# Graduate Workshop in Algebraic Geometry

## for Women and Mathematicians of Minority Genders

### A workshop held at Harvard and MIT on February 17-18 2018

## Speakers

Jennifer Balakrishnan, Boston University

Melody Chan, Brown University

Angela Gibney, Rutgers University

Brooke Ullery, Harvard University

## Components

Each day of the workshop involved

- Morning session: two 90-minute mini-courses, which will be accessible to graduate students who have taken an introductory class in algebraic geometry.
- Lunch session: 5-minute math talks by participants on a topic they find interesting. We encourage all participants to give a talk, even if they haven't begun research yet.
- Afternoon session: concurrent TA-led sessions to work on concrete examples and open-ended problems contributed by the speakers.

On Saturday evening there was also an informal professional development discussion over dinner.

## Minicourses

### Jennifer Balakrishnan: Explicit Coleman integration for curves

The Coleman integral is a *p*-adic line integral that allows for the computation of a number of interesting invariants in arithmetic geometry, including *p*-adic heights and regulators. Coleman integration also plays a key role in techniques for computing rational points on curves and torsion points on their Jacobians. I will present some of these applications and then describe how to carry out explicit Coleman integration on curves, based on recent joint work with Jan Tuitman. I will also give numerical examples using our Coleman integration algorithm to compute rational points on curves, as in the Chabauty-Coleman method.

### Melody Chan: Tropical and algebraic curves and their moduli spaces

This will be a mini-course on tropical and algebraic curves and their moduli spaces, with an emphasis on the relationship between the two. Tropical geometry is a modern degeneration technique in algebraic geometry; it is a degeneration in which algebraic objects are replaced by entirely combinatorial ones. The theory of tropical curves is currently the one that has been fleshed out the most; we will use it as a window to look at applications of tropical geometry to enumerative and topological questions on algebraic curves and their moduli.

### Angela Gibney: Vector bundles of conformal blocks on the moduli space of curves

The moduli space of stable n-pointed curves of genus g has played an important role in the literature: as a means of learning about smooth curves and their degenerations, as a model for moduli spaces generally, and as a test variety for developing theories in algebraic geometry. Conformal blocks are invariants of a curve attached to a Lie group. In particular, vector spaces of conformal blocks for G at any stable curve C can be identified with global sections of an ample line bundle on the moduli stack of G-bundles on C. These vector spaces fit together to form vector bundles, and we can use these bundles as a tool to study the moduli space of curves.

During my lecture I'll introduce the moduli space of curves and define these bundles. In the afternoon session, I'll illustrate our interest in these bundles on the moduli space through a set of problems about cones of positive cycles on the moduli space of curves.

### Brooke Ullery: Measures of irrationality for algebraic varieties

The gonality of a smooth projective curve is the smallest degree of a map from the curve to the projective line. There are a few different definitions that attempt to generalize the notion of gonality to higher dimensional varieties. The intuition is that the higher these numbers, the further the variety is from being rational. We will discuss these measures of irrationality and various methods of calculating and bounding them. We’ll mainly focus on the examples of hypersurfaces and, more generally, complete intersections in projective space.

## Funding and Registration

There will be a limited amount of funding for participants outside the greater Boston area. We expect to be able to provide partial travel support for some participants from within the United States. Accommodation for funded participants will be arranged by the organizers for the nights of Friday, February 16, and Saturday, February 17.

Funding will be allotted with preference to graduate students in the early years.

We would like to especially solicit applications for funding from mathematicians from under-represented groups and students at smaller schools.

The deadlines for funding applications and registration have passed.

## Schedule

### Friday, February 16

- Dessert social for early arrivals (MIT 2-290): 7-8:30pm

### Saturday, February 17

- Breakfast (MIT 2-449): 8-8:30am
- Jennifer Balakrishnan, part 1 (MIT 4-237): 8:30-9:15
- Jennifer Balakrishnan, part 2 (MIT 4-237): 9:30-10:15
- Angela Gibney, part 1 (MIT 4-237): 10:30-11:15
- Angela Gibney, part 2 (MIT 4-237): 11:30-12:15
- Lunch with participant talks (MIT 2-290): 12:30-2pm
- Group work (4 parallel): 2:30-4pm
- Coffee break (MIT 2-449): 4-4:30pm
- Group work (4 parallel): 4:30-6pm
- Break: 6-6:30pm
- Dinner (MIT 2-290): 6:30-8pm

### Sunday, February 18

- Breakfast: 8-8:30am
- Melody Chan, part 1 (Science Center Hall A): 8:30-9:15
- Melody Chan, part 2 (Science Center Hall A): 9:30-10:15
- Brooke Ullery, part 1 (Science Center Hall A): 10:30-11:15
- Brooke Ullery, part 2 (Science Center Hall A): 11:30-12:15
- Lunch with participant talks: 12:15-1:15pm
- Group work (4 parallel): 1:30-4:30pm

## Practical Information

Activities on Saturday February 17th will be held in the Simons Building (building 2) and the adjacent building 4 at MIT; a room number w-xyz indicates that the room is in building w and on floor x (so for example 2-449 is in building 2 on the 4th floor). See this page for information on getting to the Simons Building, as well as the interactive MIT campus map here. Activities on Sunday February 18th will be held in the Science Center at Harvard University. See this page for information on getting to the Science Center. Hall A is on the ground floor of the Science Center.

Accommodations for funded participants will be arranged by the organizers after funding decisions have been made. If you are arranging your own accommodations for the workshop, we suggest looking into Hosteling International Boston (bed in a shared room), the Porter Square Hotel, and Airbnb for affordable options.

Organized by Rohini Ramadas and Isabel Vogt

Faculty Sponsors: Bjorn Poonen and Gigliola Staffilani

*Photo by Lstrong2k at English Wikipedia [CC BY-SA 1.0]*