Lecture Series
이정인 (Ajou University)
Title: Unlikely intersection problems in positive characteristic
Abstract: The principle of unlikely intersections asserts that when the intersection of two geometric objects is larger than expected, there should be an underlying geometric reason explaining this phenomenon. Motivated by the study of unlikely intersection problems in arithmetic geometry, numerous analogous problems have emerged in the context of arithmetic dynamics.
In the first talk, we introduce these problems in arithmetic dynamics and highlight the distinctions between the characteristic zero and positive characteristic settings. In the second talk, we present a proof of the converse of dynamical Mordell–Lang conjecture in positive characteristic and discuss height functions. In the third talk, we apply height functions to the study of simultaneously preperiodic points for families of polynomials in positive characteristic. The second and third talks are based on joint works with GyeongHyeon Nam.
쾨니히 요아힘 (Korea National University of Education)
Title: Dynamical Galois groups of polynomials
Abstract: The dynamical Galois group of a polynomial (or more generally, of a rational function) is a central object of interest when studying the arithmetic properties of polynomial iterates. Dynamical Galois groups and the associated Galois representations (a.k.a. ``arboreal representations") have recently received a lot of interest, although many of their properties remain mysterious. In this series of three lectures, I will introduce some of the basics of dynamical Galois groups over number fields. I will present important families of examples, review group-theoretical and number theoretical background as well as the basics of (dynamical) monodromy groups, and explain the role of some key notions, notably the special role of PCF maps. As we proceed, I will review some expectations and known results concerning key notions such as the ``largeness" of dynamical Galois groups and ``dynamical irreducibility". In the last lecture, I will focus on the implications of group-theoretical properties of dynamical Galois groups for certain arithmetic questions. I will discuss connections of ``largeness" with these arithmetic problems, and if time permits, I will introduce some of my own recent results.
한지영 (Pusan National University)
Title: Oppenheim conjecture-type problems for a system of a quadratic form and a linear form
Abstract: After Margulis proved Oppenheim conjecture, a question relative to the distribution of images of a quadratic form at integer vectors, numerous problems rooted from number theory have been studied by homogeneous dynamical tools and methods. In the first talk, we will examine how one can deal with such a problem in homogeneous dynamics through Gorodnik’s theorem (2004): Let Q be a quadratic form and L be a linear form. If
(a) the restriction of Q on the kernel space of L is indefinite;
(b) Any nontrivial linear combination of Q and the square of L is irrational,
Then the image of (Q,L) at integer vectors is dense in the real vector plane. Quantifying such a problem demands more developed tools in homogeneous dynamics, such as equidistribution theorems. In the second and third talks, we review the homogeneous dynamical ideas for proving the quantitative version of Gorodnik’s theorem. This is a joint work with Seonhee Lim and Keivan Mallahi-Karai.
Research Talks
김영균 (Seoul National University)
Title: Finite orbits of unramified morphisms
Abstract: We review results on upper bounds for the sizes of preperiodic orbits in arithmetic dynamical systems. We then show that if an integral algebraic dynamical system admits an unramified reduction, the size of its preperiodic orbits is controlled by the corresponding dynamical system modulo that reduction. Using this, we establish an upper bound for finite orbits in integral dynamical systems whose morphisms have unramified reduction.
남경현 (Aalto University)
Title: On simultaneously preperiodic points for one-parameter families of polynomials in characteristic p
Abstract: In 2011, Baker and DeMarco discovered an interesting property of simultaneously preperiodic points: two complex numbers a and b are preperiodic for the map x^d + c (with d > 1) for infinitely many complex numbers c if and only if a^d = b^d. This result was generalised to arbitrary polynomials by Ghioca, Hsia, and Tucker in 2013. More recently, Ghioca and Hsia studied this problem over a field of characteristic p under certain special conditions. Our project aims to address cases not covered by Ghioca and Hsia. Furthermore, we obtain a result concerning the colliding orbits problem, which can be derived easily from the result on simultaneously preperiodic points. This is a joint work with Jungin Lee.
이영준 (Seoul National University)
Title: Applications of arithmetic dynamics and algebraic dynamics to orbit Zariski closure problem
Abstract: I demonstrate the algorithm to compute orbit Zariski closure given an explicit rational action on $\mathbb{A}_{\overline{\mathbb{Q}}}$ and a point of the affine plane. It follows by combining results of arithmetic and algebraic dynamics. For the arithmetic dynamics, we have the result by Whang which gives criterion to decide whether the orbit is finite or not. For the algebraic dynamics, we have the geometric group theoretic description of the automorphism group of the affine plane provided by Stephane Lamy.
한민식 (University of Rochester)
Title: Finite Index Theorems for Iterated Galois Groups
Abstract: When a rational map of degree d defined over a field K is iterated, it generates a tree of preimages for a given base point in K. The absolute Galois group of K acts on this infinite d-ary rooted tree T, and this action defines a group homomorphism (called an "arboreal Galois representation") into Aut(T), the full automorphism group of T. Inspired by the work of Odoni (1985), there has been active research on the index of the image of an arboreal Galois representation in Aut(T). In this talk, I will briefly survey the main results on this topic and introduce present recent joint work with Thomas J. Tucker on the image of the arboreal Galois representations of unicritical polynomials.