Project acronym 100 Archaic Genomes
Project Genome sequences from extinct hominins
Researcher (PI) Svante PÄÄBO
Host Institution (HI) MAX-PLANCK-GESELLSCHAFT ZUR FORDERUNG DER WISSENSCHAFTEN EV
Call Details Advanced Grant (AdG), LS2, ERC-2015-AdG
Summary Neandertals and Denisovans, an Asian group distantly related to Neandertals, are the closest evolutionary relatives of present-day humans. They are thus of direct relevance for understanding the origin of modern humans and how modern humans differ from their closest relatives. We will generate genome-wide data from a large number of Neandertal and Denisovan individuals from across their geographical and temporal range as well as from other extinct hominin groups which we may discover. This will be possible by automating highly sensitive approaches to ancient DNA extraction and DNA libraries construction that we have developed so that they can be applied to many specimens from many sites in order to identify those that contain retrievable DNA. Whenever possible we will sequence whole genomes and in other cases use DNA capture methods to generate high-quality data from representative parts of the genome. This will allow us to study the population history of Neandertals and Denisovans, elucidate how many times and where these extinct hominins contributed genes to present-day people, and the extent to which modern humans and archaic groups contributed genetically to Neandertals and Denisovans. By retrieving DNA from specimens that go back to the Middle Pleistocene we will furthermore shed light on the early history and origins of Neandertals and Denisovans.
Summary
Neandertals and Denisovans, an Asian group distantly related to Neandertals, are the closest evolutionary relatives of present-day humans. They are thus of direct relevance for understanding the origin of modern humans and how modern humans differ from their closest relatives. We will generate genome-wide data from a large number of Neandertal and Denisovan individuals from across their geographical and temporal range as well as from other extinct hominin groups which we may discover. This will be possible by automating highly sensitive approaches to ancient DNA extraction and DNA libraries construction that we have developed so that they can be applied to many specimens from many sites in order to identify those that contain retrievable DNA. Whenever possible we will sequence whole genomes and in other cases use DNA capture methods to generate high-quality data from representative parts of the genome. This will allow us to study the population history of Neandertals and Denisovans, elucidate how many times and where these extinct hominins contributed genes to present-day people, and the extent to which modern humans and archaic groups contributed genetically to Neandertals and Denisovans. By retrieving DNA from specimens that go back to the Middle Pleistocene we will furthermore shed light on the early history and origins of Neandertals and Denisovans.
Max ERC Funding
2 350 000 €
Duration
Start date: 2016-11-01, End date: 2021-10-31
Project acronym ACCOMPLI
Project Assembly and maintenance of a co-regulated chromosomal compartment
Researcher (PI) Peter Burkhard Becker
Host Institution (HI) LUDWIG-MAXIMILIANS-UNIVERSITAET MUENCHEN
Call Details Advanced Grant (AdG), LS2, ERC-2011-ADG_20110310
Summary "Eukaryotic nuclei are organised into functional compartments, – local microenvironments that are enriched in certain molecules or biochemical activities and therefore specify localised functional outputs. Our study seeks to unveil fundamental principles of co-regulation of genes in a chromo¬somal compartment and the preconditions for homeostasis of such a compartment in the dynamic nuclear environment.
The dosage-compensated X chromosome of male Drosophila flies satisfies the criteria for a functional com¬partment. It is rendered structurally distinct from all other chromosomes by association of a regulatory ribonucleoprotein ‘Dosage Compensation Complex’ (DCC), enrichment of histone modifications and global decondensation. As a result, most genes on the X chromosome are co-ordinately activated. Autosomal genes inserted into the X acquire X-chromosomal features and are subject to the X-specific regulation.
We seek to uncover the molecular principles that initiate, establish and maintain the dosage-compensated chromosome. We will follow the kinetics of DCC assembly and the timing of association with different types of chromosomal targets in nuclei with high spatial resolution afforded by sub-wavelength microscopy and deep sequencing of DNA binding sites. We will characterise DCC sub-complexes with respect to their roles as kinetic assembly intermediates or as representations of local, functional heterogeneity. We will evaluate the roles of a DCC- novel ubiquitin ligase activity for homeostasis.
