In this talk, I will review the notion of stability appearing in number of different mathematical contexts, and discuss how they are linked, especially through mirror symmetry. Of particular interest are special Lagrangians and stable quiver representations.

For a given homogeneous polynomial, one can define the module of logarithmic derivations whose sheafification is called the logarithmic tangent sheaf on the projective space. Ever since the sheaf was originally introduced by P. Deligne to define the mixed Hodge structure on the complement of the corresponding divisor, there have been many studies on the object, relating other areas of mathematics to algebraic geometry. In this talk, we introduce two main problems, the Terao conjecture and Torelli problem, and see the current state of the art in terms of vector bundles. 

The SYZ conjecture suggests a folklore that ``Lagrangian multi-sections are mirror to holomorphic vector bundles". In this paper, we prove this folklore for Lagrangian multi-sections inside cotangent bundle of vector spaces, which are equivariantly mirror to complete toric varieties by the work of Fang-Liu-Treumann-Zaslow. We also introduce the Lagrangian realization problem, which asks whether one can construct an unobstructed Lagrangian multi-section with asymptotic condition prescribed by a tropical Lagrangian multi-section. We solve the realization problem for 2-fold tropical Lagrangian multi-sections over a complete 2-dimensional fan that satisfy the so called alternating slope condition. As an application, we show that every rank 2 toric vector bundle on the projective plane is mirror to a Lagrangian multi-section.

Assume that S is a finite set of points in n-dimensional space. In algebraic geometry, it is interesting to ask when the points of the set S impose independent linear conditions on polynomials of degree at most d. The most basic and useful is to take points in n-dimensional complex projective space and to ask about homogeneous forms of degree d instead of polynomials of degree at most d. 

In commutative algebra, a unique factorization domain is a commutative ring in which every element can be uniquely written as a product of prime elements, analogous to the fundamental theorem of arithmetic for the integers. Most rings familiar from elementary mathematics are UFDs: the integers, the polynomial rings over a field, the ring of functions in a fixed number of complex variables holomorphic at the origin etc. However, most factor rings of a polynomial ring are not UFDs. An affine algebraic variety is called factorial if its coordinate ring is UFD. For a projective algebraic variety, one can define the factoriality in a similar way. For a 3-fold with mild singularities, the factoriality problem was investigated by Clemens. He showed that the factoriality of many singular 3-folds can be expressed in terms of the number of independent linear conditions that their singular points impose on the homogeneous forms of certain degree. We plan to investigate how the factoriality of singular 3-folds depends on the number of singular points. This problem can be studied by methods of commutative algebra, topology, differential geometry and algebraic geometry.

We introduce Legendrian links, and their invariants of DGA type and sheaf theoretic type. We discuss the relationship between Lagrangian fillings for Legendrian links and the above invariants. We also review how the cluster structure appears on the space of Lagrangian fillings if time permits. 

Sums of squares are a certificate of nonnegativity of real polynomials because any sums of squares is obviously nonnegative. Hilbert classified the cases that any nonnegative (homogeneous) polynomials is sum of squares in terms of the number of variables and the degree. For example, any bivariate homogeneous polynomials (i.e. binary forms) of degree 2d is a sum of squares of degree d binary forms. One may ask whether any nonnegative binary forms is a sum of squares if we don't allow a monomial. 

In this talk, I will introduce the Hankel index that quantifies the structural difference between the set of sums of squares and nonnegative polynomials on variety. As a result, we can answer the question by investigating the Hankel index on corresponding variety. (If time is allowed, we will introduce and discuss some geometric questions related to Hankel index and rank of points with respect to varieties.)

It is a fundamental question in symplectic topology in how many ways a given contact manifold can be written as the boundary of a symplectic manifold. In this talk, we briefly introduce some classification results on symplectic fillings of contact manifolds in high dimensions. In particular we are interested in uniqueness results of symplectic fillings in terms of J-holomorphic curves and Floer theory.

In this talk, we generalize the notion of rational singularities for any reflexive sheaf of rank 1 and prove generalizations of standard facts about rational singularities. Moreover, we introduce the notion of (B q+1 ) as a dual notion of well- known Serre’s notion of (S q+1 ) and prove a theorem about q-birational morphism.