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Course Description

Dear all, we are starting the zoom course, the aim of this course is to explain modern physics for mathematicians and provide physicists with adequate mathematical tools for description of the Standard model and gravity in a unified way. In this semester we will discuss classical theory, and we will come to quantum theory in the next semester.

In the main line physics for mathematicians we will cover (in a relatively nonstandard way) the following topics.

1. Modern view on Newton’s mechanics, missed zeroth law and renormalization.

2. Differential geometry arising from motion of machines and least action principle.

3. Motion of particles in a curved space-time, conserved quantity and Einstein’s formula. Why it is impossible to move faster than light.

4. Supermanifolds and homological vector fields, de Rham differential graded algebra of differential forms and cohomology.

5. Berezin integral and Hodge *-operation as an odd Fourier transform, Laplacians and quadratic actions for p-forms, abelian gauge symmetry and propagation of waves.

6. Electromagnetism as a theory of 1-form, Maxwell’s equations and generalization of electromagnetic duality to the theory of p-forms in d-dimensional spaces. Why electromagnetism for d=3 is equivalent to ether theory.

7.Unified description of gravity and Standard model as SU(3)xSU(2)xU(1)x Spin(3,1) gauge theory. Quarks, leptons, Higgs fields, gauge fields and graviquarks (Einstein-Cartan theory of gravity)

8. Spin of fermions, Dirac actions in different dimensions, coupling to gravity and to Higgs fields.

9. Mesons made out of quarks and metric as a gravimeson.

10. Hilbert-Einstein versus Einstein-Cartan theories of gravity, tiny difference due to fermions.

11. Higgs phenomena in Standard model, Z and W bosons, electromagnetism as a stabilizer of Higgs field. Computation of charges of quarks and leptons.

12. Gravihiggs phenomena due to graviquark. What actually spins in electron.

13. Baryons (proton and neutron) made out of quarks, cosmological constant term as a baryon.

Mathematics for physicists would support physics for mathematicians program and include:

External and internal differential geometry, manifolds, tangent and cotangent bundles, vector fields, metric and differential forms, homology and cohomology, real Hodge theory, Lie groups and Lie algebras, principal G-bundles, invariant and horizontal forms, connection and curvature, torsion and Levi-Civita connection, Hilbert action and Hilbert-Einstein equations, Clifford algebra and its representation, Spinor and Fundamental representations of Spin(d) and Spin(d,1) group, particular case of Spin(3)=SU(2) and Spin(3,1), and SO(3)=SU(2)/Z_2, representation theory for SU(2) and elements of representation theory for SU(3).

We will have two lectures every week and two seminars.

Welcome to our program!