Grant: 20-14736S
from 01/01/2020
to 31/12/2022
Grantor: Czech Science Foundation
Hysteresis modeling in mathematical engineering
Objectives:
Rate-independent hysteresis memory is known to occur in many physical processes such as magnetization of ferromagnetic materials, fluid flow through porous media, and phase transitions. Theoretical understanding of hysteresis mechanisms is of a key importance in engineering applications, where neglecting dissipative hysteresis effects in numerical predictions may lead to error accumulation and discrepancies with experiments. Most of the modern multifunctional materials used for high-precision devices exhibit hysteresis, which has to be taken into account in mathematical modeling. Surprisingly, hysteresis is also present in economic models. We focus here on mathematical and computational aspects of hysteresis in the whole range of applications. Special attention will be paid to the theoretical analysis of typical problems arising in dealing with smart materials, water-ice phase transitions in porous solids, and economics.
Grant: GA20-01074S
from 01/01/2020
to 31/12/2022
Grantor: Czech Science Foundation
Adaptive methods for the numerical solution of partial differential equations: analysis, error estimates and iterative solvers
Objectives:
The project deals with the numerical solution of several types of partial differential equations (PDEs) describing various practical phenomena and problems. The aim is to develop reliable and efficient numerical methods allowing to obtain approximate solutions of PDEs under the given tolerance using a minimal number of arithmetic operations. The whole process includes the proposals and analysis of discretization schemes together with suitable solvers for the solution of arising algebraic systems, a posteriori error estimation including algebraic errors and adaptive techniques balancing various error contributions. We focus on the use of adaptive higher-order schemes which allow to reduce significantly the number of necessary degrees of freedom required for the achievement of the prescribed accuracy. The adaptive mesh refinement must also take into account the properties of the resulting algebraic systems. The expected outputs of this projects are adaptive reliable and efficient numerical methods for the solution of the considered types of PDEs.
Symmetries, dualities and approximations in derived algebraic geometry and representation theory
Objectives:
The project focuses on new trends in homological algebra, represenation theory and algebraic geometry. In particular, we aim at studying and developing a theory of the exotic versions of derived categories and equivalences of these, studying derived commutative algebra, algebraic geometry and representation theory, and studying the homological algebra of and the structure theory for contramodules over topological rings, which were discovered only a few year ago. The applicants with collaborators recently published their results in distinguished mathematical journals (J. reine angew. Math., Invent. Math., Adv. Math., Mem. Amer. Math. Soc. and others), and the proposed project naturally builds on these results.
Grant: GJ20-17488Y
from 01/01/2020
to 31/12/2022
Grantor: Czech Science Foundation
Applications of C*-algebra classification: dynamics, geometry, and their quantum analogues
Objectives:
The 3 year project will bring together 4 promising mathematicians to build upon the successes of the classification programme for C*-algebras. There are 3 main goals:
A-Study structural properties and establish classification theorems for topological dynamical systems and their C*-algebras
B-Examine quantum group actions and the structure of their associated homogeneous spaces
C-Apply metric and geometric structures in C*-algebras to understand the fine structure of classifiable C*-algebras
Each goal consists of 3-4 concrete objectives. The expected output of 12-15 high quality research papers is outlined in an achievable schedule, divided into 4 month blocks. Already home to several researchers in C*-algebras, the Project's successful funding would position the Host Institute as a major European centre for C*-algebras. The team is highly active in the C*- algebra community and collaborates with world-leading experts, promising a successful outcome and efficient dissemination of results.
Grant: GX20-31529X
from 01/01/2020
to 31/12/2024
Grantor: Czech Science Foundation
Abstract convergence schemes and their complexities
Objectives:
Abstract convergence schemes are basic category-theoretic structures which serve as universes for studying infinite evolution-like processes and their limiting behavior. Convergence schemes endowed with extra structures provide an applicable framework for studying both discrete and continuous processes as well as their random variants.
