Project acronym WallCrossAG
Project Wall-Crossing and Algebraic Geometry
Researcher (PI) Arend BAYER
Host Institution (HI) THE UNIVERSITY OF EDINBURGH
Country United Kingdom
Call Details Consolidator Grant (CoG), PE1, ERC-2018-COG
Summary We will establish stability conditions and wall-crossing in derived categories as a standard methodology for a wide range of fundamental problems in algebraic geometry. Previous work based on wall-crossing, in particular my joint work with Macri, has led to breakthroughs on the birational geometry of moduli spaces and related varieties. Recent advances have made clear that the power of stability conditions extends far beyond this setting, allowing us to study vanishing theorems or bounds on global sections, Brill-Noether problems, or moduli spaces of varieties.
The Brill-Noether problem is one of the oldest and most fundamental questions of algebraic geometry, aiming to classify possible degrees and embedding dimensions of embeddings of a given variety into projective spaces. Recent work by myself, a post-doc (Chunyi Li) and a PhD student (Feyzbakhsh) of mine has established wall-crossing as a powerful new method for such questions. We will push this method further, all the way towards a proof of Green's conjecture, and the Green-Lazarsfeld conjecture, for all smooth curves.
We will use similar methods to prove new Bogomolov-Gieseker type inequalities for Chern classes of stable sheaves and complexes on higher-dimensional varieties. In addition to constructing stability conditions on projective threefolds---the biggest open problem within the theory of stability conditions, we will apply them to study moduli spaces of sheaves on higher-dimensional varieties, and to characterise special abelian varieties.
We will use the construction of stability conditions for families of varieties in my current joint work to systematically study the geometry of Fano threefolds and fourfolds, in particular their moduli spaces, by establishing relations between different moduli spaces, and describing their Torelli maps. Finally, we will study rationality questions, with a particular view towards a wall-crossing proof of the irrationality of the very general cubic fourfold.
Summary
We will establish stability conditions and wall-crossing in derived categories as a standard methodology for a wide range of fundamental problems in algebraic geometry. Previous work based on wall-crossing, in particular my joint work with Macri, has led to breakthroughs on the birational geometry of moduli spaces and related varieties. Recent advances have made clear that the power of stability conditions extends far beyond this setting, allowing us to study vanishing theorems or bounds on global sections, Brill-Noether problems, or moduli spaces of varieties.
The Brill-Noether problem is one of the oldest and most fundamental questions of algebraic geometry, aiming to classify possible degrees and embedding dimensions of embeddings of a given variety into projective spaces. Recent work by myself, a post-doc (Chunyi Li) and a PhD student (Feyzbakhsh) of mine has established wall-crossing as a powerful new method for such questions. We will push this method further, all the way towards a proof of Green's conjecture, and the Green-Lazarsfeld conjecture, for all smooth curves.
We will use similar methods to prove new Bogomolov-Gieseker type inequalities for Chern classes of stable sheaves and complexes on higher-dimensional varieties. In addition to constructing stability conditions on projective threefolds---the biggest open problem within the theory of stability conditions, we will apply them to study moduli spaces of sheaves on higher-dimensional varieties, and to characterise special abelian varieties.
We will use the construction of stability conditions for families of varieties in my current joint work to systematically study the geometry of Fano threefolds and fourfolds, in particular their moduli spaces, by establishing relations between different moduli spaces, and describing their Torelli maps. Finally, we will study rationality questions, with a particular view towards a wall-crossing proof of the irrationality of the very general cubic fourfold.
Max ERC Funding
1 999 840 €
Duration
Start date: 2019-06-01, End date: 2024-05-31
Project acronym WALLXBIRGEOM
Project Wall-crossing and Birational Geometry
Researcher (PI) Arend Bayer
Host Institution (HI) THE UNIVERSITY OF EDINBURGH
Country United Kingdom
Call Details Starting Grant (StG), PE1, ERC-2013-StG
Summary We will use modern techniques in algebraic geometry, originating from string theory and mirror symmetry, to study fundamental problems of classical flavour. More concretely, we apply wall-crossing in the derived category to the birational geometry of moduli spaces.
Bridgeland stability is a notion of stability for complexes in the derived category. Wall-crossing describes how moduli spaces of stable complexes change under deformation of the stability condition, often via a birational surgery occurring in its minimal model program (MMP). This relates wall-crossing to the most basic question of algebraic geometry, the classification of algebraic varieties.
