Shift polynomial is a complex polynomial that every critical point escape to infinity under iteration and the shift locus of quadratic polynomial is a collection of $c$ that $z^2 + c$ became a shift polynomial. The Julia set of shift polynomial has nice properties. It is a Cantor set and also its dynamics is conjugate to one-sided shift of $0$-$1$ infinite sequence. Julia set of shift polynomial permute itself whenever the parameter wanders around in the shift locus. In this talk, I will introduce 1) how the Julia set permute by introducing related works done by Blanchard, Devaney, Keen, and recently Calegari, 2) what happens if the parameter passes the parabolic points, and 3) its lift into $Mod(S^2 - \{\text{Cantor set}\})$.

One can understand the space of lattices in the d-dimensional vector space with unit covolume as the quotient space SL(d,R)/SL(d,Z). By studying orbits of subgroups of SL(d,R) in this space, one can approach many problems of counting lattice points. Among these problems, we are interested in Oppenheim conjecture-type problems, which are about exploring the distribution of images of integral lattice points under quadratic forms. In this talk, let us introduce equidistribution theorems on homogeneous spaces, and how these theorems are connected to these Oppenheim conjecture-type problems.

For a given finitely generated group G, we can construct a geodesic space called an asymptotic cone and G acts on its asymptotic cone isometrically. But it is known that this action is not faithful generally. We proved that if G is non-elementary hyperbolic, then its kernel is the same as the following three subgroups, the kernel of action on its Gromov boundary, the unique maximal finite normal subgroup, and FC(G), the set of elements having only finitely many conjugacy classes. Actually, we extended this result to a non-elementary relatively hyperbolic group replacing the Gromov boundary by the Bowditch boundary, with more conditions on ultrafilter and sequence. This work is joint with my advisor Hyungryul Baik.

We prove that the (2,1)-cable of the figure-eight knot is not smoothly slice, thereby solving a question posed by Kawauchi in 1980. This is a joint work with Irving Dai, Abhishek Mallick, JungHwan Park, and Matthew Stoffregen.

We discuss translation lengths of special types of pseudo-Anosov maps on curve graphs, where maps are constructed from multiple of Dehn twists of filling multicurves. With enough power provided to each Dehn twists, we give bounds of stable translation lengths directly from it's construction. We also present cases where stable translation length is integer, and can be directly computed. From this, we give application to minimal translation length word problem and ratio-optimizer. This is a joint work with Hyungryul Baik.

Quantization of the Teichmüller space of a Riemann surface was suggested as an approach to three dimensional quantum gravity in 1980. Its first major results were established in 1990's by Kashaev and by Chekhov and Fock, and it was later generalized to the theory of quantum cluster varieties by Fock and Goncharov. I will give an introduction to this subject in elementary terms, and explain some results and unsolved problems, as well as its generalizations.

Asymptotic dimension is one of a quasi-isometric invariant in metric geometry and geometric group theory. Gromov showed that the difference of the asymptotic dimension of a hyperbolic group and the topological dimension of its Gromov boundary is exactly one. In this talk, we investigate the analogy of Gromov's theorem with relatively hyperbolic groups, which is a generalized notion of hyperbolic groups. We also introduce some nice geometric objects as candidates of satisfying relative version of Gromov's theorem.

A laminar group is a subgroup of orientation preserving circle homeomorphisms preserving circle laminations. Laminar group theory is motivated by Thurston's universal circle theorem. The theorem says that a tautly foliated three manifold group acts on the (universal) circle preserving a pair of circle laminations. Laminar group theory studies the converse of this theorem. In this talk, I will introduce some basic notions and recent progress. This work is joint with Hyungryul Baik and Hongtaek Jung.

In this talk, we consider geometric invariants on noncompact hyperbolic 3-manifolds. The talk consists of three parts: the renormalization of volume for hyperbolic 3-manifolds having infinite volume, a brief introduction to Chern-Simons invariant, and its renormalization for hyperbolic 3-manifolds.

In the first part, we will introduce problems in Diophantine appoximation that can be solved using dynamics of group actions on homogeneous spaces. We will then focus on specific families of problems in Diophantine approximation, namely Hausdorff dimension of badly approximable vectors (The talk is based on the joint works with Uri Shapira, Nicolas de Saxce, and with Wooyeon Kim and Taehyeong Kim.)

Find explicit Hierarchically hyperbolic space(HHS) structure for a given metric space or group is usually challanging question, due to complicated definition of HHS. Recently Behrstock-Hagen-Martin-Sisto introduce a simpler combinatorial criterion for curve graph, called the combinatorial HHS (CHHS), to verify whether a given hierarchy of curve graphs of subspaces induces HHS structure. Such criterion is, however, not the optimal one as it cannot cover the case of the curve graph of CAT(0) cube complex, although its HHS structure combinatorially encoded in the curve graph. In this talk, we will briefly review the (combinatorial) HHS theory and explain the modification of the combinatorial HHS to cover cubical curve graphs and using this, extend the factor system machinary to the CAT(0) prism complex.

We answer a question of Livingston from 1982 by producing Seifert surfaces of the same genus for a knot in the 3-sphere that do not become isotopic when their interiors are pushed into the 4-ball. We give examples where the surfaces are not topologically isotopic in the 4-ball, as well as examples that are topologically but not smoothly isotopic. These latter surfaces are distinguished by their associated cobordism maps on Khovanov homology, and our calculations demonstrate the stability and computability of these maps under certain satellite operations. This is joint work with Kyle Hayden, Seungwon Kim, Maggie Miller, and Isaac Sundberg.

Given a simplicial complex K, one can define a topological space Z(K) called the moment-angle complex. The cohomology of Z(K) is captured by the Tor-algebra corresponding to the face ring of K. In this talk, we introduce a certain differential the cohomology of Z(K) to make it a chain complex. This leads us to define a double cohomology of Z(K), which is a new combinatorial invariant of K. Then, we discuss how it comes in the standard pipeline of the topological data analysis (TDA). This is a joint work (in progress) with A. Bahri, I. Limonchenko, T. Panov and D. Stanley.

Self-distributive algebraic structures provide an important class of solutions of the set-theoretic Yang-Baxter equation. Since C. S. Peirce emphasized the importance of self-distributivity in algebraic structures in 1880, these structures have been actively studied by a number of scholars, including Mayer, Toyoda, Bruck, Galkin, etc. While homology theories of associative structures, such as groups and rings, have been extensively studied, starting with the work of Eileenberg and Hochschild, it has not been long since the study of homology theories of non-associative distributive structures began to be active. In this talk, we introduce homology theories of self-distributive structures and how they are related to knot theory.