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Key research themes
1. How can tame ramification and resolution of singularities principles be leveraged to establish existential decidability and Ax-Kochen-Ershov principles in valued fields with infinite ramification?
This theme investigates the extension of model-theoretic results, notably the Ax-Kochen-Ershov (AKE) principles and decidability of valued fields, from classical settings (residue characteristic zero and unramified mixed characteristic) to tamely ramified fields including infinitely ramified extensions. The work focuses on the conditional use of forms of resolution of singularities to derive existential decidability results and transfer principles that circumvent the limitations posed by infinite ramification.
Key finding: Under the assumption of a particular form of resolution of singularities (Conjecture R), the paper proves a general existential Ax-Kochen-Ershov principle for henselian valued fields that are tamely ramified over a discretely... Read more
2. What structural and functorial properties of ramified extensions and formal moduli spaces enable arithmetic transfer conjectures in the presence of ramification?
This research theme focuses on the development of arithmetic transfer (AT) conjectures relating orbital integrals to arithmetic intersection numbers on formal moduli spaces in ramified contexts. It examines the construction of regular formal moduli spaces of p-divisible groups associated to unitary groups over ramified quadratic extensions, considerations of morphisms between them, and the formulation and partial resolution of conjectures extending Gan-Gross-Prasad and arithmetic fundamental lemma (AFL) frameworks to ramified settings, including exotic smoothness phenomena.
Key finding: The paper constructs various regular formal moduli spaces (Rapoport-Zink spaces) of p-divisible groups associated with unitary groups under ramified quadratic extensions, including cases exhibiting exotic smoothness where... Read more
Key finding: This work formulates and proves a local arithmetic transfer (AT) conjecture in the presence of ramification for unitary groups over ramified quadratic p-adic extensions, focusing on exotic smooth Rapoport-Zink spaces. For odd... Read more
3. How can ramification theory and structure of wild ramified automorphisms be characterized via formal power series, ramification filtrations, and conditions on iterates?
This theme explores the ramification properties of wildly ramified automorphisms in local fields of positive characteristic through the analytic and algebraic structure of associated formal power series (e.g., Nottingham groups). It centers on criteria identifying b-ramified power series via the behavior of iterates, computation of first nontrivial coefficients in such iterates, and the characterization of ramification filtrations in terms of compositional dynamics of power series. Applications include arithmetic dynamics and understanding fine structure of wild inertia groups.
Key finding: Under the hypothesis p > b^2, the paper computes explicitly the first nontrivial coefficient of the p-th iterate of a formal power series f = X + Σ a_i X^{b+i} over the residue field k, linking this to the ramification... Read more
4. What are the limitations and techniques for minimizing ramification in finite Galois extensions of number fields, especially for semiabelian groups and ramification group decompositions?
This theme addresses the algebraic and arithmetic problem of realizing finite groups as Galois groups over Q with minimal ramification, focusing on tamely ramified extensions and the role of semiabelian groups. It discusses the minimal number of ramified primes required (in relation to generators of the group), functorial properties of wreath products affecting group construction and ramification, and solutions to the minimal ramification problem for large classes of groups, including nilpotent and p-groups.
Key finding: Generalizing previous results, the paper proves that any finite semiabelian group G can be realized as a Galois group over Q of a tamely ramified extension with exactly wl(G) rational primes ramified, where wl(G) is the... Read more
5. How can temporal description logics be employed to represent and computationally solve the ramification problem of indirect effects in dynamic systems?
This theme investigates the application of temporal description logics (TDL) to capture and reason about indirect, non-persistent effects of actions (the ramification problem) in dynamic artificial intelligence systems. By combining interval-based temporal operators with expressive DL syntaxes, the work provides a formalism that models integrity constraints, temporally-scoped actions, and their indirect consequences, further offering algorithms for generating and evaluating static and dynamic effects, including effects altering historical knowledge.
Key finding: The paper develops a temporal extension of description logics, integrating interval-based temporal constructs and action representations that can express temporal integrity constraints and non-persistent effects. Using this... Read more
6. What are the implications of ramification behavior on the arithmetic, geometric, and model-theoretic properties of extensions of local fields and formal moduli spaces, including inertia group structure, stable vector bundles, and canonical lifting of curves?
This theme encompasses the study of ramification structures in number fields and local fields, focusing on the interaction between ramification and geometric objects such as vector bundles and moduli spaces, and on arithmetic consequences like canonical lifts and reductions of Galois representations. It includes investigations of inertia groups in Galois covers, the influence of ramification on stability of vector bundles under pullback, canonical lifting computations for genus 2 curves in characteristic two, and non-vanishing phenomena in arithmetic geometry connected with ramification.
Key finding: The paper introduces the notion of genuinely ramified finite separable surjective maps between irreducible normal projective varieties as those inducing surjective étale fundamental group homomorphisms. It proves that such... Read more
Key finding: The article provides a computational method for constructing canonical lifts of genus 2 hyperelliptic curves defined over fields of characteristic two, specifically addressing the intricacies posed by ramification arising in... Read more
Key finding: Under certain weight congruences and slope conditions on modular eigenforms, the paper constructs new eigenforms whose mod p Galois representations are twists by the cyclotomic character of the original. Employing... Read more
Key finding: The paper studies the reduction of the wild part of inertia groups in Galois covers of the projective line branched only at infinity over a field of characteristic p. Under conditions involving jumps in the ramification... Read more