Abstract: Existence of massive gauge bosons breaks down the local gauge invariance of Yang-Maills Lagrangians. In this lecture we shall look at one method to deal with this situation through the introduction of Higgs fields.

Abstract: Following Chapter 4 of the HoTT book, we continue our journey in the various characterizations of equivalences in Type Theory. We also show how Function Extensionality follows from the Univalence Axiom.

Abstract: Given a Calabi-Yau manifold $(M,\omega,\Omega)$ of complex dimension $n$, a Special Lagrangian submanifold (SL-submanifold) $L\subset M$ is a real $n$ dimensional submanifold calibrated by $\text{Re}\:\Omega$. These type of submanifolds are Lagrangian with respect to the symplectic structure $\omega$ and minimal with respect to the Calabi-Yau metric of the ambient space. Singular SL-submanifolds are particularly important as they play a fundamental role in mirror symmetry. In this talk I will survey the results obtained in the last years on deformation and moduli spaces of SL-submanifolds with conical singularities. Moreover I will introduce SL-submanifolds with higher order singularities (in particular edge singularities) and I will explain the approach used by the speaker to study moduli spaces of such type of singularities and some results obtained about the moduli space.