ERC FUNDED PROJECTS

Title: Discrete and convex geometry: challenges, methods, applications Abstract: Research in discrete and convex geometry, using tools from combinatorics, algebraic topology, probability theory, number theory, and algebra, with applications in theoretical computer science, integer programming, and operations research. Algorithmic aspects are emphasized and often serve as motivation or simply dictate the questions. The proposed problems can be grouped into three main areas: (1) Geometric transversal, selection, and incidence problems, including algorithmic complexity of Tverberg's theorem, weak epsilon-nets, the k-set problem, and algebraic approaches to the Erdos unit distance problem. (2) Topological methods and questions, in particular topological Tverberg-type theorems, algorithmic complexity of the existence of equivariant maps, mass partition problems, and the generalized HeX lemma for the k-coloured d-dimensional grid. (3) Lattice polytopes and random polytopes, including Arnold's question on the number of convex lattice polytopes, limit shapes of lattice polytopes in dimension 3 and higher, comparison of random polytopes and lattice polytopes, the integer convex hull and its randomized version.

1 298 012 €

Start date: 2011-04-01, End date: 2017-03-31

This interdisciplinary research project, relying on mutually complementary historical, anthropological and folklore investigations, will examine continuities and transformations in vernacular religion in the border-zone between Eastern and Western Christianity. The project will have three foci: 1) the role of the religious worldview and norms in past and present communities; 2) change and religious modernisation including the intertwining of the breaking up of the traditional worldview and the appearance of consumer-type attitudes of New Age religiosity; 3) the role of religion in identity formation and the emergence of religious pluralism and co-operation as well as of religious antagonism and conflict between different denominations and nationalities in the region. Members of the project will study these questions in Hungarian, Romanian, Serbian, Ukrainian and Croatian communities of mixed religion. Thematically the research will be organised around exploring symbolic exchange relationships (demonology and witchcraft) sacred communication (shrines, visions, miracles, saints) and healing using both historical sources and contemporary anthropological field work. The project builds on two previous long-term historical/folkloristic research projects led by PI Éva Pócs and will expand and complement their findings through contemporary anthropological field research and continued archival work. Integrating the results of the current and earlier projects through an innovative electronic document collection, embedded in a geographical information system, will enhance the impact of both sets of materials. The research will bring us closer to understanding a) inter-religious relationships between Catholic, Protestant and Orthodox believers, b) problems of national identity underlying religious antagonisms, and c) how religious and cultural border zones separate and unite, generate conflict and create mutual understanding, potentially promoting peaceful co-existence.

2 079 485 €

Start date: 2013-09-01, End date: 2018-08-31

Frontal cortical areas are responsible for a wide range of executive and cognitive functions. Frontal cortices communicate with the thalamus via bidirectional pathways and these connections are indispensable for frontal cortical operations. Still, we have very little information about the specificity of connections, synaptic interactions and plasticity between frontal cortex and thalamus and the roles of these interactions in frontal cortical functions. In the present proposal, we will test the hypothesis that frontal cortical areas developed a highly specialized connectivity pattern with the thalamus. This supports unique interactions between the cortex and the thalamus according to the specific requirements of frontal cortical activity, including experience-dependent plastic changes. The project will use cell type-specific viral tracing in mice and 3D electron microscopic reconstructions in mice and humans to identify circuit motifs that are evolutionarily conserved, yet, still specific to fronto-thalamic pathways. The physiological approach will employ in vivo optogenetics combined with intra-, juxta- and extracellular recordings. We will perform behavioral experiments by bidirectional modulation of well-defined elements in the network, in learning paradigms, which depend on the integrity of frontal cortex. The project is the first systematic approach which aims to understand the nature of interaction between the frontal cortex and the thalamus. It will not only fill the tremendous gap in our knowledge regarding these pathways but will help us elucidate the functional organization of non-sensory thalamus in general. Frontal cortices are involved in a wide range of major neurological disorders (e.g. Parkinson’s disease, epilepsy, schizophrenia, chronic pain) which affect executive functions and involve fronto-thalamic pathways. We believe that understanding fronto-thalamic interactions will lead to fundamentally novel insight into the nature of these diseases.

1 597 575 €

Start date: 2017-09-01, End date: 2022-08-31

"In recent years there has been spectacular progress in studying growth in groups. A central result in this new area, obtained by Pyber-Szabo' (with a similar result proved by Breuillard-Green-Tao), shows that powers of generating subsets of finite simple groups of ""bounded dimension"" grow fast. Extending this Product Theorem Szabo' and the PI also proved a weaker version of a conjecture of Helfgott-Lindenstrauss. The Product Theorem has deep consequences in the study of groups, number theory and random walks. A central open question of the area is to remove the dependence on dimension in our Product Theorem. The PI formulated a new Conjecture, as a step forward. The way to further progress is via combining techniques from asymptotic group theory and probability theory. It is from this perspective that the current GROGandGIN proposal addresses issues concerning random walks. We examine how recent probabilistic arguments for random walks in the symmetric group may be transferred to matrix groups. While the first results in the subject of growth concern matrix groups we see an evolving theory of growth in permutation groups. This relies on earlier work of Babai and the PI which aims at finding proofs which do not use the Classification of Finite Simple Groups (CFSG). Similarly, Babai's famous Quasipolynomial Graph Isomorphism Algorithm builds on ideas from CFSG-free proofs due to him. The PI has recently removed CFSG from the analysis of Babai's algorithm. Our method goes ""halfway"" towards removing CFSG from proofs of growth results for permutation groups, currently a major open problem. The GROGandGIN initiative plans to improve various other parts of Babai's paper, working with several people who look at it from different angles, with an eye towards obtaining a Polynomial Graph Isomorphism algorithm. The GROGandGIN team will also study growth in Lie groups since the theory of random walks in Lie groups has been revitalised using analogues of our Product Theorem."

1 965 340 €

Start date: 2017-08-01, End date: 2022-07-31