The main goal of the project is unifying and extending several concepts from model theory, algebra, topology and analysis, related to generic structures. We propose studying selected topics within the framework of abstract convergence schemes, addressing questions on their complexity and classification. One of our inspirations is the theory of universal homogeneous models, where convergence of finite structures is involved. Another motivation is set-theoretic forcing, where a convergence scheme is simply a partially ordered set of approximations of some ``unreachable" objects, living outside of the universe of set theory.
Matematický ústav AV ČR usiluje o HR Award - zavedení profesionálního řízení lidských zdrojů
Matematický ústav je přední českou věřejnou organizací, kterých posláním je vědecký výzkum v oblastech matematiky a jejích aplikací. Pro posílení jeho konkurenceschopnosti v mezinárodním kontextu je klíčové uvést dosavadní strategii řízení a rozvoje lidských zdrojů do souladu s Evropskou chartou pro výzkumné pracovníky, a tím umožnit získání ocenění HR Award. Pro ústav jde o mimořádnou příležitost, jak zkvalitnit a zprofesionalizovat péči o lidské zdroje, které jsou alfou a omegou jeho úspěchu.
Tento projekt je podpořen z operačního programu Výzkum, vývoj a vzdělávání, Výzva č. 02_18_054 pro Rozvoj kapacit pro výzkum a vývoj II v prioritní ose 2 OP, reg. č. CZ.02.2.69/0.0/0.0/18_054/0014664.
Grant: GJ19-05271Y
from 01/01/2019
to 31/12/2021
Grantor: Czech Science Foundation
Groups and their actions, operator algebras, and descriptive set theory
Objectives:
The goal of the project is to to find new connections and prove some interesting conjectures on the boundaries of three, currently very attractive mathematical disciplines - geometric group theory, operator algebras, and descriptive set theory.
Grant: GJ19-07129Y
from 01/01/2019
to 31/12/2021
Grantor: Czech Science Foundation
Linear-analysis techniques in operator algebras and vice versa
Objectives:
The theory of normed spaces and their operators is at the core of the linear analysis. The idea of employing algebras of operators acting on infinite-dimensional spaces originated in quantum physics and was further successfully integrated with the theory of unitary representations of locally compact groups. An operator algebra, which is also a normed space, carries intrinsically a much richer structure and therefore operator algebras are not usually viewed from the perspective of linear analysis. Nevertheless, the transfer of ideas from Banach spaces can be very fruitful as illustrated by the notion of nuclearity that was recognised as an approximation property with respect to a certain class of finite-rank operators. On the other hand, operator Ktheory
was almost unknown in Banach space theory until it was spectacularly applied in the seminal work of Gowers and Maurey. Consequently, there is tremendous potential in transferring ideas between these two areas. The very aim of the project is a closer reconciliation of these two theories by interchanging ideas between them.
Grant: GA19-05497S
from 01/01/2019
to 31/12/2021
Grantor: Czech Science Foundation
Complexity of mathematical proofs and structures
Objectives:
We study weak logical systems, guided by the question: what is the weakest natural theory in which we can prove a mathematical statement? This question is often fundamentally complexity theoretic in nature, as proofs in such weak systems can be associated with feasible computations. We will study this and related topics in a range of settings, including bounded arithmetic, model theory, algebraic complexity, bounded set theory, and nonclassical logics.
Grant: GA19-04243S
from 01/01/2019
to 31/12/2021
Grantor: Czech Science Foundation
Partial differential equations in mechanics and thermodynamics of fluids
Objectives:
Partial differential equations in mechanics and thermodynamics of fluids are a mathematical tool which models the time evolution of basic physical quantities. The goal of this project is to study these systems of partial differential equations from the point of view of mathematical and numerical analysis and to compare these results with the properties of their numerical solutions. We will mostly deal with solvability of the problems (existence of solutions for different formulations, possibly their uniqueness), qualitative properties of the solutions, analysis of the adequate numerical methods and numerical solutions of these problems. The proposal of the project assumes a close collaboration of specialists from different branches. Such a collaboration stimulates positively the developement of all participating mathematical disciplines.
