Georgia Institute of Technology College of Sciences

Primary decomposition is an indispensable tool in commutative algebra, both theoretically and computationally in practice. While primary decomposition of ideals is ubiquitous, the case for general modules is less well-known. I will give a comprehensive exposition of primary decomposition for modules, starting with a gentle review of practical symbolic algorithms, leading up to recent developments including differential primary decomposition and numerical primary decomposition. Based on joint works with Yairon Cid-Ruiz, Marc Harkonen, Robert Krone, and Anton Leykin.

In the talk, I will present an equidistribution result for families of (non-degenerate) subvarieties in a (general) family of abelian varieties. This extends a result of DeMarco and Mavraki for curves in fibered products of elliptic surfaces. Using this result, one can deduce a uniform version of the classical Bogomolov conjecture for curves embedded in their Jacobians, namely that the number of torsion points lying on them is uniformly bounded in the genus of the curve. This has been previously only known in few cases by work of David--Philippon and DeMarco--Krieger--Ye. Finally, one can obtain a rather uniform version of the Mordell-Lang conjecture as well by complementing a result of Dimitrov--Gao--Habegger: The number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian. Again, this was previously known only under additional assumptions (Stoll, Katz--Rabinoff--Zureick-Brown).

Algebraic geometry (AG) is a major generalization of linear algebra which is fairly influential in mathematics. Since the 1980's with the development of computer algebra systems like Mathematica, AG has been leveraged in areas of STEM as diverse as statistics, robotic kinematics, computer science/geometric modeling, and mirror symmetry. Part one of my talk will be a brief introduction to AG, to two notions of taking powers of ideals (regular vs symbolic) in Noetherian commutative rings, and to the ideal containment problem that I study in my thesis. Part two of my talk will focus on stating the main results of my thesis in a user-ready form, giving a "comical" example or two of how to use them. At the risk of sounding like Paul Rudd in Ant-Man, I hope this talk will be awesome.

BlueJeans link: https://bluejeans.com/851535338

In the last years, connections between graphs and Riemann surfaces have been
discovered on several different levels. In particular, graphs are closely related
to singular Riemann surfaces and the boundary in the Deligne–Mumford com-
pactification of moduli spaces. Moreover, in both settings there is a notion of a
canonical measure (the Arakelov–Bergman and Zhang measures) which reflects
crucial geometric information.
In this talk, we focus on the following question: what is the limit of the canon-
ical measures along a family of Riemann surfaces? Combining the canonical
measures on Riemann surfaces and metric graphs, we obtain a full description
and a new compactification of the moduli space of Riemann surfaces in terms
of hybrid curves.

Based on joint work with Omid Amini (École polytechnique).

BlueJeans link: https://bluejeans.com/476849994