Crucial to the recruitment of the DCC and its distribution to target genes are non-coding roX RNAs that are transcribed from the X. We will determine the secondary structure ‘signatures’ of roX RNAs in vitro and determine the binding sites of the protein subunits in vivo. By biochemical and cellular reconstitution will test the hypothesis that roX-encoded RNA aptamers orchestrate the assembly of the DCC and contribute to the exquisite targeting of the complex."
Summary
"Eukaryotic nuclei are organised into functional compartments, – local microenvironments that are enriched in certain molecules or biochemical activities and therefore specify localised functional outputs. Our study seeks to unveil fundamental principles of co-regulation of genes in a chromo¬somal compartment and the preconditions for homeostasis of such a compartment in the dynamic nuclear environment.
The dosage-compensated X chromosome of male Drosophila flies satisfies the criteria for a functional com¬partment. It is rendered structurally distinct from all other chromosomes by association of a regulatory ribonucleoprotein ‘Dosage Compensation Complex’ (DCC), enrichment of histone modifications and global decondensation. As a result, most genes on the X chromosome are co-ordinately activated. Autosomal genes inserted into the X acquire X-chromosomal features and are subject to the X-specific regulation.
We seek to uncover the molecular principles that initiate, establish and maintain the dosage-compensated chromosome. We will follow the kinetics of DCC assembly and the timing of association with different types of chromosomal targets in nuclei with high spatial resolution afforded by sub-wavelength microscopy and deep sequencing of DNA binding sites. We will characterise DCC sub-complexes with respect to their roles as kinetic assembly intermediates or as representations of local, functional heterogeneity. We will evaluate the roles of a DCC- novel ubiquitin ligase activity for homeostasis.
Crucial to the recruitment of the DCC and its distribution to target genes are non-coding roX RNAs that are transcribed from the X. We will determine the secondary structure ‘signatures’ of roX RNAs in vitro and determine the binding sites of the protein subunits in vivo. By biochemical and cellular reconstitution will test the hypothesis that roX-encoded RNA aptamers orchestrate the assembly of the DCC and contribute to the exquisite targeting of the complex."
Max ERC Funding
2 482 770 €
Duration
Start date: 2012-02-01, End date: 2017-01-31
Project acronym AGALT
Project Asymptotic Geometric Analysis and Learning Theory
Researcher (PI) Shahar Mendelson
Host Institution (HI) TECHNION - ISRAEL INSTITUTE OF TECHNOLOGY
Call Details Starting Grant (StG), PE1, ERC-2007-StG
Summary In a typical learning problem one tries to approximate an unknown function by a function from a given class using random data, sampled according to an unknown measure. In this project we will be interested in parameters that govern the complexity of a learning problem. It turns out that this complexity is determined by the geometry of certain sets in high dimension that are connected to the given class (random coordinate projections of the class). Thus, one has to understand the structure of these sets as a function of the dimension - which is given by the cardinality of the random sample. The resulting analysis leads to many theoretical questions in Asymptotic Geometric Analysis, Probability (most notably, Empirical Processes Theory) and Combinatorics, which are of independent interest beyond the application to Learning Theory. Our main goal is to describe the role of various complexity parameters involved in a learning problem, to analyze the connections between them and to investigate the way they determine the geometry of the relevant high dimensional sets. Some of the questions we intend to tackle are well known open problems and making progress towards their solution will have a significant theoretical impact. Moreover, this project should lead to a more complete theory of learning and is likely to have some practical impact, for example, in the design of more efficient learning algorithms.
Summary
In a typical learning problem one tries to approximate an unknown function by a function from a given class using random data, sampled according to an unknown measure. In this project we will be interested in parameters that govern the complexity of a learning problem. It turns out that this complexity is determined by the geometry of certain sets in high dimension that are connected to the given class (random coordinate projections of the class). Thus, one has to understand the structure of these sets as a function of the dimension - which is given by the cardinality of the random sample. The resulting analysis leads to many theoretical questions in Asymptotic Geometric Analysis, Probability (most notably, Empirical Processes Theory) and Combinatorics, which are of independent interest beyond the application to Learning Theory. Our main goal is to describe the role of various complexity parameters involved in a learning problem, to analyze the connections between them and to investigate the way they determine the geometry of the relevant high dimensional sets. Some of the questions we intend to tackle are well known open problems and making progress towards their solution will have a significant theoretical impact. Moreover, this project should lead to a more complete theory of learning and is likely to have some practical impact, for example, in the design of more efficient learning algorithms.
