1월 28일 (수)
황원태 (전북대)
제목: An introductory lecture on the Kawaguchi-Silverman conjecture (KSC)
초록: We introduce one of the famous conjectures, so called, the Kawaguchi-Silverman conjecture in arithmetic dynamics.
권용재 (공주대)
제목: Characterization of minimal rational functions on $\mathbb{Z}_{p}$
초록: In this talk, we characterize the minimality of rational functions defined on the ring of $p$-adic integers, $\mathbb{Z}_{p}$. We restrict our attention to rational functions whose denominators have no zeros modulo $p$. For the primes $p=2$ and $p=3, we employ linearization techniques to reduce the minimality problem of a rational function to that of an associated polynomial. This method enables us to derive explicit criteria for minimality entirely in terms of the coefficients. For the general case $p\ge5$, we utilize $p$-adic Hermite interpolation to provide an algorithmic criterion based on the reduction modulo $p$ and the valuation of the perturbation term. Our results generalize the existing minimality theory for polynomials to the class of rational functions.
1월 29일 (목)
황원태 (전북대)
제목: An introductory lecture on the Dynamical Manin-Mumford conjecture (DMM)
초록: We introduce one of the famous conjectures, so called, the Dynamical Manin-Mumford conjecture in arithmetic dynamics.
송종백 (부산대)
제목: Introduction to toric geometry and the cohomology of toric varieties
초록: The g-conjecture, proved by Billera and Lee (1981) and Stanley (1980), provides a comprehensive combinatorial characterization of simplicial (or simple) polytopes in terms of their face numbers. Stanley’s contribution to this conjecture employed "toric geometry", specifically leveraging the connection between the face numbers of lattice polytopes and a certain topological invariant of (rationally) smooth toric varieties. In this first lecture, we will outline the key ideas behind this relationship and discuss possible extensions to singular toric varieties. This lecture will be an introduction to the toric geometry from the combinatorial view point, without complicated algebraic geometry and algebraic topology. You only need to know elementary polynomial computations and how to count face numbers of a polytope like number of vertices, edges and so on.
정재우 (대구경북과학기술원)
제목: An introduction to toric geometry and sums of squares cones on toric varieties
초록: Projective toric varieties form an important class of algebraic varieties that are completely described by combinatorial data such as lattice polytopes and fans. They provide a natural bridge between algebraic geometry, convex geometry, and combinatorics. In particular, projective spaces, Veronese varieties, and Segre varieties arise as basic examples of toric varieties. In this talk, I will introduce the basic ideas of toric geometry from the viewpoint of monomial parametrizations and lattice polytopes. I will explain why these fundamental projective models can be understood as toric varieties and how they correspond to lattice polytopes. I will also discuss problems concerning the structure of sums of squares cones on toric varieties, including known results and some open problems.
조계환 (전북대)
제목: Rational points on elliptic curves from a beginner’s point of view
초록: This talk presents an introduction to rational points on elliptic curves. We will first review basic terminology and concepts regarding elliptic curves. We will then briefly mention the Birch and Swinnerton-Dyer (BSD) conjecture. Finally, if time allows, we will briefly discuss the Mock Heegner points introduced by P. Monsky.
김혜정 (전북대)
제목: The classification of links up to one-two-way pass-move
초록: We define a local move for links called the one-two-way pass-move, abbreviated briefly as the 1-2-move. The 1-2-move is motivated from the pass-move and the #-move, and it is a hybrid of them. We give a complete classification of links up to 1-2-move. In fact, we show that the equivalence under the 1-2-move is the same as that of the pass-move. On the other hand, we show that the number of 1-2-moves behaves differently from the number of pass-moves.
1월 30일 (금)
윤영한 (아주대)
제목: Bier spheres in toric topology
초록: Bier spheres form a distinguished class of simplicial spheres and provide infinitely many non-polytopal examples. It is known that every Bier sphere can be realized as a smooth complete fan, which naturally connects them to toric topology. In this talk, we study the homotopy types of full subcomplexes of Bier spheres. We explain how these homotopy types can be described and computed, and how the results lead to information on topological invariants arising in toric topology. This is joint work with Suyoung Choi and Seonghyeon Yu.
송종백 (부산대)
제목: Introduction to toric geometry and the cohomology of toric varieties
초록: While the integral cohomology ring of a smooth toric variety has a nice presentation as a truncated polynomial ring, thanks to the works of Danilov and Jurkiewicz, the structure remains largely unknown for singular toric varieties. In this second lecture, I will present a computational result specifically for the class of toric varieties with vanishing odd-degree cohomology groups. We will then apply this result to toric surfaces to derive a concrete presentation of their cohomology rings. This lecture will involve some equivariant algebraic topology, but essentially you only need polynomial computations again.