(This is a reading seminar given by the PhD Student Taeyoon Woo.) In this reading seminar, I will go through the construction of the un/stable motivic homotopy categories and their basic properties. A brief review of the topological side will help an overview. Nisnevich topology and simplicial homotopy theory of sheaves will be the main notions for presenting an ∞-topos of motivic spaces. The unstable motivic homotopy is then defined as A^1-localization, which is modeled by a Bousfield localization. Here I will sketch a proof of the purity theorem after some basic properties. If possible, I will discuss stabilization and the representability of algebraic K-theory.

(This is a seminar talk given by an undergraduate student, Mr. Rayhyun Kim, after his individual reading course studies.) In this seminar, I want to discuss about some basic notions of homological algebra that appears in category of sheaves. Many useful functors which are not exact, and the failure of exactness often contains important information. In the sheaves category this point of view naturally leads to sheaf cohomology. Going one step further, I will also discuss how sheaf cohomology is related to higher direct images. From this perspective, I would like to explain how inverse and direct image functors, their adjunction, and the derived functors of left exact functors fit together in a natural example. TBA

(This is a seminar talk given by Mr. Joon Song, an undergraduate student, after his individual reading studies.) In many situations one meets the same phenomenon: a short exact sequence gives rise to a long exact sequence in cohomology. However, the construction of such long exact sequences differs from case to case. In particular, the construction of the long exact sequence in sheaf theory is different from that of complexes. Is there a single framework where all long exact sequence arise in the same way? This leads to distinguished triangles which generalize the short exact sequence. In this presentation, I will introduce abelian categories, resolutions, and the derived category with their basic properties, and show how we can use the derived category briefly, through derived functor.

(This is a seminar talk given by an undergraduate student, Mr. Dohyun Kwon, reporting on his reading course studies.) This talk aims to provide a geometric analysis of hyperelliptic curves within the framework of Riemann surface theory. In the beginning, the fundamental tools in Riemann surface theory, such as the Riemann-Roch theorem, Serre duality and the Hurwitz formula will be introduced briefly. With these tools, we will first compute the genus of hyperelliptic curves and provide an explicit basis for the space of holomorphic 1-forms. Then, we will focus on the relation between the canonical map and hyperelliptic curves. The main goal is to examine the canonical map of the compact Riemann surface for cases of genus 2 or greater, and understand why it characterizes the hyperelliptic case when the canonical map fails to be an embedding. In particular, we will explicitly observe the canonical map in genus 2 and 3 cases.

(This is a reading seminar given by the PhD student Taeyoon Woo.) In this reading seminar, I will go through the construction of the un/stable motivic homotopy categories and their basic properties. A brief review of the topological side will help an overview. Nisnevich topology and simplicial homotopy theory of sheaves will be the main notions for presenting an ∞-topos of motivic spaces. The unstable motivic homotopy is then defined as A^1-localization, which is modeled by a Bousfield localization. Here I will sketch a proof of the purity theorem after some basic properties. If possible, I will discuss stabilization and the representability of algebraic K-theory.

(This is a reading seminar given by the PhD student Taeyoon Woo.) In this reading seminar, I will go through the construction of the un/stable motivic homotopy categories and their basic properties. A brief review of the topological side will help an overview. Nisnevich topology and simplicial homotopy theory of sheaves will be the main notions for presenting an ∞-topos of motivic spaces. The unstable motivic homotopy is then defined as A^1-localization, which is modeled by a Bousfield localization. Here I will sketch a proof of the purity theorem after some basic properties. If possible, I will discuss stabilization and the representability of algebraic K-theory.