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HEИндивидуална стипендия2023–2025

DefHyp · Deformation Theory of infinite-type hyperbolic manifolds

„Хоризонт Европа“ — Действия „Мария Склодовска-Кюри“

Период
2023-07-01 → 2025-07-31
Финансиране от ЕС
175 920 €
Участници
3
Схема
HORIZON-TMA-MSCA-PF-EF

Линиите свързват координатора с партньорите.

Накратко на български

Хиперболичните пространства с безкраен тип се анализират чрез промяна на тяхната геометрия и форма. Тези изследвания помагат за разбирането на квантовата гравитация и връзката между различните физични модели на вселената.

Този кратък обзор е генериран от изкуствен интелект

Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.

Резултати накратко

Deformation Theory of infinite-type hyperbolic manifolds

The DefHyp project aimed to explore the deformation theory of infinite-type hyperbolic 3-manifolds, building on the researcher's pioneering work in this emerging area of geometry and topology. The main research objectives, as outlined in Annex 1, include: 1. Parametrize quasi-conformal deformations of hyperbolic structures on infinite-type 3-manifolds Q1.A). 2. Investigate quasi-conformal rigidity in infinite-type hyperbolic manifolds (Q2). 3. Explore uniqueness and rigidity questions in the spirit of the Mostow Rigidity and Ending Lamination Theorems (Q3, Q4). 4. Analyze the renormalised volume and its implications for AdS/CFT correspondence and quantum gravity (Q5, Q6). Good progress has been made toward Q1 and Q2, culminating in several key results now published or submitted by the researcher and he has now an almost completed pre-print. These works lay foundational tools for understanding deformation spaces of infinite-type manifolds, addressing previously open questions. Progress on Q3 and Q4 has been more limited, but new research directions have been opened thanks to the participation at the BIRS: Blooming Beast. Important progress on Q5 has been done culminating in two papers and third one being written. The second question on renormalized volume has partial result. The researcher and collaborators have proven most of the result but are trying to prove a stronger convergence, work up to date has been written in various manuscripts and opened interesting new directions and re-covered more explicit bounds on classical results of Thurston by further developing the deformation theory of hyperbolic 3-manifolds.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз

Цел на проекта

Hyperbolic geometry, and its connection to 3-dimensional geometry, have been a key topic in contemporary mathematics, leading for instance to the resolution of the Poincaŕe Conjecture (2006) and to the Fields medals awarded to Thurston (1982), McMullen (1998), Perelman (2006, declined) and Mirzakhani (2014). The project will enter the unexplored territory that opens when removing this fundamental hypothesis. Specifically, the PI plans to attack the very challenging problem of understanding the space of hyperbolic metrics on 3-manifolds that have a non-finitely generated fundamental group. The project lies at the intersection between the study of the topology and geometry of hyperbolic 3-manifolds and is well-suited to the complementary expertise of the PI and his supervisor, professor Schlenker, the PI being an expert of the topology of infinite-type 3-manifolds and the supervisor being an expert in hyperbolic geometry. An essential aspect of this research program is understanding the interplay between the topology and the geometry of infinite-type hyperbolic manifolds with the goal to borrow insights from each side to address issues in the other. One of the first objectives is to understand, by looking at topological properties, how much of the rich theory of the finite-type setting extends to the case where the fundamental group is not finitely generated. The second objective is more geometric and plans to study infinite-type 3-manifolds by seeing them as geometric ‘limits’ of finite-type hyperbolic 3-manifolds and looking at which geometric, or topological, aspects survive in the limit. The second part of the project will involve, under the direction of professor Krasnov, is to investigate the AdS-CFT correspondence, an important conjectural relationship linking quantum gravity (formulated as M-theory) in M and conformal field theories (CFT) in the boundary of M, using tools from hyperbolic geometry, e.g. renormalised volume.

Оригинален текст от CORDIS (на английски).

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Данни: CORDIS, © Европейски съюз