Juliette Bruce

Experimental Talks in AG - Exploring ways to virtually communicate math

The recent shift from in-person to online talks provides us with an opportunity to explore new and interesting ways to communicate and learn things related to algebraic geometry. The goal of this seminar is to provide a space for speakers and audience members to do this exploration.

Each speaker will be given 50 minutes to create and lead an environment where topics related to algebraic geometry can be communicated and learned. How the speaker uses their time and creates such an environment is entirely up to them. They may try mixing pre-recorded and live aspects of a talk, they might try leading group work, who knows, it is entirely up to the speaker. The only request made of speakers is that they think critically about new ways to best use their time to promote the communication and learning of math. After the 50 minutes, a short group discussion will be held discussing how this form of online math communication went, and how it might be built upon. Everyone participating is asked to keep in mind that this is a place for experimentation.

This event is being organized by Juliette Bruce in association with the Algebraic Geometry Syndicate (AGS) discord server. Updates about the talks will be posted both here and on the AGS discord server. If you would like to join the AGS discord server please email juliette.bruce 'at' math.wisc.edu.

Date (2:00pm (CST) - 3:00pm (CST)) Speaker Title
May 27, 2020 Sachi Hashimoto - Boston University An obstruction to weak approximation on a Calabi-Yau threefold
June 3, 2020 Soumya Sankar - UW Madison Counting elliptic curves with a rational N-isogeny
June 10, 2020 #ShutDownSTEM N/A
June 17, 2020 Kristin DeVleming - UC San Diego Moduli spaces of plane curves
June 24, 2020 Takumi Murayama - Princeton University Every variety is birational to a weakly normal hypersurface
July 1, 2020 Andrew Kobin - UC Santa Cruz Zeta functions in number theory, algebraic geometry and beyond
July 8, 2020 Madeline Brandt - UC Berkeley Limits of Voronoi and Delaunay Cells
In this talk, we investigate the arithmetic structure of a class of Calabi-Yau threefolds. These threefolds were constructed over the complex numbers by Hosono and Takagi as a linear section of a double quintic symmetroid, and have a beautiful and simple story in the geometry of quadrics over the rational numbers. In forthcoming work with Honigs, Lamarche, and Vogt, we exhibit an obstruction to weak approximation on these threefolds. For the "experimental" nature of this seminar, we will conclude by working through a demonstration in cocalc. Attendees are asked to make cocalc accounts to participate fully; no prior coding experience necessary!

This talk can be accessed via Zoom. The password is the number of lines on a cubic surface.
The problem of counting elliptic curves over Q with a rational N isogeny can be rephrased as a question of counting rational points on the moduli stacks X_0(N). In this talk, I will discuss heights on projective varieties and a generalization to stacks of certain kinds, based on upcoming work of Ellenberg, Satriano and Zureick-Brown. We will then use this to count points on X_0(N) for low N. This is joint work with Brandon Boggess.

This talk can be accessed via Zoom. The password is the number of lines on a cubic surface.
Compactifying moduli spaces has been a fundamental problem in algebraic geometry that has been richly developed in the past 50 years. In that time, many different perspectives have been studied and these have resulted in many different compactifications. Starting from an audience discussion, we will consider the moduli space of plane curves of fixed degree, some potential compactifications, and how they fit together. Based on that discussion, I will mention a few of my favorite proper moduli spaces of plane curves, discuss their relationships, and pose some open questions.

This talk can be accessed via Zoom. The password is the number of lines on a cubic surface.
Classically, it is known that every variety is birational to a projective hypersurface. For curves and surfaces, this hypersurface can be taken to have at worst nodal and at worst ordinary singularities, respectively. We will prove that in arbitrary dimension, this hypersurface can be taken to be weakly normal, and for smooth projective varieties of dimension at most five, this hypersurface can be taken to have semi-log canonical singularities. These results are due to Roberts and Zaare-Nahandi and to Doherty in characteristic zero, respectively, and to Rankeya Datta and myself in positive characteristic. Attendees will be asked to do some concrete computations with polynomials.

This talk can be accessed via Zoom. The password is the number of lines on a cubic surface.
Participants will have a chance to fondly recall their favourite zeta functions. Together, we will discuss how different examples relate to/generalize each other. Then I will describe a general framework for studying zeta functions using decomposition spaces from homotopy theory.

This talk can be accessed via Zoom. The password is the number of lines on a cubic surface.
Voronoi diagrams of finite point sets partition space into regions. Each region contains all points which are nearest to one point in the finite point set. Voronoi diagrams (and their generalizations and variations) have been an object of interest for hundreds of years by mathematicians spanning many fields, and they have numerous applications across the sciences. Recently, Cifuentes, Ranestad, Sturmfels, and Weinstein defined Voronoi cells of varieties, in which the finite point set is replaced by a real algebraic variety. Each point y on the variety has a cell of points in the ambient space corresponding to those points which are closer to y than any other point on the variety. In this talk, we present the limiting behavior of Voronoi diagrams of finite sets, where the finite sets are sampled from the variety and the sample size increases. In this setting, we observe that many interesting features of the variety can be seen in a Voronoi Diagram, including its medial axis, curvatures, normals, reach, and singularities.

This talk can be accessed via Zoom. The password is the number of lines on a cubic surface.