Non-commutative differential geometry (NCG) generalizes Riemannian geometry and provides a framework within which many interesting Yang-Mills theories coupled to Einstein-Hilbert gravity may be elegantly reformulated. One physical value of this reformulation is that it is subject to a number of geometric constraints which severely limit the allowable physical models that may be constructed. Surprisingly the standard model of particle physics coupled to Einstein-Hilbert gravity is one such model, and so it might be possible to use the rigidity of the NCG formalism to single out possible extensions to explore. In this talk I will provide a first introduction to the NCG standard particle model from the perspective of a particle model builder, focusing on the construction itself rather than its phenomenology.

It is a central problem concerning division algebras to determine which finite dimensional division algebras are crossed products, that is, admit a maximal subfield that is Galois over the center. We shall discuss a simple crossed product criterion for tamely ramified valued division algebras, and apply it to describe the location of noncrossed products in the Brauer group of Henselian fields.

Based on joint work with Timo Hanke and Adrian Wadsworth.

We will continue the discussion of path induction, and use it to produce further operations on paths, such as applying functions to paths, and the transport between fibers. Then, we will discuss the notions of homotopy between functions and equivalence between types.

We will give an introduction to the ideas of the Langlands program, through the classical problems in algebraic number theory, elliptic curves and L-functions.

In this talk we give a selected overview of the major developments of the Langlands program so far, and suggest some questions as speculations on the future possibilities of the program.