A well-known problem in algebraic geometry is to construct smooth projective Calabi-Yau varieties $Y$. In the smoothing approach, we construct first a degenerate (reducible) Calabi-Yau scheme $V$ by gluing pieces. Then we aim to find a family $f\colon X \to C$ with special fiber $X_0 = f^{-1}(0) \cong V$ and smooth general fiber $X_t = f^{-1}(t)$. In this talk, we see how infinitesimal logarithmic deformation theory solves the second step of this approach: the construction of a family out of a degenerate fiber $V$. This is achieved via the logarithmic Bogomolov-Tian-Todorov theorem as well as its variant for pairs of a log Calabi-Yau space $f_0\colon X_0 \to S_0$ and a line bundle $\mathcal{L}_0$ on $X_0$.

There is a well-known class of knots, called torus knots, which are those that can be drawn on a "standardly embedded" torus (one that separates the 3-sphere into two solid tori). A fairly natural property of other knots to consider is the genus necessary for that knot to be drawn on a standardly embedded genus g surface. This knot invariant has been studied under the name "embeddability". The goal of this talk is to introduce the invariant, look at some upper and lower bounds in terms of other invariants, and examine its behavior under connected sum.

The BNSR Invariant is a classical geometric invariant that encodes the finite generation of all coabelian subgroups of a given finitely generated group. The aim of this talk is to present a conjecture about the structure of the BNSR invariant of an Artin group and to present a new family in which the conjecture is true in terms of graph colorings.

We introduce ultrasolid modules, a variant of complete topological vector spaces. In this setting, we will prove some results in commutative algebra and apply them to the deformation of algebraic varieties in the language of derived algebraic geometry.