Max ERC Funding
750 000 €
Duration
Start date: 2009-03-01, End date: 2014-02-28
Project acronym AGELESS
Project Comparative genomics / ‘wildlife’ transcriptomics uncovers the mechanisms of halted ageing in mammals
Researcher (PI) Emma Teeling
Host Institution (HI) UNIVERSITY COLLEGE DUBLIN, NATIONAL UNIVERSITY OF IRELAND, DUBLIN
Call Details Starting Grant (StG), LS2, ERC-2012-StG_20111109
Summary "Ageing is the gradual and irreversible breakdown of living systems associated with the advancement of time, which leads to an increase in vulnerability and eventual mortality. Despite recent advances in ageing research, the intrinsic complexity of the ageing process has prevented a full understanding of this process, therefore, ageing remains a grand challenge in contemporary biology. In AGELESS, we will tackle this challenge by uncovering the molecular mechanisms of halted ageing in a unique model system, the bats. Bats are the longest-lived mammals relative to their body size, and defy the ‘rate-of-living’ theories as they use twice as much the energy as other species of considerable size, but live far longer. This suggests that bats have some underlying mechanisms that may explain their exceptional longevity. In AGELESS, we will identify the molecular mechanisms that enable mammals to achieve extraordinary longevity, using state-of-the-art comparative genomic methodologies focused on bats. We will identify, using population transcriptomics and telomere/mtDNA genomics, the molecular changes that occur in an ageing wild population of bats to uncover how bats ‘age’ so slowly compared with other mammals. In silico whole genome analyses, field based ageing transcriptomic data, mtDNA and telomeric studies will be integrated and analysed using a networks approach, to ascertain how these systems interact to halt ageing. For the first time, we will be able to utilize the diversity seen within nature to identify key molecular targets and regions that regulate and control ageing in mammals. AGELESS will provide a deeper understanding of the causal mechanisms of ageing, potentially uncovering the crucial molecular pathways that can be modified to halt, alleviate and perhaps even reverse this process in man."
Summary
"Ageing is the gradual and irreversible breakdown of living systems associated with the advancement of time, which leads to an increase in vulnerability and eventual mortality. Despite recent advances in ageing research, the intrinsic complexity of the ageing process has prevented a full understanding of this process, therefore, ageing remains a grand challenge in contemporary biology. In AGELESS, we will tackle this challenge by uncovering the molecular mechanisms of halted ageing in a unique model system, the bats. Bats are the longest-lived mammals relative to their body size, and defy the ‘rate-of-living’ theories as they use twice as much the energy as other species of considerable size, but live far longer. This suggests that bats have some underlying mechanisms that may explain their exceptional longevity. In AGELESS, we will identify the molecular mechanisms that enable mammals to achieve extraordinary longevity, using state-of-the-art comparative genomic methodologies focused on bats. We will identify, using population transcriptomics and telomere/mtDNA genomics, the molecular changes that occur in an ageing wild population of bats to uncover how bats ‘age’ so slowly compared with other mammals. In silico whole genome analyses, field based ageing transcriptomic data, mtDNA and telomeric studies will be integrated and analysed using a networks approach, to ascertain how these systems interact to halt ageing. For the first time, we will be able to utilize the diversity seen within nature to identify key molecular targets and regions that regulate and control ageing in mammals. AGELESS will provide a deeper understanding of the causal mechanisms of ageing, potentially uncovering the crucial molecular pathways that can be modified to halt, alleviate and perhaps even reverse this process in man."
Max ERC Funding
1 499 768 €
Duration
Start date: 2013-01-01, End date: 2017-12-31
Project acronym Agglomerates
Project Infinite Protein Self-Assembly in Health and Disease
Researcher (PI) Emmanuel Doram LEVY
Host Institution (HI) WEIZMANN INSTITUTE OF SCIENCE
Call Details Consolidator Grant (CoG), LS2, ERC-2018-COG
Summary Understanding how proteins respond to mutations is of paramount importance to biology and disease. While protein stability and misfolding have been instrumental in rationalizing the impact of mutations, we recently discovered that an alternative route is also frequent, where mutations at the surface of symmetric proteins trigger novel self-interactions that lead to infinite self-assembly. This mechanism can be involved in disease, as in sickle-cell anemia, but may also serve in adaptation. Importantly, it differs fundamentally from aggregation, because misfolding does not drive it. Thus, we term it “agglomeration”. The ease with which agglomeration can occur, even by single point mutations, shifts the paradigm of how quickly new protein assemblies can emerge, both in health and disease. This prompts us to determine the basic principles of protein agglomeration and explore its implications in cell physiology and human disease.
