Sergei Gukov is a professor of mathematics and theoretical physicist. Gukov graduated from Moscow Institute of Physics and Technology (MIPT) in Moscow, Russia before obtaining a doctorate in physics from Princeton University under the supervision of Edward Witten.[1]He held a Long-term Prize fellowship of Clay Mathematics Institute at Harvard University (2001-2006) and during 2007-2008 was a member of the school of mathematics at the Institute for Advanced Study, Princeton. Since 2007, he has been professor of mathematics and theoretical physics at the California Institute of Technology (Caltech). Starting 2010, Gukov was elected as an external scientific member of the Max Planck Society at the MPIM, Bonn.Peter Koroteev is a lecturer at the Department of Mathematics at Univeristy of California Berkeley. He got his PhD from University of Minnesota in 2012 under supervision of Prof. Arkady Vainshtein. Since then he worked as Postdoctoral Researcher at Perimeter Institute for Theoretical Physics 2012-2016 and as Visiting Assistant Professor at University of California, Davis, 2016-2019. Peter’s work is focused on the interplay between representation theory, algebraic geometry, and mathematical physics. He is also an instructor at Berkeley and Stanford Math Circles.Satoshi Nawata currently serves as an associate professor of physics at Fudan University. He received a Bachelor of Science at Tokyo Institute of Technology. He went on to achieve a PhD from the University of Wisconsin-Milwaukee. Following this, Satoshi Nawata had the opportunity to hold a postdoctoral position at several notable institutions, including the Tata Institute of Fundamental Research, Perimeter Institute for Theoretical Physics, NIKHEF, University of Warsaw, and the California Institute of Technology among others. He also spent time as a visiting researcher at the Max Planck Institute for Mathematics and IHES. Since 2016, Satoshi Nawata has been imparting knowledge and driving research as an Associate Professor at Fudan University.Du Pei is currently an associate professor of quantum mathematics at the University of Southern Denmark. He obtained his Ph.D. in theoretical physics in 2016 from Caltech, and has conducted research in theoretical physics and mathematical physics at Harvard University, MSRI, Caltech and Aarhus University. His main interests concern the geometric, algebraic and categorical properties of quantum field theory.Ingmar Saberi is a Senior Researcher at Ludwig-Maximilians-Universität München, Germany at Ilka Brunner's research group. Previously he was a postdoc at the University of Heidelberg, Germany in Johannes Walcher's group at the Mathematisches Institut. Ingmar got his PhD at California Institute of Technology under supervision of Prof. Sergei Gukov.

• Introduction.- 2d sigma-models and DAHA.- 3d theories and modularity.-4 d theories, fivebranes, and M-theory.- Appendix.

Sergei Gukov is a professor of mathematics and theoretical physicist. Gukov graduated from Moscow Institute of Physics and Technology (MIPT) in Moscow, Russia before obtaining a doctorate in physics from Princeton University under the supervision of Edward Witten.[1]He held a Long-term Prize fellowship of Clay Mathematics Institute at Harvard University (2001-2006) and during 2007-2008 was a member of the school of mathematics at the Institute for Advanced Study, Princeton. Since 2007, he has been professor of mathematics and theoretical physics at the California Institute of Technology (Caltech). Starting 2010, Gukov was elected as an external scientific member of the Max Planck Society at the MPIM, Bonn.Peter Koroteev is a lecturer at the Department of Mathematics at Univeristy of California Berkeley. He got his PhD from University of Minnesota in 2012 under supervision of Prof. Arkady Vainshtein. Since then he worked as Postdoctoral Researcher at Perimeter Institute for Theoretical Physics 2012-2016 and as Visiting Assistant Professor at University of California, Davis, 2016-2019. Peter’s work is focused on the interplay between representation theory, algebraic geometry, and mathematical physics. He is also an instructor at Berkeley and Stanford Math Circles.Satoshi Nawata currently serves as an associate professor of physics at Fudan University. He received a Bachelor of Science at Tokyo Institute of Technology. He went on to achieve a PhD from the University of Wisconsin-Milwaukee. Following this, Satoshi Nawata had the opportunity to hold a postdoctoral position at several notable institutions, including the Tata Institute of Fundamental Research, Perimeter Institute for Theoretical Physics, NIKHEF, University of Warsaw, and the California Institute of Technology among others. He also spent time as a visiting researcher at the Max Planck Institute for Mathematics and IHES. Since 2016, Satoshi Nawata has been imparting knowledge and driving research as an Associate Professor at Fudan University.Du Pei is currently an associate professor of quantum mathematics at the University of Southern Denmark. He obtained his Ph.D. in theoretical physics in 2016 from Caltech, and has conducted research in theoretical physics and mathematical physics at Harvard University, MSRI, Caltech and Aarhus University. His main interests concern the geometric, algebraic and categorical properties of quantum field theory.Ingmar Saberi is a Senior Researcher at Ludwig-Maximilians-Universität München, Germany at Ilka Brunner's research group. Previously he was a postdoc at the University of Heidelberg, Germany in Johannes Walcher's group at the Mathematisches Institut. Ingmar got his PhD at California Institute of Technology under supervision of Prof. Sergei Gukov.

