Discrete isometry groups of symmetric spaces

This survey is based on a series of lectures that we gave at MSRI in Spring 2015 and on a series of papers, mostly written jointly with Joan Porti. Our goal here is to: 1. Describe a class of discrete subgroups $\Gamma<G$ of higher rank semisimple Lie groups, which exhibit some "rank 1 behavior". 2. Give different characterizations of the subclass of Anosov subgroups, which generalize convex-cocompact subgroups of rank 1 Lie groups, in terms of various equivalent dynamical and geometric properties (such as asymptotically embedded, RCA, Morse, URU). 3. Discuss the topological dynamics of discrete subgroups $\Gamma$ on flag manifolds associated to $G$ and Finsler compactifications of associated symmetric spaces $X=G/K$. Find domains of proper discontinuity and use them to construct natural bordifications and compactifications of the locally symmetric spaces $X/\Gamma$.Comment: 77 page

Simple Subject-Verb Agreement: a Morphosyntactic Path to Arabic Variations

The analytic object of this dissertation is to formally model the Arabic subject-verb agreement aspects, more particularly, the verbal agreement with simple subject DPs. It aims to define how φ-agreement is formally manifested across the Arabic varieties, more specifically, Standard Arabic and the current dialects, and hopes to draw the latter varieties’ interrelation. In other words, this thesis hopes to advance the overall understanding of subject-verb agreement in Arabic and contribute to a clearer and simpler view of a number of specific syntactic phenomena. Most important of all, the subject DP relative order with respect to the verbal predicate influences the possible subject-verb-agreement choices attested in Standard Arabic (SA), whereby a subject-verb (SV) order shows full agreement in all φ-features, but a verb-subject (VS) order shows only partial agreement, typically, in Gender and Person. Nonetheless, full subject-verb agreement in VS order is robustly found in different dialects of the Arab world, in which the Number feature is obligatory. Remarkably, not only is the partial agreement attested in SA absent in the modern dialects, but also Gender and Number morphology distinctions may often be minimized. On the one hand, a masculine agreement is syncretic whenever the agreement relation is established between a verbal predicate and dual or plural subject DPs, whether they are masculine or feminine. On the other hand, plural and dual nouns trigger plural agreement on the agreeing verbal predicate; the plural number is syncretic whenever the subject DP is plural or dual. What’s more, the Arabic (traditional) texts have an abundance of examples that do not conform to the SA norm of agreement and whose well-formedness is unquestionable, suggesting that the agreement asymmetry may not be absolute. These observations urge an in-depth investigation, assuming that they may present profound paradoxes when analyzed via the standard Agree-based mechanism. Despite the dissimilarity between SA and the modern dialects in terms of subject-verb agreement, these varieties are mostly alike in other matters. For these reasons, I believe that any account to the subject-verb agreement must take these points into consideration. To my knowledge, there has been no detailed analysis devoted to the interrelation between the standard variety and the modern dialects in terms of subject-verb agreement. So, believing that any syntactic account to the subject-verb agreement in Arabic ought to be flexible to cover the various agreement phenomena, I argue that the various (often outwardly non-canonical) agreement patterns in Arabic are manifestations of the core syntactic Agree mechanism. Their agreement behavior is often attributed to a fundamental mismatch between the syntactic and morphological components, subject to variety/dialect-specific requirements. In simple terms, taking the core properties of the Agree-based system to feature valuation (Chomsky, 2000 et seq.), the assumptions in Distributed Morphology (Halle & Marantz, 1993; 1994; Halle, 1994, among others), and the feature geometry advocated by Harley & Ritter (2002), among others, I posit that these agreement patterns attest very general conditions on the agreement and φ-feature manifestations in Arabic, defined in terms of restrictions on T’s φ-Probe that agrees with the subject DP. Overall, given the formulation of the conditions advanced, the agreement facts across the Arabic varieties, I believe, arise naturally and predictably from the interaction of Agree, conditions on T’s φ-Probe, and postsyntactic requirements

Mathematical Logic: Proof Theory, Constructive Mathematics

[no abstract available

How Knowledge Affects Obligations:a Study on the Logic of Knowledge-Based Obligations

A doctor generally does not have an obligation to treat a person if the doctor does not know that he/she is ill. But when the doctor knows the person is ill, the doctor ought to treat him/her. So what we ought to do depends a lot on what we know at that moment. This thesis provides a logical formalization of the concept of knowledge-based conditional obligation which paraphrases `A ought to be achieved by an agent under the condition that he/she knows B'. This notion is characterized by a sound and strongly complete axiomatization in this thesis. Furthermore, the framework is dynamified in order to show how knowledge change affects obligation change. Accordingly, those situations where agents' obligations are updated by their dynamic information can be modelled. Besides information change, norm change may also lead to new obligations. The notion of relativized conditional obligations is proposed for conceptualizing obligations induced by different normative systems. Several types of updates on normative systems are studied. They bring about new obligations or defeat original obligations in different ways. In the final chapters, this thesis jumps out of the approach of `ought-to-be' obligations and conceptualizes the notion of knowledge-based `ought-to-do' obligation based on dynamic epistemic logic. The basic idea of defining this notion is that an action ought to be done if and only if the action always improves the initial situation to a better situation. A sound and strongly complete axiom system is also established