The lecture series gives a view on computational methods and their some applications to existence and classfication problems. In the first lectures I will introduce Groebner basis and their basic applications in commutative algebra such as computing kernel and images of morphism between finitely presented modules over polynomial rings. As a theoretical application of Groebner basis I will give Petri's analysis of the equations of a canonical curve. The second topic will be Computer aided existence and unirationality proofs of algebraic varieties and their moduli spaces. In case of curves liaison theory is needed, which will be developed. For existence proofs random searches over finite fields is a technique that has not been exploited very much. I will illustrate this technique in a number of examples, in particular for the construction of certain surfaces. Classification of non-minimal surfaces uses adjunction theory. We will discuss this from a computational point of view.

Physics-Informed Neural Networks (PINNs) have emerged as a promising method for solving partial differential equations (PDEs) by embedding physical laws directly into the learning process. However, a critical question remains: How do we validate that PINNs accurately solve these PDEs? This talk explores the types of mathematical validation required to ensure that PINNs can reliably approximate solutions to PDEs. We will discuss the conditions under which PINNs can converge to the correct solution, the relationship between minimizing residuals and achieving accurate results, and the role of optimization algorithms in this process. Our goal is to provide a clear understanding of the theoretical foundations needed to trust PINNs in practical applications while addressing the challenges in this emerging field.

We consider the Euler-Poisson system, which describes the ion dynamics in electrostatic plasmas. In plasma physics, the pressureless model is often employed to simplify analysis. However, the behavior of solutions to the pressureless model generally differs from that of the isothermal model, both qualitatively and quantitatively - for instance, in the case of blow-up solutions. In previous work, we investigated a class of initial data leads to finite-time C^1 blow-up solutions. In order to understand more precise blow-up profiles, we construct blow-up solutions converging to the stable self-similar blow-up profile of the Burgers equation. For the isothermal model, the density and velocity exhibit C^{1/3} regularity at the blow-up time. For the pressureless model, we provide the exact blow-up profile of the density function, showing that the density is not a Dirac measure at the moment of blow-up. We also consider the peaked traveling solitary waves, which are not differentiable at a point. Our findings show that the singularities of these peaked solitary waves have nothing to do with the Burgers blow-up singularity. We study numerical solutions to the Euler-Poisson system to provide evidence of whether there are solutions whose blow-up nature is not shock-like. This talk is based on collaborative work with Junho Choi (KAIST), Yunjoo Kim, Bongsuk Kwon, Sang-Hyuck Moon, and Kwan Woo (UNIST)

The lecture series gives a view on computational methods and their some applications to existence and classfication problems. In the first lectures I will introduce Groebner basis and their basic applications in commutative algebra such as computing kernel and images of morphism between finitely presented modules over polynomial rings. As a theoretical application of Groebner basis I will give Petri's analysis of the equations of a canonical curve. The second topic will be Computer aided existence and unirationality proofs of algebraic varieties and their moduli spaces. In case of curves liaison theory is needed, which will be developed. For existence proofs random searches over finite fields is a technique that has not been exploited very much. I will illustrate this technique in a number of examples, in particular for the construction of certain surfaces. Classification of non-minimal surfaces uses adjunction theory. We will discuss this from a computational point of view.

The lecture series gives a view on computational methods and their some applications to existence and classfication problems. In the first lectures I will introduce Groebner basis and their basic applications in commutative algebra such as computing kernel and images of morphism between finitely presented modules over polynomial rings. As a theoretical application of Groebner basis I will give Petri's analysis of the equations of a canonical curve. The second topic will be Computer aided existence and unirationality proofs of algebraic varieties and their moduli spaces. In case of curves liaison theory is needed, which will be developed. For existence proofs random searches over finite fields is a technique that has not been exploited very much. I will illustrate this technique in a number of examples, in particular for the construction of certain surfaces. Classification of non-minimal surfaces uses adjunction theory. We will discuss this from a computational point of view.

The lecture series gives a view on computational methods and their some applications to existence and classfication problems. In the first lectures I will introduce Groebner basis and their basic applications in commutative algebra such as computing kernel and images of morphism between finitely presented modules over polynomial rings. As a theoretical application of Groebner basis I will give Petri's analysis of the equations of a canonical curve. The second topic will be Computer aided existence and unirationality proofs of algebraic varieties and their moduli spaces. In case of curves liaison theory is needed, which will be developed. For existence proofs random searches over finite fields is a technique that has not been exploited very much. I will illustrate this technique in a number of examples, in particular for the construction of certain surfaces. Classification of non-minimal surfaces uses adjunction theory. We will discuss this from a computational point of view.

We propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for numerical solution of partial differential equations. We start with a detailed explanation of the method for the Poisson equation and then extend the study to other PDEs. We shall show that the numerical solution can approximate the exact PDE solution very well. Then we present a large amount of numerical experimental results to demonstrate the performance of the method over the two- and three-dimensional settings.

