Summary on Recent Research in Algebraic Geometry

# Algebraic Geometry

Algebraic geometry is a branch of mathematics that studies solutions to algebraic equations and their geometric properties.

## Summary of Recent Research Papers

1. **Equiresidual Algebraic Geometry I: The Affine Theory (2019)** by Jean Barbet
   - This work generalizes classical algebraic geometry to non-algebraically closed fields.
   - It develops equiresidual algebraic geometry (EQAG), which works over any commutative field.
   - Key concepts include normic forms, a generalization of Hilbert's Nullstellensatz, and special kinds of algebras and radicals that connect to model-theoretic algebraic geometry.
   - The theory connects with scheme theory through a novel notion of a prime spectrum.

2. **Machine Learning Algebraic Geometry for Physics (2022)** by Jiakang Bao et al.
   - Reviews machine learning applications in algebraic geometry and physics.
   - Algebraic geometry problems can be reformulated as tensor mappings, enabling supervised and unsupervised learning techniques.
   - The paper addresses how AI can assist in doing mathematics through learning geometrical structures.

3. **Algorithms in Real Algebraic Geometry: A Survey (2014)** by Saugata Basu
   - Surveys algorithmic developments in real algebraic geometry, including effective quantifier elimination and computing topological invariants of semi-algebraic sets.
   - Discusses complexity aspects and computational hardness of problems in real algebraic geometry.
   - Covers numerical approaches such as semi-definite programming alongside exact algorithms.

## Relevant Papers

- Jean Barbet, "Equiresidual algebraic geometry I: The affine theory," 2019. [PDF](https://arxiv.org/pdf/1912.00347v3)
- Jiakang Bao, Yang-Hui He, Elli Heyes, Edward Hirst, "Machine Learning Algebraic Geometry for Physics," 2022. [PDF](https://arxiv.org/pdf/2204.10334v1)
- Saugata Basu, "Algorithms in Real Algebraic Geometry: A Survey," 2014. [PDF](https://arxiv.org/pdf/1409.1534v1)

This summary provides a glimpse into different facets of algebraic geometry research including foundational theory, computational techniques, and newer interdisciplinary approaches involving AI and machine learning.