Our previous results additionally provide a very direct connection between Bridgeland stability conditions and positivity of divisors, the main tool of modern birational geometry. This makes the above link significantly more effective, precise and useful. We will exploit this in the following long-term projects:
1. Prove a Bogomolov-Gieseker type inequality for threefolds that we conjectured previously. This would provide a solution in dimension three to well-known open problems of seemingly completely different nature: the existence of Bridgeland stability conditions, Fujita's conjecture on very ampleness of adjoint line bundles, and projective normality of toric varieties.
2. Study the birational geometry of moduli space of sheaves via wall-crossing, adding more geometric meaning to their MMP.
3. Prove that the MMP for local Calabi-Yau threefolds is completely induced by deformation of Bridgeland stability conditions. The motivation is a derived version of the Kawamata-Morrison cone conjecture, classical questions on Chern classes of stable bundles, and mirror symmetry.
4. Answer major open questions on the birational geometry of the moduli space of genus zero curves (for example, the F-conjecture) using exceptional collections in the derived category and wall-crossing.
Summary
We will use modern techniques in algebraic geometry, originating from string theory and mirror symmetry, to study fundamental problems of classical flavour. More concretely, we apply wall-crossing in the derived category to the birational geometry of moduli spaces.
Bridgeland stability is a notion of stability for complexes in the derived category. Wall-crossing describes how moduli spaces of stable complexes change under deformation of the stability condition, often via a birational surgery occurring in its minimal model program (MMP). This relates wall-crossing to the most basic question of algebraic geometry, the classification of algebraic varieties.
Our previous results additionally provide a very direct connection between Bridgeland stability conditions and positivity of divisors, the main tool of modern birational geometry. This makes the above link significantly more effective, precise and useful. We will exploit this in the following long-term projects:
1. Prove a Bogomolov-Gieseker type inequality for threefolds that we conjectured previously. This would provide a solution in dimension three to well-known open problems of seemingly completely different nature: the existence of Bridgeland stability conditions, Fujita's conjecture on very ampleness of adjoint line bundles, and projective normality of toric varieties.
2. Study the birational geometry of moduli space of sheaves via wall-crossing, adding more geometric meaning to their MMP.
3. Prove that the MMP for local Calabi-Yau threefolds is completely induced by deformation of Bridgeland stability conditions. The motivation is a derived version of the Kawamata-Morrison cone conjecture, classical questions on Chern classes of stable bundles, and mirror symmetry.
4. Answer major open questions on the birational geometry of the moduli space of genus zero curves (for example, the F-conjecture) using exceptional collections in the derived category and wall-crossing.
Max ERC Funding
1 282 912 €
Duration
Start date: 2013-12-01, End date: 2018-11-30
Project acronym Waterscales
Project Mathematical and computational foundations for modeling cerebral fluid flow.
Researcher (PI) Marie Elisabeth ROGNES
Host Institution (HI) SIMULA RESEARCH LABORATORY AS
Country Norway
Call Details Starting Grant (StG), PE1, ERC-2016-STG
Summary Your brain has its own waterscape: whether you are reading or sleeping, fluid flows through the brain tissue and clears waste in the process. These physiological processes are crucial for the well-being of the brain. In spite of their importance we understand them but little. Mathematics and numerics could play a crucial role in gaining new insight. Indeed, medical doctors express an urgent need for multiscale modeling of water transport through the brain, to overcome limitations in traditional techniques. Surprisingly little attention has been paid to the numerics of the brain's waterscape however, and fundamental knowledge is missing.
In response, the Waterscales ambition is to establish the mathematical and computational foundations for predictively modeling fluid flow and solute transport through the brain across scales -- from the cellular to the organ level. The project aims to bridge multiscale fluid mechanics and cellular electrophysiology to pioneer new families of mathematical models that couple macroscale, mesoscale and microscale flow with glial cell dynamics. For these models, we will design numerical discretizations that preserve key properties and that allow for whole organ simulations. To evaluate predictability, we will develop a new computational platform for model adaptivity and calibration. The project is multidisciplinary combining mathematics, mechanics, scientific computing, and physiology.
If successful, this project enables the first in silico studies of the brain's waterscape across scales. The new models would open up a new research field within computational neuroscience with ample opportunities for further mathematical and more applied study. The processes at hand are associated with neurodegenerative diseases e.g. dementia and with brain swelling caused by e.g. stroke. The Waterscales project will provide the field with a sorely needed, new avenue of investigation to understand these conditions, with tremendous long-term impact.
