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.

This talk is about the complex dynamics, which cares the iteration of holomorphic map (usually a rational map on the Riemann sphere), and the shift locus is a nice set of polynomials that all critical points escape to infinity under iteration. Understanding the shape and topology of shift locus is a challenge for decades, and accumulated works are done by Blanchard, Branner, Hubbard, Keen, McMullen, and recently Calegari introduce a nice lamination model. In this talk I will explain the basic complex dynamics and introduce the topology of the shift locus of cubic polynomials done by Calegari's paper 'Sausages and Butcher paper' and if time allows, I will end this talk with the connection to the Big mapping class group, the MCG of Sphere - Cantor set.