• Introduction.- 2d sigma-models and DAHA.- 3d theories and modularity.-4 d theories, fivebranes, and M-theory.- Appendix.

Sergei Gukov is a professor of mathematics and theoretical physicist. Gukov graduated from Moscow Institute of Physics and Technology (MIPT) in Moscow, Russia before obtaining a doctorate in physics from Princeton University under the supervision of Edward Witten.[1]He held a Long-term Prize fellowship of Clay Mathematics Institute at Harvard University (2001-2006) and during 2007-2008 was a member of the school of mathematics at the Institute for Advanced Study, Princeton. Since 2007, he has been professor of mathematics and theoretical physics at the California Institute of Technology (Caltech). Starting 2010, Gukov was elected as an external scientific member of the Max Planck Society at the MPIM, Bonn.Peter Koroteev is a lecturer at the Department of Mathematics at Univeristy of California Berkeley. He got his PhD from University of Minnesota in 2012 under supervision of Prof. Arkady Vainshtein. Since then he worked as Postdoctoral Researcher at Perimeter Institute for Theoretical Physics 2012-2016 and as Visiting Assistant Professor at University of California, Davis, 2016-2019. Peter’s work is focused on the interplay between representation theory, algebraic geometry, and mathematical physics. He is also an instructor at Berkeley and Stanford Math Circles.Satoshi Nawata currently serves as an associate professor of physics at Fudan University. He received a Bachelor of Science at Tokyo Institute of Technology. He went on to achieve a PhD from the University of Wisconsin-Milwaukee. Following this, Satoshi Nawata had the opportunity to hold a postdoctoral position at several notable institutions, including the Tata Institute of Fundamental Research, Perimeter Institute for Theoretical Physics, NIKHEF, University of Warsaw, and the California Institute of Technology among others. He also spent time as a visiting researcher at the Max Planck Institute for Mathematics and IHES. Since 2016, Satoshi Nawata has been imparting knowledge and driving research as an Associate Professor at Fudan University.Du Pei is currently an associate professor of quantum mathematics at the University of Southern Denmark. He obtained his Ph.D. in theoretical physics in 2016 from Caltech, and has conducted research in theoretical physics and mathematical physics at Harvard University, MSRI, Caltech and Aarhus University. His main interests concern the geometric, algebraic and categorical properties of quantum field theory.Ingmar Saberi is a Senior Researcher at Ludwig-Maximilians-Universität München, Germany at Ilka Brunner's research group. Previously he was a postdoc at the University of Heidelberg, Germany in Johannes Walcher's group at the Mathematisches Institut. Ingmar got his PhD at California Institute of Technology under supervision of Prof. Sergei Gukov.

• Introduction.- 2d sigma-models and DAHA.- 3d theories and modularity.-4 d theories, fivebranes, and M-theory.- Appendix.