Abstract:
In these two talks, we first discuss the construction of V. Voevodsky's motivic categories. They carry a natural t-structure, but unfortunately this t-structure does not satisfy the desired properties. Therefore, people produced some alternatives. Among those, one can mention the slice filtration constructed by A. Huber and B. Kahn. After discussing some results in the theory of slice filtration, we will talk about the motive associated to certain moduli spaces (and their local models) that arise in the arithmetic of function fìelds.

Abstract:
In this talk, we will review some of the basic facts on the Moduli space of genus g>=2 algebraic curves and consider the problem of counting rational points on the Moduli space of genus two more specifically. Then, we speak about the weighted heights of algebraic varieties in weighted projective spaces and describe how weighted heights simplify the problem of counting rational points on the Moduli space of genus two.

Abstract:
In general, Hilbert schemes parameterize closed subschemes of a given scheme and represent a functor originally introduced by Grothendieck as a special case of a more general object known as the quote functor. In this series of lectures, we will first provide an overview of the construction of Hilbert schemes and then focus on the more specific case of the Hilbert scheme of points on the affine plane $A^2$, denoted as HIlb$(A^2)^ n$.
Even in the case of the affine plane $A^2$, the Hilbert scheme of points possesses a rich geometric structure with intriguing connections and applications to various fields, including combinatorics, representation theory, and even physics. For instance, M. Haiman utilized the geometry of this particular Hilbert scheme to establish the Macdonald n!-conjecture for symmetric functions. If time permits, we will also present some results and theorems related to the Hilbert scheme of points on a projective smooth surface.

Abstract:
In this series of talks, we will explore modular curves through various perspectives, highlighting their rich structures and application.
We begin by defining modular curves as Riemann surfaces, where they naturally serve as the domain for modular forms. This leads to a discussion of modular forms and Hecke operators. Next, we introduce modular curves as moduli spaces of elliptic curves with level structures, providing an algebro-geometric viewpoint that connects them to moduli theory and making a deep connection between the geometry of the modular curves and the arithmetic of elliptic curves. Finally, we approach modular curves from the adelic viewpoint, which provides a framework to study their global properties and connections to automorphic forms. Along the way, we will discuss the rational and integral models of modular curves, and Hecke correspondences. By examining these different perspectives, we aim to provide a multifaceted understanding of modular curves, bridging complex analysis, algebraic geometry, and number theory.

Abstract:
Classifying and understanding the structure of vector  bundles  on a (smooth) projective variety is an old and important problem in algebraic geometry. Recently, a class of vector bundles on smooth projective varieties, so called Ulrich bundles, have been of particular interest and investigated or studied  over some  nicest smooth projective varieties by a handful of authors including Eisenbud, Schreyer, Beauville, and others. In this expository talk, I will discuss Ulrich bundles and their existing problem as long as time and my knowledge permits. The origin of  Ulrich bundles comes from commutative algebra, therefore I will  begin by discussing  their relevant  importance to some conjectures in commutative algebra.