### Algebraic and semi-algebraic invariants on quadrics

- Series
- Dissertation Defense
- Time
- Friday, July 22, 2022 - 08:30 for 2 hours
- Location
- Skiles 006 and Zoom meeting (https://gatech.zoom.us/j/96755126860)
- Speaker
- Jaewoo Jung – Georgia Institute of Technology – jjung325@gatech.edu

Dissertation defense information

Date and Time: July 22, 2022, 08:30 am - 10:30 am (EST)

Location:

- Skiles 006 (In-person)
- Zoom meeting (Online): https://gatech.zoom.us/j/96755126860

Summary

This dissertation consists of two topics concerning algebraic and semi-algebraic invariants on quadrics.

The ranks of the minimal graded free resolution of square-free quadratic monomial ideals can be investigated combinatorially. We study the bounds on the algebraic invariant, Castelnuovo-Mumford regularity, of the quadratic ideals in terms of properties on the corresponding simple graphs. Our main theorem is the graph decomposition theorem that provides a bound on the regularity of a quadratic monomial ideal. By combining the main theorem with results in structural graph theory, we proved, improved, and generalized many of the known bounds on the regularity of square-free quadratic monomial ideals.

The Hankel index of a real variety is a semi-algebraic invariant that quantifies the (structural) difference between nonnegative quadrics and sums of squares on the variety. This project is motivated by an intriguing (lower) bound of the Hankel index of a variety by an algebraic invariant, the Green-Lazarsfeld index, of the variety. We study the Hankel index of the image of the projection of rational normal curves away from a point. As a result, we found a new rank of the center of the projection which detects the Hankel index of the rational curves. It turns out that the rational curves are the first class of examples that the lower bound of the Hankel index by the Green-Lazarsfeld index is strict.

Advisor: Dr. Grigoriy Blekherman, School of Mathematics, Georgia Institute of Technology

Committee:

- Dr. Matthew Baker, School of Mathematics, Georgia Institute of Technology
- Dr. Anton Leykin, School of Mathematics, Georgia Institute of Technology
- Dr. Rainer Sinn, Institute of Mathematics, Universität Leipzig
- Dr. Josephine Yu, School of Mathematics, Georgia Institute of Technology