Summary
Your brain has its own waterscape: whether you are reading or sleeping, fluid flows through the brain tissue and clears waste in the process. These physiological processes are crucial for the well-being of the brain. In spite of their importance we understand them but little. Mathematics and numerics could play a crucial role in gaining new insight. Indeed, medical doctors express an urgent need for multiscale modeling of water transport through the brain, to overcome limitations in traditional techniques. Surprisingly little attention has been paid to the numerics of the brain's waterscape however, and fundamental knowledge is missing.
In response, the Waterscales ambition is to establish the mathematical and computational foundations for predictively modeling fluid flow and solute transport through the brain across scales -- from the cellular to the organ level. The project aims to bridge multiscale fluid mechanics and cellular electrophysiology to pioneer new families of mathematical models that couple macroscale, mesoscale and microscale flow with glial cell dynamics. For these models, we will design numerical discretizations that preserve key properties and that allow for whole organ simulations. To evaluate predictability, we will develop a new computational platform for model adaptivity and calibration. The project is multidisciplinary combining mathematics, mechanics, scientific computing, and physiology.
If successful, this project enables the first in silico studies of the brain's waterscape across scales. The new models would open up a new research field within computational neuroscience with ample opportunities for further mathematical and more applied study. The processes at hand are associated with neurodegenerative diseases e.g. dementia and with brain swelling caused by e.g. stroke. The Waterscales project will provide the field with a sorely needed, new avenue of investigation to understand these conditions, with tremendous long-term impact.
Max ERC Funding
1 500 000 €
Duration
Start date: 2017-04-01, End date: 2022-03-31
Project acronym WordMeasures
Project Word Measures in Groups and Random Cayley Graphs
Researcher (PI) Doron Puder
Host Institution (HI) TEL AVIV UNIVERSITY
Country Israel
Call Details Starting Grant (StG), PE1, ERC-2019-STG
Summary Recent years brought immense progress in the study of Cayley graphs of finite groups, with many new results concerning their expansion, diameter, girth etc. Yet, many central open questions remain. These questions, especially those concerning random Cayley graphs, are major motivation to this proposed research.
Central in this project is the study of word measures in finite and compact groups. A word w in a free group F induces a measure on every finite or compact group G as follows: substitute every generator of F with an independent Haar-random element of G and evaluate the product defined by w to obtain a random element in G. The main goal here is to expose the invariants of the word w which control different aspects of these measures. The study of word measures, mostly by the PI and collaborators, has proven useful not only for analyzing random Cayley and Schreier graphs of G, but also for many questions revolving around free groups and their automorphism groups. Moreover, the study of word measures has exposed a deep and beautiful mathematical theory with surprising connections to objects in combinatorial and geometric group theory and in low dimensional topology. This theory is still in its infancy, with many beautiful open questions and intriguing challenges ahead.
Another line of research revolves around a few irreducible representations of a finite group which control the spectral gap of Cayley graphs. The proof of Aldous' conjecture in 2010 showed that this happens more commonly than one could have naïvely guessed. There is additional evidence, some of which found by the PI and collaborators, that Aldous' conjecture is only the tip of the iceberg, especially for Cayley graphs of the symmetric group Sym(n). Our most optimistic conjectures here have extremely strong consequences for these Cayley graphs.
We intend to use our progress in the above two directions in order to answer some intriguing open questions concerning Cayley and Schreier graphs.
Summary
Recent years brought immense progress in the study of Cayley graphs of finite groups, with many new results concerning their expansion, diameter, girth etc. Yet, many central open questions remain. These questions, especially those concerning random Cayley graphs, are major motivation to this proposed research.
Central in this project is the study of word measures in finite and compact groups. A word w in a free group F induces a measure on every finite or compact group G as follows: substitute every generator of F with an independent Haar-random element of G and evaluate the product defined by w to obtain a random element in G. The main goal here is to expose the invariants of the word w which control different aspects of these measures. The study of word measures, mostly by the PI and collaborators, has proven useful not only for analyzing random Cayley and Schreier graphs of G, but also for many questions revolving around free groups and their automorphism groups. Moreover, the study of word measures has exposed a deep and beautiful mathematical theory with surprising connections to objects in combinatorial and geometric group theory and in low dimensional topology. This theory is still in its infancy, with many beautiful open questions and intriguing challenges ahead.
Another line of research revolves around a few irreducible representations of a finite group which control the spectral gap of Cayley graphs. The proof of Aldous' conjecture in 2010 showed that this happens more commonly than one could have naïvely guessed. There is additional evidence, some of which found by the PI and collaborators, that Aldous' conjecture is only the tip of the iceberg, especially for Cayley graphs of the symmetric group Sym(n). Our most optimistic conjectures here have extremely strong consequences for these Cayley graphs.
