During my graduate studies, I worked extensively with algebras and their corresponding geometric objects, as outlined below. The symbol “\( \leftrightarrow \)” represents a one-to-one correspondence between isomorphism classes.

Key Algebras and Their Geometric Correspondences

1. Lie Algebras and Lie Groups
   \( \{ \) Lie algebras \( \} \)
   \( \leftrightarrow \) \( \{ \) connected, simply connected Lie groups \( \} \)
2. Jordan Algebras and Symmetric Cones
   \( \{ \) Euclidean Jordan algebras (formally real Jordan algebras) \( \} \)
   \( \leftrightarrow \) \( \{ \) symmetric cones (homogeneous and self-dual open convex cones) \( \} \)
3. Clans and Homogeneous Domains/Cones
   • \( \{ \) clans (compact normal left symmetric algebras) \( \} \)
     \( \leftrightarrow \) \( \{ \) homogeneous (regular open convex) domains \( \} \)
   • \( \{ \) clans with a unit element \( \} \)
     \( \leftrightarrow \) \( \{ \) \(T\)-algebras \( \} \)
     \( \leftrightarrow \) \( \{ \) \(N\)-algebras \( \} \)
     \( \leftrightarrow \) \( \{ \) homogeneous (regular open convex) cones \( \} \)
4. Normal \(j\)-Algebras and Siegel Domains
   • \( \{ \) normal \(j\)-algebras obtained from clans \( \} \)
     \( \leftrightarrow \) \( \{ \) homogeneous tube domains over regular open convex cones (homogeneous Siegel domains of the first kind) \( \} \)
   • \( \{ \) normal \(j\)-algebras \( \} \)
     \( \leftrightarrow \) \( \{ \) homogeneous Siegel domains (of the second kind) \( \} \)

Properties of Algebras and Their Geometric Significance

• \(N\)-algebras are associative, while other algebras are non-associative.
• Jordan algebras are commutative, while other algebras are non-commutative.

Although these algebras are computationally challenging, they serve as powerful tools for studying geometric objects algebraically.

Applications in Manifolds and Analysis

In addition to my work in algebra, I conducted research in real and complex multivariable analysis on manifolds. This area plays a crucial role in deepening the relationship between algebra and geometry.