♦ Byung-Hak Hwang

Title: Formalizing mathematics: why and how

Abstract: Formalizing mathematics involves translating mathematical statements from natural language into a precise formal language that computers can interpret. In this talk, I will provide a brief introduction to the concept of formalization, discuss its significance, and offer practical ways to explore this process firsthand.

♦ Mingu Jung

Title: A metric characterization of Radon-Nikodým Lipschitz operators

Abstract: The study of Lipschitz-free spaces lies at the intersection of Functional Analysis and Metric Geometry and is considered a fundamental object in non-linear Functional Analysis. Due to the linearization universal property of Lipschitz-free spaces, every Lipschitz map between metric spaces can be linearized to a bounded linear operator between the corresponding Lipschitz-free spaces. These bounded linear operators, known as Lipschitz operators, provide an intriguing framework to explore the relationship between the category of metric spaces and that of (Lipschitz-free) Banach spaces through this linearization functor. In this talk, we will discuss a metric characterization of Lipschitz operators that are Radon-Nikodým operators. This work is based on joint research with colleagues from Besançon, France.

♦ Chanho Kim

Title: On the adjoint Selmer groups of semi-stable elliptic curves and Flach's zeta elements

Abstract: We explicitly construct the rank one primitive Stark (equivalently, Kolyvagin) system extending a constant multiple of Flach's zeta elements for semi-stable elliptic curves. As its arithmetic applications, we obtain the equivalence between a specific behavior of the Stark system and the minimal modularity lifting theorem, and we also discuss the cyclicity of the adjoint Selmer groups. This Stark system construction yields a more refined interpretation of the collection of Flach's zeta elements than the "geometric Euler system" approach due to Flach, Wiles, Mazur, and Weston.

♦ Jeong-Seop Kim

Title: Positivity of tangent bundles

Abstract: The tangent bundle is one of the most natural vector bundles that can be defined for a smooth projective variety. As the positivity imposed on the tangent bundle strengthens, the geometry of the variety becomes more constrained. This raises the question of how we can characterize the variety according to the positivity on its tangent bundle. In this talk, I will introduce key results and conjectures related to this question, along with various notions of positivity in algebraic geometry, and then present some recent progress.

♦ Sanghoon Kwak

Title: Graphs, Surfaces, and their Symmetries

Abstract: Graphs and surfaces are fundamental objects in mathematics. In this expository talk, we will take a gentle journey through these objects from the perspective of geometric group theory, culminating in a discussion of recent joint work with George Domat and Hannah Hoganson on rigidity theorems—specifically, exploring when the symmetry group of an object uniquely determines the object itself.

♦ Kwangwoo Lee

Title: Lucas numbers, Pell's equations, and automorphisms of K3 surfaces

Abstract: We find some correspondences of Lucas numbers, Pell's equations, and automorphisms of K3 surfaces of Picard number 2. Using this relation, we solve some Diophantine equations.

♦ Carl-Fredrik Nyberg Brodda

Title: Some combinatorial problems in algebra

Abstract: I will discuss some important combinatorial problems in algebra, focussing on groups, semigroups, inverse semigroups, and their presentations via generators and relations. I will go through the history of algorithmic, homological, and growth-related problems for these objects, and some recent progress on a problem that has remained open since 1914. This will include joint work with I. Foniqi, R. D. Gray (East Anglia, UK) and M. Kambites, N. Szakács, R. Webb (Manchester, UK).

♦ Jaeseong Oh

Title: The Ubiquity of $q,t$-Catalan Numbers

Abstract: A fundamental goal in algebraic combinatorics is to provide combinatorial interpretations for numerical sequences arising in various fields of mathematics. The Catalan numbers are a prime example. In this talk, I will introduce the $q,t$-Catalan numbers, a significant refinement of the Catalan numbers. I will highlight their wide-ranging occurrences in fields such as combinatorics, representation theory, knot theory, geometry, and probability, and more.

♦ Hyungsung  Yun

Title: Recent progress on regularity theory for fully nonlinear degenerate parabolic equations

Abstract: This talk presents recent progress on regularity theory for fully nonlinear degenerate/singular parabolic equations. Nonlinear equations appear in many application problems, and we specifically discuss fully nonlinear parabolic equations whose degeneracy is of the form σ(Du,u,x). These degenerate types are divided into degenerate equations on the flat boundary, porous medium type, and p-Laplace type depending on the function σ. For each type of σ, we discuss the regularity results and techniques for fully nonlinear degenerate equations corresponding to σ.