We propose an interdisciplinary research program bridging atomic and cellular scales to explore agglomeration in three aims: (i) Map the landscape of protein agglomeration in response to mutation in endogenous yeast proteins; (ii) Characterize how yeast physiology impacts agglomeration by changes in gene expression or cell state, and, conversely, how protein agglomerates impact yeast fitness. (iii) Analyze agglomeration in relation to human disease via two approaches. First, by predicting single nucleotide polymorphisms that trigger agglomeration, prioritizing them using knowledge from Aims 1 & 2, and characterizing them experimentally. Second, by providing a proof-of-concept that agglomeration can be exploited in drug design, whereby drugs induce its formation, like mutations can do.
Overall, through this research, we aim to establish agglomeration as a paradigm for protein assembly, with implications for our understanding of evolution, physiology, and disease.
Summary
Understanding how proteins respond to mutations is of paramount importance to biology and disease. While protein stability and misfolding have been instrumental in rationalizing the impact of mutations, we recently discovered that an alternative route is also frequent, where mutations at the surface of symmetric proteins trigger novel self-interactions that lead to infinite self-assembly. This mechanism can be involved in disease, as in sickle-cell anemia, but may also serve in adaptation. Importantly, it differs fundamentally from aggregation, because misfolding does not drive it. Thus, we term it “agglomeration”. The ease with which agglomeration can occur, even by single point mutations, shifts the paradigm of how quickly new protein assemblies can emerge, both in health and disease. This prompts us to determine the basic principles of protein agglomeration and explore its implications in cell physiology and human disease.
We propose an interdisciplinary research program bridging atomic and cellular scales to explore agglomeration in three aims: (i) Map the landscape of protein agglomeration in response to mutation in endogenous yeast proteins; (ii) Characterize how yeast physiology impacts agglomeration by changes in gene expression or cell state, and, conversely, how protein agglomerates impact yeast fitness. (iii) Analyze agglomeration in relation to human disease via two approaches. First, by predicting single nucleotide polymorphisms that trigger agglomeration, prioritizing them using knowledge from Aims 1 & 2, and characterizing them experimentally. Second, by providing a proof-of-concept that agglomeration can be exploited in drug design, whereby drugs induce its formation, like mutations can do.
Overall, through this research, we aim to establish agglomeration as a paradigm for protein assembly, with implications for our understanding of evolution, physiology, and disease.
Max ERC Funding
2 574 819 €
Duration
Start date: 2019-04-01, End date: 2024-03-31
Project acronym AMAREC
Project Amenability, Approximation and Reconstruction
Researcher (PI) Wilhelm WINTER
Host Institution (HI) WESTFAELISCHE WILHELMS-UNIVERSITAET MUENSTER
Call Details Advanced Grant (AdG), PE1, ERC-2018-ADG
Summary Algebras of operators on Hilbert spaces were originally introduced as the right framework for the mathematical description of quantum mechanics. In modern mathematics the scope has much broadened due to the highly versatile nature of operator algebras. They are particularly useful in the analysis of groups and their actions. Amenability is a finiteness property which occurs in many different contexts and which can be characterised in many different ways. We will analyse amenability in terms of approximation properties, in the frameworks of abstract C*-algebras, of topological dynamical systems, and of discrete groups. Such approximation properties will serve as bridging devices between these setups, and they will be used to systematically recover geometric information about the underlying structures. When passing from groups, and more generally from dynamical systems, to operator algebras, one loses information, but one gains new tools to isolate and analyse pertinent properties of the underlying structure. We will mostly be interested in the topological setting, and in the associated C*-algebras. Amenability of groups or of dynamical systems then translates into the completely positive approximation property. Systems of completely positive approximations store all the essential data about a C*-algebra, and sometimes one can arrange the systems so that one can directly read of such information. For transformation group C*-algebras, one can achieve this by using approximation properties of the underlying dynamics. To some extent one can even go back, and extract dynamical approximation properties from completely positive approximations of the C*-algebra. This interplay between approximation properties in topological dynamics and in noncommutative topology carries a surprisingly rich structure. It connects directly to the heart of the classification problem for nuclear C*-algebras on the one hand, and to central open questions on amenable dynamics on the other.
