The final goal of my research is to solve tt*-equations. In my research, I use two methods for solving the tt*-equations. In the first way, I try to construct a new solution to the tt*-equation from global solutions to the tt*-Toda equation (Guest-Its-Lin). Regarding the first way, I investigated intrinsic properties of the tt*-Toda equation. I gave an intrinsic description of the tt*-Toda equation as a flat bundle, and I classified the flat bundles. From the classification, I gave an equivalence relation on restrictive conditions called "anti-symmetry conditions" of the tt*-Toda equation, and we can show that there are essentially only two types of these conditions. As an application, we observed the relation between the tt*-Toda equations and representations of the vertex algebra following the results of Fredrickson and Neizke. By using the global solutions to the tt*-Toda equation, I also gave a solution to the tt*-equation on the quantum cohomology ring of the Grassmannian. The solution is given as the exterior product of solutions to the tt*-Toda equation. In the second way, I consider an isomonodromic deformation of the tt*-equation with certain conditions and I try to solve the tt*-equation by using the Vanishing Lemma. In this method, the existence of solutions to the tt*-equation is equivalent to the existence of a Riemann-Hilbert problem and there is a correspondence between the solutions and upper unitriangular matrices. However, upper triangular matrices do not always correspond to solvable solutions. Physicists claimed that the ADE-type Cartan matrices give solvable solutions to the tt*-equations. We gave the mathematical proof for the claim regarding upper triangular matrices by using Vanishing Lemma. I expect that this method could be applied to more general cases. In physics, it was claimed that upper unitriangular matrices with certain conditions give solvable solutions. Regarding papers, one paper was accepted, and one paper is in application. I am preparing papers about my results above. Regarding presentations, I gave 4 presentations (3 talks and 1 poster presentation). Regarding other research activities, I visited Taiwan from 2/27 to 3/4, and I discussed with Prof. Martin Guest, who is the visitor to the National Taiwan University. We mainly discussed the paper about intrinsic properties of the tt*-Toa equation.