Date:
Wed, 01/01/202511:00-13:00
Location:
Ross 70
Zoom link: https://huji.zoom.us/j/82871443680?pwd=jMrhvccCZ8DvREjVaxJVQY1JareJKg.1
Meeting ID: 828 7144 3680
Passcode: 660682
Title: Strongly minimal reducts of ACVF, a positive answer.
Abstract: Zilber's Trichotomy states that any strongly minimal structure is either trivial, locally modular, or interprets an infinite field. Locally modular, non-trivial, strongly minimal structures are essentially vector spaces, so Zilber's Trichotomy says that any strongly minimal structure is either trivial, a vector space, or interprets an infinite field. This was proven false by Hrushovski in 1993. However, some instances of the trichotomy hold true and have been applied, notably in Hrushovski's proof of the Mordell-Lang Conjecture. As a result, the conjecture has become an important classification idea that tends to hold in tame enough geometric settings.
In this context, A. Hasson and D. Sustretov proved the trichotomy to be true when the strongly minimal set is definable in an algebraically closed field (ACF) and the universe is one-dimensional. More recently, B. Castle proved it under the condition that the strongly minimal structure is definable in a model of ACF_0, with no assumptions on the dimension of the universe.
IIn this talk we will introduce the trichotomy restricted to strongly minimal structures definable in an algebraically closed valued field (ACVF). In this setting, we prove the trichotomy assuming the existence of a definable group that is "locally isomorphic" to either the additive or multiplicative group of the valued field. We will outline the key ideas used in this proof, which was recently employed by B. Castle, A. Hasson, and J. Ye to provide a full positive answer to the trichotomy for ACVF, and, consequently, for ACF.
Meeting ID: 828 7144 3680
Passcode: 660682
Title: Strongly minimal reducts of ACVF, a positive answer.
Abstract: Zilber's Trichotomy states that any strongly minimal structure is either trivial, locally modular, or interprets an infinite field. Locally modular, non-trivial, strongly minimal structures are essentially vector spaces, so Zilber's Trichotomy says that any strongly minimal structure is either trivial, a vector space, or interprets an infinite field. This was proven false by Hrushovski in 1993. However, some instances of the trichotomy hold true and have been applied, notably in Hrushovski's proof of the Mordell-Lang Conjecture. As a result, the conjecture has become an important classification idea that tends to hold in tame enough geometric settings.
In this context, A. Hasson and D. Sustretov proved the trichotomy to be true when the strongly minimal set is definable in an algebraically closed field (ACF) and the universe is one-dimensional. More recently, B. Castle proved it under the condition that the strongly minimal structure is definable in a model of ACF_0, with no assumptions on the dimension of the universe.
IIn this talk we will introduce the trichotomy restricted to strongly minimal structures definable in an algebraically closed valued field (ACVF). In this setting, we prove the trichotomy assuming the existence of a definable group that is "locally isomorphic" to either the additive or multiplicative group of the valued field. We will outline the key ideas used in this proof, which was recently employed by B. Castle, A. Hasson, and J. Ye to provide a full positive answer to the trichotomy for ACVF, and, consequently, for ACF.