek, Wilson, ... Sourav Chatterjee Yang{Mills for probabilists Fundamental forces I The four fundamental forces: I The electromagnetic force. Interaction of light and matter. I The weak force. Interactions between sub-atomic particles. I The strong force. Force that holds together quarks that form protons and neutrons. I Gravity. I Physical theories: I Electromagnetic force: Quantum electrodynamics (QED). Dirac, Feynman, Tomonaga, Schwinger, Dyson, ... I Weak force: Electroweak theory. Glashow, Salam, Weinberg, ... I Strong force: Quantum chromodynamics (QCD). Yang, Mills, Gross, Politzer, Wilczek, Wilson, ... I Gravity: General relativity (GR). Einstein. Sourav Chatterjee Yang{Mills for probabilists I The various quantum field theories are unified into one body known as the Standard Model. I The first quantum field theory was quantum electrodynamics. I Gave astonishingly accurate predictions, matching up to 10 decimal places with experimental data! I The other theories have similar success stories. I However, there is no rigorous mathematical foundation for these theories. (Clay millennium problem of Yang{Mills existence.) I Even from the point of view of theoretical physicists, there are very important unsolved theoretical problems | quark confinement, mass gap, etc. Quantum field theories I The theories explaining the first three forces are known as quantum field theories. Sourav Chatterjee Yang{Mills for probabilists I The first quantum field theory was quantum electrodynamics. I Gave astonishingly accurate predictions, matching up to 10 decimal places with experimental data! I The other theories have similar success stories. I However, there is no rigorous mathematical foundation for these theories. (Clay millennium problem of Yang{Mills existence.) I Even from the point of view of theoretical physicists, there are very important unsolved theoretical problems | quark confinement, mass gap, etc. Quantum field theories I The theories explaining the first three forces are known as quantum field theories. I The various quantum field theories are unified into one body known as the Standard Model. Sourav Chatterjee Yang{Mills for probabilists I Gave astonishingly accurate predictions, matching up to 10 decimal places with experimental data! I The other theories have similar success stories. I However, there is no rigorous mathematical foundation for these theories. (Clay millennium problem of Yang{Mills existence.) I Even from the point of view of theoretical physicists, there are very important unsolved theoretical problems | quark confinement, mass gap, etc. Quantum field theories I The theories explaining the first three forces are known as quantum field theories. I The various quantum field theories are unified into one body known as the Standard Model. I The first quantum field theory was quantum electrodynamics. Sourav Chatterjee Yang{Mills for probabilists I The other theories have similar success stories. I However, there is no rigorous mathematical foundation for these theories. (Clay millennium problem of Yang{Mills existence.) I Even from the point of view of theoretical physicists, there are very important unsolved theoretical problems | quark confinement, mass gap, etc. Quantum field theories I The theories explaining the first three forces are known as quantum field theories. I The various quantum field theories are unified into one body known as the Standard Model. I The first quantum field theory was quantum electrodynamics. I Gave astonishingly accurate predictions, matching up to 10 decimal places with experimental data! Sourav Chatterjee Yang{Mills for probabilists I However, there is no rigorous mathematical foundation for these theories. (Clay millennium problem of Yang{Mills existence.) I Even