We intend to use our progress in the above two directions in order to answer some intriguing open questions concerning Cayley and Schreier graphs.
Max ERC Funding
1 470 875 €
Duration
Start date: 2020-02-01, End date: 2025-01-31
Project acronym WORDS
Project Words and Waring type problems
Researcher (PI) Aner Shalev
Host Institution (HI) THE HEBREW UNIVERSITY OF JERUSALEM
Country Israel
Call Details Advanced Grant (AdG), PE1, ERC-2009-AdG
Summary Hilbert's solution to Waring problem in Number Theory shows that every positive integer is a sum of g(n) nth powers. Surprising non-commutative analogues of this phenomenon were discovered recently in Group Theory, where powers are replaced by general words. Moreover, the study of group words occurs naturally in important contexts, such as the Burnside problems, Serre's problem on profinite groups, and finite simple group theory. We propose a systematic study of word maps on groups, their images and kernels, as well as related Waring type problems. These include a celebrated conjecture of Thompson, problems regarding covering numbers and mixing times of random walks, as well as probabilistic identities in finite and profinite groups. This is a highly challenging project in which we intend to utilize a wide spectrum of tools, including Representation Theory, Algebraic Geometry, Number Theory, computational group theory, as well as probabilistic methods and Lie methods. Moreover, we aim to establish new results on representations and character bounds, which would be very useful in various additional contexts. Apart from their intrinsic interest, the problems and conjectures we propose have exciting applications to other fields, and the project is likely to shed new light not just in group theory but also in combinatorics, probability and geometry.
Summary
Hilbert's solution to Waring problem in Number Theory shows that every positive integer is a sum of g(n) nth powers. Surprising non-commutative analogues of this phenomenon were discovered recently in Group Theory, where powers are replaced by general words. Moreover, the study of group words occurs naturally in important contexts, such as the Burnside problems, Serre's problem on profinite groups, and finite simple group theory. We propose a systematic study of word maps on groups, their images and kernels, as well as related Waring type problems. These include a celebrated conjecture of Thompson, problems regarding covering numbers and mixing times of random walks, as well as probabilistic identities in finite and profinite groups. This is a highly challenging project in which we intend to utilize a wide spectrum of tools, including Representation Theory, Algebraic Geometry, Number Theory, computational group theory, as well as probabilistic methods and Lie methods. Moreover, we aim to establish new results on representations and character bounds, which would be very useful in various additional contexts. Apart from their intrinsic interest, the problems and conjectures we propose have exciting applications to other fields, and the project is likely to shed new light not just in group theory but also in combinatorics, probability and geometry.
Max ERC Funding
1 197 800 €
Duration
Start date: 2010-01-01, End date: 2014-12-31
Project acronym ZETA-FM
Project Zeta functions and Fourier-Mukai transforms
Researcher (PI) Lenny Taelman
Host Institution (HI) UNIVERSITEIT VAN AMSTERDAM
Country Netherlands
Call Details Consolidator Grant (CoG), PE1, ERC-2019-COG
Summary Arithmetic geometry and the study of derived categories of coherent sheaves are two central areas of research in algebraic geometry. Despite their many points of contact, they have until recently remained largely disjoint.
The zeta function of an algebraic variety over a finite field is one of the most studied invariants in arithmetic geometry, and a conjecture of Orlov predicts that this invariant can be detected by the derived category of coherent sheaves on the variety. In this project, I will prove this for large classes of varieties.
To achieve this, I will enrich a wide range of techniques from arithmetic geometry with ideas that have classically been used in the study of derived categories. In this way, this project will also serve as a catalyst for further interaction between arithmetic geometry and derived categories.
Summary
Arithmetic geometry and the study of derived categories of coherent sheaves are two central areas of research in algebraic geometry. Despite their many points of contact, they have until recently remained largely disjoint.
The zeta function of an algebraic variety over a finite field is one of the most studied invariants in arithmetic geometry, and a conjecture of Orlov predicts that this invariant can be detected by the derived category of coherent sheaves on the variety. In this project, I will prove this for large classes of varieties.
To achieve this, I will enrich a wide range of techniques from arithmetic geometry with ideas that have classically been used in the study of derived categories. In this way, this project will also serve as a catalyst for further interaction between arithmetic geometry and derived categories.
Max ERC Funding
2 000 000 €
Duration
Start date: 2020-09-01, End date: 2025-08-31
