# Mathematics Colloquia and Seminars

Lagrangians are important submanifolds of symplectic manifolds, and Legendrians are important submanifolds of contact manifolds. We will be studying surfaces in the $4$-ball that satisfy additional geometric conditions (they are exact Lagrangian) and which bound smooth links in $S^3$ that also satisfy additional geometric conditions (they are Legendrian links). Such a surface is called a filling of the link. In the last two years, Casals and Gao found Legendrians that have infinitely many distinct exact Lagrangian fillings that are smoothly isotopic but not Hamiltonian isotopic. Given an infinite family of fillings of a Legendrian, there are now various tools to show whether these fillings are Hamiltonian isotopic or not. One of these tools is the Legendrian contact homology differential graded algebra. In particular, exact embedded Lagrangian fillings correspond to augmentations of this algebra. We will discuss how "non-equivalent" augmentations correspond to non-Hamiltonian isotopic fillings, allowing us to translate this geometric question into a more algebraic question. We will provide explicit examples of Legendrian links with infinitely many fillings distinguished with augmentations. We will also discuss recent work in progress with Casals where we distinguish fillings using Laurent polynomials over $Z_2$ obtained from augmentations. In particular, we make use of invariants of Laurent polynomials such as the volumes of Newton polytopes.