Let C be an algebraic curve over Q, i.e., a 1-dimensional complex manifold defined by polynomial equations with rational coefficients.   A celebrated result of Faltings implies that all algebraic points on C come in families of bounded degree, with finitely many exceptions.  These exceptions are known as isolated points. We explore how these isolated points behave in families of curves and deduce consequences for the arithmetic of elliptic curves.  We also explore how the beginnings of how the geometry of the  parameterized points can be used to explore the arithmetic of the curve  This talk is on joint work with A. Bourdon, Ö. Ejder, Y. Liu, and F. Odumodu, with I. Vogt,  and with work in progress with I. Balçik, S. Chan, and Y. Liu.

Free resolutions, introduced by David Hilbert over a century ago, provide us with a tool to study relations between polynomials via a sequence of free modules, or vector spaces. The numerical information provided by free resolutions is invaluable in the study of solution sets of those polynomials. The study of free resolutions is a central problem is commutative algebra, and depending on the type of polynomials in question, different tools are applied to study or construct free resolutions or find bounds on their numerical invariants. Some examples are, combinatorics, geometry, topology, homological and computational algebra. This talk will give a gentle introduction to resolutions of ideals generated by monomials, and the methods of discrete topology used to study them.  Our focus will be taking powers of these ideals, and the combinatorics associated to these powers. We will report on joint work with (subsets of): Trung Chau, Susan M. Cooper, Art Duval, Sabine El Khoury, Tai Ha, Thiago Holleben, Takayuki Hibi, Sarah Mayes-Tang, Susan Morey, Liana M. Sega, Sandra Spiroff.