부산대학교  기하학, 위상수학  세미나
(PNU Geometry and Topology seminar)

Title. The face numbers of lattice polytopes and toric varieties.
Abstract. The fundamental theorem in toric geometry establishes a one-to-one correspondence between the category of projective toric varieties and the category of lattice polytopes. This raises the question of how to read off the topological and geometric information of a toric variety from the corresponding lattice polytope. In this talk, we will look at the cohomology of a certain family of toric varieties and how it relates to the face numbers of the associated polytope. This is based on the joint work with S. Park. 

Title. Almost complex torus manifolds - graphs and problem of Petrie type
Abstract. Let a $k$-dimensional torus $T^k$ act on a $2n$-dimensional compact connected almost complex manifold $M$ with isolated fixed points. We show that there exists a (directed labeled) multigraph that encodes the fixed point data of $M$. If in addition $k=n$, i.e., $M$ is an almost complex torus manifold, the multigraph is a graph; it has no multiple edges. Petrie's conjecture asserts that if a homotopy $\mathbb{CP}^n$ admits a non-trivial circle action, its Pontryagin class agrees with that of $\mathbb{CP}^n$. Petrie proved this conjecture if instead it admits a $T^n$-action. We prove that if a $2n$-dimensional almost complex torus manifold $M$ only shares the Euler number with the complex projective space $\mathbb{CP}^n$, an associated graph agrees with that of a linear $T^n$-action on $\mathbb{CP}^n$; consequently $M$ has the same weights at the fixed points, Chern numbers, equivariant cobordism class, Hirzebruch $\chi_y$-genus, Todd genus, and signature as $\mathbb{CP}^n$. If furthermore $M$ is equivariantly formal, the equivariant cohomology and the Chern classes of $M$ and $\mathbb{CP}^n$ also agree. 

Title. On a generalization of Forelli's theorem
Abstract. One of important issues in several complex variables is to establish criteria to determine holomorphicity of complex functions. Forelli's analyticity theorem (1977), which is perhaps second only to Hartogs' theorem (1906), has been generalized to various directions until recently. In this talk, we introduce a recent work (Cho, Kim, 2021) on the topic which generalizes the previous works of Joo, Kim, Schmalz (2013), Chirka (2006), as well as the classical Forelli's theorem. We shall also introduce interesting history behind the work.

Title. Myers-type theorems for Bakry-Emery Ricci tensor
Abstract. One of the natural and important topics in Riemannian geometry is the relation between curvature and topology. Examples of good explanations of the relationship between curvature and topology include comparison geometry, Cheeger-Gromoll splitting theorems, Myers theorems, etc. In particular, Myers theorems show that positive Ricci curvature has strong topological consequences. In this talk, we focus on Myers theorems. We review the classical Myers theorem and related theorems for the Ricci curvature and we also look at the results in the Bakry-Emery Ricci tensor.

Title. Cylinders in Del Pezzo Surfaces
Abstract. An open subset in a normal projective variety is called a cylinder if it is isomorphic to A^1 × Z for some affine variety Z . With polarity condition for given ample divisor, such A^1-rulings of the variety has deep connections to group actions of the corresponding generalized cone. Indeed, certain cylinders on underlying projective variety ensures a nontrivial unipotent group actions on the cone. Even more, a cylindrical covering of the variety implies an infinitely transitive group action generated by all one-parameter unipotent subgroups on the cone. As an interesting algebro-geometric correspondence, there have been many attempts to find any construction of ample polar cylinders in del Pezzo surfaces which is 2-dimensional Fano varieties. In this talk, we present some remarkable results for del Pezzo surfaces known so far and introduce my recent research in this field. We work over an algebraically closed field of characteristic zero.

Title. Polygon spaces along polytopes
Abstract. Polygons and related spaces are classical subjects in geometry and topology. In this talk, we have an introduction to geometry and topology of the spaces of polygons and characterization of chamber structures along the A-type polytopes and chambers in them. We also discuss the relations between polygon spaces and Grassmannian spaces of real, complex and quaternions. This is joint work with Eunjeong Lee.

Title. Strichartz estimates and the well-posedness of nonlinear Schrodinger equation.
Abstract. Strichartz estimate is known as one of the fundamental tools to study the nonlinear dispersive PDEs 

and has been fruitfully used to prove the well-posedness of their Cauchy problem. 

In this talk, we study the known results of the classical Stricharz estimates for the Schrodinger equation.

We then discuss the weighted Strichartz estimates introduced in our recent work with a spatial power weight.

In an application of these weighted estimates, we provide the well-posedness result for the inhomogeneous nonlinear Schrodinger equation (INLS) in the critical case which was left unsolved.

Title. Arithmetic of character variety of reductive groups
Abstract. Counting the number of points on a variety is a historical method for investing the variety, for example, Weil conjecture. Nowadays, it is known that we can get a polynomial about the variety via point counting, which is called the $E$-polynomial. Moreover, from this E-polynomial, we can get some arithmetic-geometric information on the variety, such as dimension, the number of irreducible components and Euler characteristic. In this talk, we will consider a specific variety, which is called character variety associated to the fundamental group of a surface. In short, we will consider some punctures on the surface with regular semisimple or regular unipotent monodromy. This variety plays a crucial role in diverse areas of mathematics, including non-abelian Hodge theory, geometric Langlands program and mathematical physics. Furthermore, we will discuss the complex representation theory of finite groups to compute the number of points in such variety.

Title. Exceptional bundles and singularities
Abstract. Exceptional sheaves are one of building blocks of derived categories as they generate semiorthogonal components. For surfaces, it has been reported that certain exceptional vector bundles correspond to singularities in degenerations. Such a phenomenon is observed firstly by Hacking, who establishes the correspondence to Wahl degenerations. I will discuss a generalization of Hacking's correspondence to Q-Gorenstein degenerations, and also introduce a very recent result of Tevelev and Urzua who consider a generalization to arbitrary cyclic quotient singularity degenerations.

Title. Geometry of the Cayley Grassmannian
Abstract. The Cayley Grassmannian is a smooth projective symmetric variety parametrizing four-dimensional subalgebras of the complexified octonions. In this talk, we discuss some interesting geometric properties of this eight-dimensional Fano manifold. First, we present its algebraic moment polytope and compute the barycenter of moment polytope with respect to the Duistermaat–Heckman measure. This implies that the Cayley Grassmannian admits a Kaehler-Einstein metric. Next, using a description of the Cayley Grassmannian as the zero locus of a general global section of the 3rd exterior power of the dual universal subbundle on Gr(4, 7), we prove the infinitesimal deformation rigidity of complex structures and give a construction of Ulrich vector bundles on it.

Title. Maurer-Cartan deformation of a Lagrangian  
Abstract. The Maurer-Cartan algebra of a Lagrangian is the algebra that encodes the deformation of its Floer complex as an A-infinity algebra. I will give a convenient description of the Maurer-Cartan algebra through a natural homological algebra language, and relate it with (a version of) Koszul duality for the Floer complex. It helps us to obtain the mirror-symmetry interpretation for the Maurer-Cartan deformation and its locality in SYZ situation. Namely, the Maurer-Cartan algebra provides a neighborhood of the point mirror to the Lagrangian, which varies in size depending on geometric types of Floer generators involved in the deformation.