Summary
Algebras of operators on Hilbert spaces were originally introduced as the right framework for the mathematical description of quantum mechanics. In modern mathematics the scope has much broadened due to the highly versatile nature of operator algebras. They are particularly useful in the analysis of groups and their actions. Amenability is a finiteness property which occurs in many different contexts and which can be characterised in many different ways. We will analyse amenability in terms of approximation properties, in the frameworks of abstract C*-algebras, of topological dynamical systems, and of discrete groups. Such approximation properties will serve as bridging devices between these setups, and they will be used to systematically recover geometric information about the underlying structures. When passing from groups, and more generally from dynamical systems, to operator algebras, one loses information, but one gains new tools to isolate and analyse pertinent properties of the underlying structure. We will mostly be interested in the topological setting, and in the associated C*-algebras. Amenability of groups or of dynamical systems then translates into the completely positive approximation property. Systems of completely positive approximations store all the essential data about a C*-algebra, and sometimes one can arrange the systems so that one can directly read of such information. For transformation group C*-algebras, one can achieve this by using approximation properties of the underlying dynamics. To some extent one can even go back, and extract dynamical approximation properties from completely positive approximations of the C*-algebra. This interplay between approximation properties in topological dynamics and in noncommutative topology carries a surprisingly rich structure. It connects directly to the heart of the classification problem for nuclear C*-algebras on the one hand, and to central open questions on amenable dynamics on the other.
Max ERC Funding
1 596 017 €
Duration
Start date: 2019-10-01, End date: 2024-09-30
Project acronym ANAMULTISCALE
Project Analysis of Multiscale Systems Driven by Functionals
Researcher (PI) Alexander Mielke
Host Institution (HI) FORSCHUNGSVERBUND BERLIN EV
Call Details Advanced Grant (AdG), PE1, ERC-2010-AdG_20100224
Summary Many complex phenomena in the sciences are described by nonlinear partial differential equations, the solutions of which exhibit oscillations and concentration effects on multiple temporal or spatial scales. Our aim is to use methods from applied analysis to contribute to the understanding of the interplay of effects on different scales. The central question is to determine those quantities on the microscale which are needed to for the correct description of the macroscopic evolution.
We aim to develop a mathematical framework for analyzing and modeling coupled systems with multiple scales. This will include Hamiltonian dynamics as well as different types of dissipation like gradient flows or rate-independent dynamics. The choice of models will be guided by specific applications in material modeling (e.g., thermoplasticity, pattern formation, porous media) and optoelectronics (pulse interaction, Maxwell-Bloch systems, semiconductors, quantum mechanics). The research will address mathematically fundamental issues like existence and stability of solutions but will mainly be devoted to the modeling of multiscale phenomena in evolution systems. We will focus on systems with geometric structures, where the dynamics is driven by functionals. Thus, we can go much beyond the classical theory of homogenization and singular perturbations. The novel features of our approach are
- the combination of different dynamical effects in one framework,
- the use of geometric and metric structures for coupled partial differential equations,
- the exploitation of Gamma-convergence for evolution systems driven by functionals.
Summary
Many complex phenomena in the sciences are described by nonlinear partial differential equations, the solutions of which exhibit oscillations and concentration effects on multiple temporal or spatial scales. Our aim is to use methods from applied analysis to contribute to the understanding of the interplay of effects on different scales. The central question is to determine those quantities on the microscale which are needed to for the correct description of the macroscopic evolution.
We aim to develop a mathematical framework for analyzing and modeling coupled systems with multiple scales. This will include Hamiltonian dynamics as well as different types of dissipation like gradient flows or rate-independent dynamics. The choice of models will be guided by specific applications in material modeling (e.g., thermoplasticity, pattern formation, porous media) and optoelectronics (pulse interaction, Maxwell-Bloch systems, semiconductors, quantum mechanics). The research will address mathematically fundamental issues like existence and stability of solutions but will mainly be devoted to the modeling of multiscale phenomena in evolution systems. We will focus on systems with geometric structures, where the dynamics is driven by functionals. Thus, we can go much beyond the classical theory of homogenization and singular perturbations. The novel features of our approach are
- the combination of different dynamical effects in one framework,
- the use of geometric and metric structures for coupled partial differential equations,
- the exploitation of Gamma-convergence for evolution systems driven by functionals.
