Let $X$ be a Calabi-Yau manifold of dimension $2n$. A special Lagrangian submanifold (SL-submanifold) is an oriented, embedded real $n$-submanifold $\psi: N \rightarrow X$ calibrated by the real part of the holomorphic volume form $\Omega$. In this talk I will explain the basics of SL-submanifolds when $X= \mathbb{C}^N$ and from the point of view of minimal submanifolds and calibrated geometry (I will explain what is a minimal submanifold and calibrations). Just basic background in differential geometry is enough to follow the talk.

I will give an overview of the field of Homotopy Type Theory. This relatively new field of mathematics is based on a realization that the formal logical system of dependent type theory can be interpreted in various homotopy-theoretic settings. After briefly discussing type theory, I will sketch the idea of its homotopical interpretation and its connection to higher category theory. In the last part of the talk, I will highlight some recent results.

Homological algebra of vector spaces is well understood. In functional analysis, many infinite dimensional vector spaces also contain analysis information. A diffeological vector space is a vector space with a compatible (generalized) smooth structure. In this talk, I will present a non-trivial example from functional analysis under the framework of diffeological vector spaces, see how the generalized smooth structure can be used to generalize a known result from analysis, as a motivation for the development of homological algebra of diffeological vector spaces. Then I will talk about the similarity and difference between this homological algebra and the homological algebra of R-modules. If time permits, some open questions will be discussed at the end.