In these talks, I will discuss various aspects of Hodge polynomials of non-algebraic complex manifolds, especially those polynomials of nilmanifolds, solvmanifolds, (quasi-)Hopf, (quasi-)Calabi-Yau and LVMB manifolds. The last part of the talks are based on several joint works (some in progress) with Ludmil Katzarkov, Ernesto Lupercio and Laurent Meersseman.

The Kostant-Weil integrality theorem states that a closed 2-form can be realized as the  curvature form of a principal circle bundle with connection if and only if it is integral. Moreover, the first cohomology group with U(1) coefficients acts freely and transitively on the set of equivalence classes of principal circle bundles with connection whose curvature is the given closed 2-form. We will review Bryslinki’s proof technique of using a short exact sequence of truncated complexes of sheaves to (re)prove this result. Then we will show how this proof can be modified to (re)prove the metaplectic-c prequiantizability criterion.

The Cayley transform provides a well known biholomorphism from the upper half plane to the unit disk. In this talk I will revisit this transformation as an octahedral symmetry, and show how the complex Lagrangian Grassmannian decomposes into orbits of the real linear symplectic group.

The polygon space is a moduli space of closed linkages formed by vectors in n-dimensional Euclidean space. The length map l  on this space, which records the side lengths of polygons, has as its image a convex polytope — a rectified simplex, which is obtained by truncating a regular simplex. The side lengths of degenerate polygons form walls in the image of the length map, creating a chamber structure on it. For n=2,  the connectivity of the fibers of this map has been studied in relation to the chamber containing the side lengths (Kapovich and Millson, 1995). This talk explores the chamber structure of the image of the length map, with a focus on its relationship with the rectified simplex and the associated polygons.

The discovery of five exceptional Lie algebras was an unexpected finding from the Killing-Cartan classification of simple Lie algebras. Although at first glance they seems very awkward and unorganized, they can be constructed in a uniform manner from octonions. In particular, the 14-dimensional exceptional complex Lie group G2 can be realized as the automorphism group of the complexified octonions. I will discuss geometric properties of several interesting algebraic varieties acted upon by G2, which include the homogeneous G2-varieties, the Cayley Grassmannian, and the double Cayley Grassmannian.

Many problems in the real world can be translated (or approximated) to problems concerning polynomial equations over the real numbers. For instance, understanding metric properties (length, distance, volume, angle, etc.) is important and practical. In this talk, we will observe two problems, namely, Euclidean distance problem and Nash equilibrium problem, and see how these problems can be studied via algebraic geometry.

Scalar curvature is one of the most fundamental curvature invariants of a Riemannian manifold, measuring the average of sectional curvatures at each point. Minimal submanifolds, in contrast, arise as critical points of the area functional and serve as natural probes of the geometry of the ambient manifold. In this talk, I will explore the interplay between scalar curvature and minimal hypersurfaces. After reviewing the basic definitions and several classical results, I will highlight central theorems due to Schoen–Yau, Gromov–Lawson, and Llarull. These works demonstrate how minimal hypersurface techniques yield topological obstructions to the existence of metrics with positive scalar curvature, resolving in particular the Geroch conjecture for tori and establishing rigidity phenomena. Finally, I will introduce a generalization of Llarull’s theorem in the spectral setting, extending the connection between scalar curvature, analysis, and geometry. This is joint work with Xiaoxiang Chai and Xueyuan Wan.