SDEs and PDEs: an approach via Semigroups and Invariant Measures

Threefold hypersurfaces with zero defect

수학강연회 | 강연자 : 홍규식 | 소속 : 전주대학교

In commutative algebra, a unique factorization domain, or simply UFD, is a commutative ring in which every element can be uniquely written as a product of prime elements, analogous to the fundamental theorem of arithmetic for the integers. UFDs are sometimes called factorial rings, following the terminology of Bourbaki. Most rings familiar from elementary mathematics are UFDs: the integers, the polynomial rings over a field, the formal power series ring over a field, the ring of functions in a fixed number of complex variables holomorphic at the origin etc. However, most factor rings of a polynomial ring are not UFDs. For a given commutative ring, it is an interesting question to decide weather it is UFD or not. Two questions above are of different natures. However, they are closely related through the topology of mildly singular algebraic threefolds. In mathematics, an affine algebraic variety is essentially a set of common zeroes of a set of polynomials. Similarly, a projective algebraic variety is a set of common zeroes of a set of homogeneous forms. Algebraic varieties are one of the central objects of study in classical and modern algebraic geometry. An affine algebraic variety is called factorial if its coordinate ring is UFD. For a projective algebraic variety, one can define the factoriality in a similar way. In most of cases, the factoriality of projective varieties can be expressed in terms of topological data and can be proved by using powerful tools of topology such as the Lefschetz theorem and the Poincare duality. Algebraic surfaces, i.e., algebraic varieties of complex dimension two, are usually not factorial. For most of complex projective threefolds, i.e., algebraic varieties of complex dimension three, the factoriality simply means that its topology is trivial outside of the cycles of real dimension three. For example, every smooth threefold hypersurface is factorial by the Lefschetz theorem and the Poincare duality. For threefolds with isolated singularities, we still can use the Lefschetz theorem, but the Poincare duality usually fails. For example, every smooth threefold hypersurface is factorial if and only if the Poincare duality does not fail for it. The factoriality of threefolds with isolated singularities has been considered by several authors for a long time. For a wide class of singular threefolds, the factoriality problem, introducing the concept of defect, was investigated by Clemens. He showed that the factoriality of many singular threefolds can be expressed in terms of the number of independent linear conditions that their singular points impose on the homogeneous forms of certain degree. We plan to investigate how the factoriality of singular threefolds depends on the number of singular points.

Numerical Methods for PDEs: Exploring Finite Element and Discontinuous Galerkin Approaches

수학강연회 | 강연자 : 신동욱 | 소속 : 아주대학교

Various natural and physical phenomena can be described by partial differential equations (PDEs). To solve these equations, numerical methods based on the Finite Element (FE) and Discontinuous Galerkin (DG) approaches have been widely used and developed. In this talk, we focus on FE and DG methods for solving second-order elliptic equations. We discuss their theoretical foundations, numerical implementation, and key differences, particularly in terms of accuracy, stability, and conservation properties. While these methods allow for high-order accuracy, their convergence behavior is significantly influenced by the regularity of the solution. When the solution is not sufficiently smooth, adaptive algorithms play a crucial role in improving efficiency and accuracy. Here, we introduce reliable and fully computable a posteriori error estimators, which guide adaptive mesh refinement to enhance solution accuracy while reducing computational cost. Finally, numerical results will be presented to validate the theoretical findings and demonstrate the effectiveness of adaptive algorithms.

Prekosmic Grothendieck/Galois categories

BK21 FOUR Rookies Pitch | 강연자 : 이재혁 | 소속 : 서울대학교

In this talk, I will present a duality between mathematical objects equipped with group structures and the categories of their internal representations. We establish the duality by providing a complete abstract characterization of such categories. The primary motivation is to simultaneously generalize and unify the two categorical concepts, Galois categories and Tannakian categories. I will explain the motivation and a concrete example in detail. Then I will explain that the duality exists in a general context, namely over a prekosmos. This is a joint work with Jae-Suk Park.

Introduction to crystal base theory

BK21 FOUR Rookies Pitch | 강연자 : 우루노아끼또 | 소속 : 서울대학교

The representation theory of quantum groups has become an important area of research. For an integrable highest weight module of quantum groups, there exists a remarkable basis at q = 0, known as the crystal base, which was introduced by Kashiwara. It has a nice structure with respect to tensor products, and can be viewed as a colored oriented graph, called a crystal.

In this talk, we will briefly review crystal base theory, focusing on the case of gl_n. We will also explain how the crystal of an irreducible highest weight module can be identified with the set of semistandard Young tableaux.

Now

현재 SDEs and PDEs: an approach via Semigroups and Invariant Measures

수학강연회 | 강연자 : 이해성 | 소속 : 국립금오공과대학교

In this talk, we explore Stochastic Differential Equations (SDEs) and Partial Differential Equations (PDEs) with rough coefficients, focusing on the interplay between analysis and stochastic analysis. Starting from the Ornstein-Uhlenbeck process as a key example, we investigate the role of semigroups and invariant measures in general time-homogeneous Itô-SDEs. Furthermore, based on general second-order partial differential operators, we develop an analytic framework for constructing (infinitesimally) invariant measures and regularized semigroups. This enables us to construct solutions to the corresponding SDEs with a potential candidate for an invariant measure, or even with prescribed invariant measure. The presented approach naturally extends to more challenging situations. For instance, it applies to the study of degenerate SDEs, which illustrates the fundamental role of semigroups and invariant measures in both SDEs and PDEs.

Machine learning meets scientific computing

수학강연회 | 강연자 : 홍영준 | 소속 : 수리과학부

In recent years, advances in computational power and data availability have propelled machine learning (ML) to the forefront of scientific computing, complementing and enhancing traditional methods. This lecture explores the integration of ML with numerical methods for multi-scale problems, highlighting new perspectives in scientific computing. Key topics include the synergy between ML and numerical analysis, convergence analysis from both fields, and error estimation techniques. Additionally, we will discuss how neural networks contribute to solving complex partial differential equations (PDEs) efficiently. By leveraging these approaches, we can tackle challenging scientific problems with greater accuracy and computational efficiency. If time permits, we will also touch on emerging topics such as foundation models and ML-driven material discovery.

Survey on a geography of model theory

수학강연회 | 강연자 : 이정욱 | 소속 : 창원대학교

In this survey talk, we will explore a geography of Model Theory under the guidance of Classification Theory. First, we review basic first-order logic and several dividing lines developed in Classification Theory. Second, we will focus on Classification Theory for fields and examine some interactions between Field Arithmetic and dividing lines.