Mobius II by M.C.Escher - I am also an ant living in the 3-dimensional universe trying to know its global sturucture.

My main research interests are in low dimensional topology and geometry, and gauge and Floer theoretic invariants. More specifically, my study focuses on the TQFT aspects of Ozsváth and Szabó's Heegaard Floer theory in dimensions (3+1). These include the knot concordance group, applications on smooth four manifolds of the theory as well as properties of Ozsváth-Szabó smooth 4-manifolds invariants.

 "On intersection forms of definite 4-manifolds bounded by a rational homology 3-sphere", 2017, arxiv:1704.04419 (with Dong Heon Choe) We show that, if a rational homology 3-sphere $Y$ bounds a positive definite smooth 4-manifold, then there are finitely many negative definite lattices, up to the stable-equivalence, which can be realized as the intersection form of a smooth 4-manifold bounded by $Y$. To this end, we make use of constraints on definite forms bounded by $Y$ induced from Donaldson's diagonalization theorem, and Ozsávth and Szabó's Heegaard Floer correction term. We also present some families of Seifert fibered 3-manifolds that bound both positive and negative definite smooth 4-manifolds. "An infinite-rank summand of knots with trivial Alexander polynomial", 2016, arxiv:1604.04037, to appear in Journal of Symplectic Geometry (with Min Hoon Kim) We show that there exists a $\mathbb{Z}^\infty$-summand in the subgroup of the knot concordance group generated by knots with trivial Alexander polynomial. To this end we use the invariant Upsilon $\Upsilon$ recently introduced by Ozsváth, Stipsicz and Szabó using knot Floer homology. We partially compute $\Upsilon$ of $(n,1)$-cable of the Whitehead double of the trefoil knot. For this computation of $\Upsilon$, we determine a sufficient condition for two satellite knots to have identical $\Upsilon$ for any pattern with nonzero winding number. "Some Computations and Applications of Heegaard Floer Correction Terms", 2014, Thesis In this dissertation we study some computations and applications of Heegaard Floer correction terms. In particular we explore the correction terms for the double covers of the three-sphere branched along the Whitehead doubles of knots. As a consequence we show that Whitehead double and iterated double of some classes of knots are independent in the smooth knot concordance group. We also compute the correction terms of non-trivial circle bundles over oriented surfaces and discuss how they can be applied to four-dimensional topology. "On Independence of Iterated Whitehead Doubles in the Knot Concordance Group", 2013, arxiv:1311.2050 In this paper, we have studied a question for iterated Whitehead doubles and showed that for each $m>1$ the Whitehead double and iterated Whitehead double of $(2,2m+1)$ torus knots are not smoothly concordant and that indeed they generate a $\mathbb{Z}^2$ summand in the subgroup of the smooth knot concordance group generated by topologically slice knots. Our main tool is the Heegaard Floer correction term for the double cover of $S^3$ branched along a knot. We also present some sufficient conditions for general knots to have this independence property. Additionally, for some classes of knots including $(p,q)$ torus knots, we give an algorithmic formula for testing it in terms of its Alexander polynomial.