An introduction and overview will be given of extreme events in Wetropolis flood investigator and of extreme water-wave motion in a novel wave-energy device. I will give an overview of the mathematical models and simulations of the phenomena seen in the following movies as well as related phenomena. Subsequently, mathematical and numerical aspects of the novel wave-energy device will be highlighted. As well as a discussion on the Wetropolis World proposal. The related movies which explain Wetropolis: Wetropolis-II (temp link) 2024: https://www.youtube.com/watch?v=g8znktYpxvY, Wetropolis-I 2022: https://www.youtube.com/watch?v=rNgEqWdafKk, The wave-energy device was motived by the bore-soliton-splash (2010): https://www.youtube.com/watch?v=YSXsXNX4zW0&list=FL6mc7mUa6M4Bo2VkD970urw, as well as a first proof of principle (2013): https://www.youtube.com/watch?v=SZhe_SOxBWo

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The related movies which explain Wetropolis: Wetropolis-II (temp link) 2024: https://www.youtube.com/watch?v=g8znktYpxvY, Wetropolis-I 2022: https://www.youtube.com/watch?v=rNgEqWdafKk, The wave-energy device was motived by the bore-soliton-splash (2010): https://www.youtube.com/watch?v=YSXsXNX4zW0&list=FL6mc7mUa6M4Bo2VkD970urw, as well as a first proof of principle (2013): https://www.youtube.com/watch?v=SZhe_SOxBWo

In this talk, I will talk about isotopy problems of Seifert surfaces pushed in to the 4-ball. In particular, I will prove that every Seifert surface of a non-split alternating link become isotopic in the 4-ball. This is a joint work with Maggie Miller and Jaehoon Yoo.

In the first part of the talk, I will discuss the asymptotic expansions of the Euclidean Φ^4-measure in the low-temperature regime. Consequently, we derive limit theorems, specifically the law of large numbers and the central limit theorem for the Φ^4-measure in the low-temperature limit. In the second part of the talk, I will focus on the infinite volume limit of the focusing Φ^4-measure. Specifically, with appropriate scaling, the focusing Φ^4-measure exhibits Gaussian fluctuations around a scaled solitary wave, that is, the central limit theorem. This talk is based on joint works with Benjamin Gess, Pavlos Tsatsoulis, and Philippe Sosoe.

Let S be a simply-connected rational homology complex projective plane with quotient singularities. The algebraic Montgomery-Yang problem conjectures that the number of singular points of S is at most three. In this talk, we leverage results from the study of smooth 4-manifolds, such as the Donaldson diagonalization theorem, to establish additional conditions for S. As a result, we eliminate the possibility of a rational homology complex projective plane of specific types with four singularities. We also identify infinite families of singularities that satisfy properties in algebraic geometry, including the orbifold BMY inequality, but are obstructed from being a rational homology complex projective plane due to smooth conditions. Additionally, we discuss experimental results related to this problem. This is joint work with Jongil Park and Kyungbae Park.

As part of the Langlands conjecture, it is predicted that every $\ell$-adic Galois representation attached to an algebraic cuspidal automorphic representation of $\mathrm{GL}_n$ over a number field is irreducible. In this talk, we will prove that a type $A_1$ Galois representation attached to a regular algebraic (polarized) cuspidal automorphic representation of $\mathrm{GL}_n$ over a totally real field $K$ is irreducible for all $\ell$, subject to some mild conditions. We will also prove that the attached Galois representation is residually irreducible for almost all $\ell$. Moreover, if $K=\mathbb Q$, we will prove that the attached Galois representation can be constructed from two-dimensional modular Galois representations up to twist. This is a joint work with Professor Chun-Yin Hui.

We consider the global dynamics of finite energy solutions to energy-critical equivariant harmonic map heat flow (HMHF). It is known that any finite energy equivariant solutions to (HMHF) decompose into finitely many harmonic maps (bubbles) separated by scales and a body map, as approaching to the maximal time of existence. Our main result for (HMHF) gives a complete classification of their dynamics for equivariance indices D≥3; (i) they exist globally in time, (ii) the number of bubbles and signs are determined by the energy class of the initial data, and (iii) the scales of bubbles are asymptotically given by a universal sequence of rates up to scaling symmetry. In parallel, we also obtain a complete classification of $\dot{H}^1$-bounded radial solutions to energy-critical heat equations in dimensions N≥7, building upon soliton resolution for such solutions. This is a joint work with Frank Merle (IHES and CY Cergy-Paris University).

Mr. Saqib Mushtaq Shah, a KAIX visiting graduate student from ISI Bangalore who will stay at KAIST for 8 weeks, is going to give a series of weekly talks on the Milnor K-theory from the beginning. It is part of his KAIX summer internship works.

We consider Calogero—Moser derivative NLS (CM-DNLS) equation which can be seen as a continuum version of completely integrable Calogero—Moser many-body systems in classical mechanics. Soliton resolution refers to the phenomenon where solutions asymptotically decompose into a sum of solitons and a dispersive radiation term as time progresses. Our work proves soliton resolution for both finite-time blow-up and global solutions without radial symmetry or size constraints. Although the equation exhibits integrability, our proof does not depend on this property, potentially providing insights applicable to other non-integrable models. This research is based on the joint work with Soonsik Kwon (KAIST).

For motivational purposes, we begin by explaining the classical Satake isomorphism from which we deduce the unramified local Langlands correspondence. Then we explain a geometric interpretation of the Satake isomorphism. More precisely, we explain how one can view Hecke operators as global functions on the moduli space of unramified L-parameters. This viewpoint arises from the categorical local Langlands correspondence. The main content of the talk is p-adic and mod p analogues of this interpretation, where the space of unramified L-parameters is replaced by certain loci in the moduli stack of p-adic Galois representations (so-called the Emerton-Gee stack). We will also discuss their relationship with the categorical p-adic local Langlands program.

Kahn-Sujatha's birational motive is a variant of Chow motive that synthesis the ideas of birational geometry and motives. We explain our result saying that the unramified cohomology is a universal invariant for torsion motives of surfaces. We also exhibit examples of complex varieties violating the integral Hodge conjecture. If time permits, we discuss a pathology in positive characteristic. (Joint work with Kanetomo Sato.)