Max ERC Funding
1 390 000 €
Duration
Start date: 2011-04-01, End date: 2017-03-31
Project acronym ANOPTSETCON
Project Analysis of optimal sets and optimal constants: old questions and new results
Researcher (PI) Aldo Pratelli
Host Institution (HI) FRIEDRICH-ALEXANDER-UNIVERSITAET ERLANGEN NUERNBERG
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary The analysis of geometric and functional inequalities naturally leads to consider the extremal cases, thus
looking for optimal sets, or optimal functions, or optimal constants. The most classical examples are the (different versions of the) isoperimetric inequality and the Sobolev-like inequalities. Much is known about equality cases and best constants, but there are still many questions which seem quite natural but yet have no answer. For instance, it is not known, even in the 2-dimensional space, the answer of a question by Brezis: which set,
among those with a given volume, has the biggest Sobolev-Poincaré constant for p=1? This is a very natural problem, and it appears reasonable that the optimal set should be the ball, but this has never been proved. The interest in problems like this relies not only in the extreme simplicity of the questions and in their classical flavour, but also in the new ideas and techniques which are needed to provide the answers.
The main techniques that we aim to use are fine arguments of symmetrization, geometric constructions and tools from mass transportation (which is well known to be deeply connected with functional inequalities). These are the basic tools that we already used to reach, in last years, many results in a specific direction, namely the search of sharp quantitative inequalities. Our first result, together with Fusco and Maggi, showed what follows. Everybody knows that the set which minimizes the perimeter with given volume is the ball.
But is it true that a set which almost minimizes the perimeter must be close to a ball? The question had been posed in the 1920's and many partial result appeared in the years. In our paper (Ann. of Math., 2007) we proved the sharp result. Many other results of this kind were obtained in last two years.
Summary
The analysis of geometric and functional inequalities naturally leads to consider the extremal cases, thus
looking for optimal sets, or optimal functions, or optimal constants. The most classical examples are the (different versions of the) isoperimetric inequality and the Sobolev-like inequalities. Much is known about equality cases and best constants, but there are still many questions which seem quite natural but yet have no answer. For instance, it is not known, even in the 2-dimensional space, the answer of a question by Brezis: which set,
among those with a given volume, has the biggest Sobolev-Poincaré constant for p=1? This is a very natural problem, and it appears reasonable that the optimal set should be the ball, but this has never been proved. The interest in problems like this relies not only in the extreme simplicity of the questions and in their classical flavour, but also in the new ideas and techniques which are needed to provide the answers.
The main techniques that we aim to use are fine arguments of symmetrization, geometric constructions and tools from mass transportation (which is well known to be deeply connected with functional inequalities). These are the basic tools that we already used to reach, in last years, many results in a specific direction, namely the search of sharp quantitative inequalities. Our first result, together with Fusco and Maggi, showed what follows. Everybody knows that the set which minimizes the perimeter with given volume is the ball.
But is it true that a set which almost minimizes the perimeter must be close to a ball? The question had been posed in the 1920's and many partial result appeared in the years. In our paper (Ann. of Math., 2007) we proved the sharp result. Many other results of this kind were obtained in last two years.
Max ERC Funding
540 000 €
Duration
Start date: 2010-08-01, End date: 2015-07-31
Project acronym ANTHOS
Project Analytic Number Theory: Higher Order Structures
Researcher (PI) Valentin Blomer
Host Institution (HI) GEORG-AUGUST-UNIVERSITAT GOTTINGENSTIFTUNG OFFENTLICHEN RECHTS
Call Details Starting Grant (StG), PE1, ERC-2010-StG_20091028
Summary This is a proposal for research at the interface of analytic number theory, automorphic forms and algebraic geometry. Motivated by fundamental conjectures in number theory, classical problems will be investigated in higher order situations: general number fields, automorphic forms on higher rank groups, the arithmetic of algebraic varieties of higher degree. In particular, I want to focus on
- computation of moments of L-function of degree 3 and higher with applications to subconvexity and/or non-vanishing, as well as subconvexity for multiple L-functions;
- bounds for sup-norms of cusp forms on various spaces and equidistribution of Hecke correspondences;
- automorphic forms on higher rank groups and general number fields, in particular new bounds towards the Ramanujan conjecture;
- a proof of Manin's conjecture for a certain class of singular algebraic varieties.
