Talks

title : Fields Medal, past and present

abstract :

The first Fields Medals were awarded in 1936, in accordance with the wishes of the late Canadian mathematician John Charles Fields. After an interruption due to World War II, the medals have been awarded every four years since 1950. They are widely regarded as among the most prestigious honors in mathematics, both within the mathematical community and among the broader public. The recipients are selected by a committee whose members are appointed by the Executive Committee of the International Mathematical Union (IMU). In this talk, I will discuss Fields’ original vision, the current selection process, and also share my own thoughts on how the work of the 1990 Fields Medalists has developed since then.

title : Moduli spaces of points on a line

abstract :

In math, classification is always a fundamental issue. When it is attained by a geometric space, we call it a moduli space. In this talk, I will deal with the simplest case of points on a projective line. There are still many open problems about these moduli spaces and I will report some recent progresses based on techniques from algebraic geometry, combinatorics and probability theory.  

title : Conformal Curvatures

abstract :

In this talk, I will give an overview of different ways to understand curvature. We will start with curves and surfaces in ordinary Euclidean space and then move to the general setting of Riemannian geometry. After that, we will look at conformal geometry, where only angles are preserved, and see how this changes the meaning of curvature. I will finish with a brief discussion of recent results related to Yamabe-type problems and what they tell us about curvature in the conformal setting.

title : Forms in many variables

abstract :

Let K be a number field and let  f_1, … , f_s in K[x_1, … , x_n] be forms of odd degrees. In 1957, Birch proved that if the number of variables n is sufficiently large then the forms have a non trivial zero in K^n. Apart from some small degrees, the bound on the number of variables required was of recursive type. We prove that, for any fixed degrees, n may be taken polynomial in s. Joint work with Lampert and Snowden.

title : Avoidance of instable singularity in mean curvature flow

abstract :

The mean curvature flow is an evolution of hypersurfaces by the geometric heat equation. As a solution to a parabolic equation, its singularities converges self-similar solutions after blow-ups. Indeed, the self-similar solutions are critical points of an energy function. Thus, we can study the stability of the critical points and count their Morse index. Then, we can perturb the solution to avoid unstable singularities so that we can obtain desired shape of singularities for topological surgeries.

title : KSIAM: The Host of ICIAM2031

abstract :

I will present the history and achievements of the Korean Society for Industrial and Applied Mathematics (KSIAM). Since 2004, KSIAM has grown into a thriving community of about 1,300 members, consistently drawing 400+ participants to its conferences. With Busan set to host ICIAM 2031, the society is poised for greater international recognition. Additionally, the talk will cover NIMS, the Sovereign AI initiative, and the latest government fundings in the mathematical sciences.

title : The Korean Mathematics Ecosystem : The growth and a Vision for the future

abstract :

The 80-year history of KMS is a miniature version of Korea’s growth and its contribution to the world.  Since its inception in 1946, KMS has laid the vital groundwork for Korea’s extraordinary scientific progress.  

Moving forward, KMS is committed to evolving into a ‘Global Hub for Mathematical Innovation’—bridging the gap between theory and industry, inspiring the next generation of talent, and shaping a better world through the power of mathematics.

title : Kleinian singularities and beyond

abstract :

I review Kleinian singularities and higher-dimensional analogues through symplectic singularities, together with a new approach in dimension six.

title : Dynamics on and off elliptic curves

abstract :

Felix Klein said that the study of new transcendental functions defined by differential equations was "the central problem of the whole of modern mathematics". The beginnings of this study lay in elliptic functions, which were generalised by the Painlevé transcendents. The analytic theory of their governing differential equations has a counterpart in the theory of the algebraic curves. The most famous examples are elliptic curves. What is not widely known is that the evolution of a solution of a Painlevé equation changes the underlying elliptic curve and points move from one such curve to another under its time evolution.  This leads to global results from a geometric description of the regularized projective space of initial values.

title : Learning How to Ride the Motorcycle for the Mind

abstract :

For the past years, I used computers to solve the moving sofa problem—a well-known open problem in geometric optimization—not as a calculator but as an extension of geometric intuition. I developed custom user interfaces and scripts that manipulate the shapes I want to optimize. I also developed custom software that automates a portion of the proof. Back then, the computer was my bicycle for the mind.

Now that AI has arrived, the "bicycle" has become a motorcycle. It can search vast literature beyond my knowledge, write much better working code in an instant, and solve Erdős problems (ones I still aspire to solve in my career) on its own. The question now is how to collaborate with AI effectively as a working mathematician. I will discuss my current experiments and how AI has been helping me understand the structure of problems.

title : First order rigidity of manifold diffeomorphism groups

abstract :

Two groups are elementarily equivalent if they have the same sets of true first order group theoretic sentences. We prove that for two real numbers r,s ≥ 1, and for two smooth closed manifolds M and N, the C^r diffeomorphism group of M is elementarily equivalent to the C^s diffeomorphism group of N if and only if r=s and M is diffeomorphic to N. In the case of r=s=0, we can even weaken the hypothesis so that the manifolds are only assumed to be compact, dropping the smoothability or closedness hypotheses. This strengthens (1) Whittaker’s theorem (1963) concerned with group isomorphisms in the C^0 case, and (2) Takens--Filipkiewicz theorems (1982) concerned with group isomorphisms in the C^p case with an integer p. Joint work with Thomas Koberda (UVa) and Javier de la Nuez-Gonzalez (KIAS).

title : Finding canonical bases of skein algebras

abstract :

We review the Kauffman bracket skein algebra of an oriented surface, which is a quantization of the SL2 character variety. We discuss several vector space bases of this algebra and investigate some remarkable phenomena and conjectures, in relation to various topics like quantum cluster varieties and representation theory.