In this talk, I will give a brief introduction of what a linear algebraic group is and how it is structured. Then I will talk about the Galois descent related to linear algebraic groups. At last, I will explain what a torsor is and how it is related to other algebraic structures.

Geometric group theory concerns about how to see geometric properties in finitely generated groups. Defining Cayley graph of a finitely generated group with respect to finite generating set gives a perspective to describe geometric properties of finitely generated groups. Once we get a geometric perspective, we can classify finitely generated groups via quasi-isometry, since two Cayley graphs are quasi-isometric. In this talk, we will explain some basic notions appeared in geometric group theory (for example, quasi-isometry, hyperbolic groups, Švarc–Milnor lemma) and some theorems related to (relative) hyperbolicity of groups.

In extremal graph theory, one big question is finding a condition of the number of edges that guarantees the existence of a particular substructure in a graph. In the first half of this talk, I'll talk about the history of such problems, especially focusing on clique subdivisions. In the last half of the talk, I'll introduce my recent result with Jaehoon Kim, Younjin Kim, and Hong Liu, which states that if a graph G has no dense small subgraph, then G has a clique subdivision of size almost linear in its average degree and discuss some applications and further open questions.