The underlying methods are closely related; for example, rational points on algebraic varieties
will be counted by a multiple L-series technique.
Summary
This is a proposal for research at the interface of analytic number theory, automorphic forms and algebraic geometry. Motivated by fundamental conjectures in number theory, classical problems will be investigated in higher order situations: general number fields, automorphic forms on higher rank groups, the arithmetic of algebraic varieties of higher degree. In particular, I want to focus on
- computation of moments of L-function of degree 3 and higher with applications to subconvexity and/or non-vanishing, as well as subconvexity for multiple L-functions;
- bounds for sup-norms of cusp forms on various spaces and equidistribution of Hecke correspondences;
- automorphic forms on higher rank groups and general number fields, in particular new bounds towards the Ramanujan conjecture;
- a proof of Manin's conjecture for a certain class of singular algebraic varieties.
The underlying methods are closely related; for example, rational points on algebraic varieties
will be counted by a multiple L-series technique.
Max ERC Funding
1 004 000 €
Duration
Start date: 2010-10-01, End date: 2015-09-30
Project acronym AQSER
Project Automorphic q-series and their application
Researcher (PI) Kathrin Bringmann
Host Institution (HI) UNIVERSITAET ZU KOELN
Call Details Starting Grant (StG), PE1, ERC-2013-StG
Summary This proposal aims to unravel mysteries at the frontier of number theory and other areas of mathematics and physics. The main focus will be to understand and exploit “modularity” of q-hypergeometric series. “Modular forms are functions on the complex plane that are inordinately symmetric.” (Mazur) The motivation comes from the wide-reaching applications of modularity in combinatorics, percolation, Lie theory, and physics (black holes).
The interplay between automorphic forms, q-series, and other areas of mathematics and physics is often two-sided. On the one hand, the other areas provide interesting examples of automorphic objects and predict their behavior. Sometimes these even motivate new classes of automorphic objects which have not been previously studied. On the other hand, knowing that certain generating functions are modular gives one access to deep theoretical tools to prove results in other areas. “Mathematics is a language, and we need that language to understand the physics of our universe.”(Ooguri) Understanding this interplay has attracted attention of researchers from a variety of areas. However, proofs of modularity of q-hypergeometric series currently fall far short of a comprehensive theory to describe the interplay between them and automorphic forms. A recent conjecture of W. Nahm relates the modularity of such series to K-theory. In this proposal I aim to fill this gap and provide a better understanding of this interplay by building a general structural framework enveloping these q-series. For this I will employ new kinds of automorphic objects and embed the functions of interest into bigger families
A successful outcome of the proposed research will open further horizons and also answer open questions, even those in other areas which were not addressed in this proposal; for example the new theory could be applied to better understand Donaldson invariants.
Summary
This proposal aims to unravel mysteries at the frontier of number theory and other areas of mathematics and physics. The main focus will be to understand and exploit “modularity” of q-hypergeometric series. “Modular forms are functions on the complex plane that are inordinately symmetric.” (Mazur) The motivation comes from the wide-reaching applications of modularity in combinatorics, percolation, Lie theory, and physics (black holes).
The interplay between automorphic forms, q-series, and other areas of mathematics and physics is often two-sided. On the one hand, the other areas provide interesting examples of automorphic objects and predict their behavior. Sometimes these even motivate new classes of automorphic objects which have not been previously studied. On the other hand, knowing that certain generating functions are modular gives one access to deep theoretical tools to prove results in other areas. “Mathematics is a language, and we need that language to understand the physics of our universe.”(Ooguri) Understanding this interplay has attracted attention of researchers from a variety of areas. However, proofs of modularity of q-hypergeometric series currently fall far short of a comprehensive theory to describe the interplay between them and automorphic forms. A recent conjecture of W. Nahm relates the modularity of such series to K-theory. In this proposal I aim to fill this gap and provide a better understanding of this interplay by building a general structural framework enveloping these q-series. For this I will employ new kinds of automorphic objects and embed the functions of interest into bigger families
A successful outcome of the proposed research will open further horizons and also answer open questions, even those in other areas which were not addressed in this proposal; for example the new theory could be applied to better understand Donaldson invariants.
Max ERC Funding
1 240 500 €
Duration
Start date: 2014-01-01, End date: 2019-04-30