Grant: GA19-09659S
from 01/01/2019
to 31/12/2021
Grantor: Czech Science Foundation
Exact solutions of gravity theories: black holes, radiative spacetimes and electromagnetic fields
Objectives:
Exact solutions to Einstein gravity play a crucial role in the understanding of many mathematical and physical aspects of the theory. In recent years, for several theoretical reasons, various modifications of Einstein gravity and their solutions have been studied. Due to the complexity of resulting field equations, very few exact solutions of such theories are known. We plan to construct and study exact solutions to Einstein gravity and various higher-order
gravities, such as quadratic gravity and Lovelock gravity, with a strong focus on black hole solutions, radiative spacetimes, and p-form fields. We also intend to study generic properties of certain classes of spacetimes, such as asymptotically flat spacetimes. When appropriate, we will benefit from employing mathematical methods, such as algebraic classification and a generalized GHP formalism developed in part by our team.
Grant: GX19-27871X
from 01/01/2019
to 31/12/2023
Grantor: Czech Science Foundation
Efficient approximation algorithms and circuit complexity
Objectives:
The goal of this project is to understand the role of approximation in fine-grained and parameterized complexity and create solid foundations for these areas by developing lower bound techniques capable of addressing the key unproven assumptions under-pinning these areas. We will focus on several central problems: Edit Distance, Integer Programming, Satisfiability and study their approximation and parameterized algorithms with the aim of
designing the best possible algorithms. Additionally we will focus on several methods of proving complexity lower bounds.
Operads are objects formalizing compositionsof operations with several inputs. They were invented to describe homotopy invariant structures on topological spaces. Later it turned out that they can be used as well for the study of sundry structures in geometry, algebra and mathematical physics.
The research supported by Praemium Academie is aimed at formulating a unifying paradigm for very general operadic structures, and using this emerging systematic approach for proving various results in algebra, geometry and mathematical physics. Our team is international from the very beginning, as emphasized by the planned positions for postdocs and foreign specialists.
Grant: MSM100191801
from 01/01/2018
to 31/12/2018
Grantor: Czech Academy of Sciences
Structure and localizations of the derived category of a commutative ring
Objectives:
The derived category of an abelian category is a meeting point between algebra and homotopical methods originally designed for topology. This project aims to extend the current knowledge about derived categories in the following directions. 1) Extend the recent description of equivalence classes of (big) tilting modules over an arbitrary commutative ring to the setting of silting complexes, or even compactly generated t-structures. 2) Describe the cotilting modules, or even the cosilting modules, over arbitrary Pruefer domain. Study the derived equivalences these modules induce. 3) Search for a purely tensor triangulated category construction of a non-compactly generated smashing localization based on an algebraic construction by Bazzoni and Šťovíček used for the module category of a valuation domain. The possible application could help solve the long open Telescope Conjecture for the homotopy category of spectra. The three topics are are strongly connected to each other and are expected to be studied simultaneously.
Grant: GJ18-01472Y
from 01/01/2018
to 31/12/2020
Grantor: Czech Science Foundation
Graph limits and inhomogeneous random graphs
Objectives:
Theories of dense and sparse graph limits are one of the most important recent tools of discrete mathematics. Their emergence and development have led to many breakthroughs on old problems in extremal graph theory and random graph theory, and especially have linked discrete mathematics to areas such as probability theory, functional analysis or group theory in a profound way. Recognitions related to the development of the field include the 2012 Fulkerson Prize, the 2013 Coxeter-James Prize, and the 2013 David P. Robbins Prize.
The project will study the theories of dense a sparse graph limits as well as the related theory of inhomogeneous random graphs. Specific problems in the area of inhomogeneous random graphs include questions on key graph parameters such as the chromatic number or the independence number. In the theory of sparse graph limits our main goal is to extend our understanding of local-global convergence. A further goal is to create a comprehensive theory of limits of subgraphs of hypercubes.
Grant: GC18-01953J
from 01/01/2018
to 31/12/2020
Grantor: Czech Science Foundation
Geometric methods in statistical learning theory and applications
Objectives:
Statistical learning theory is mathematical foundation of machine learning - the currently fastest growing branch of computer sciences and artificial intelligence. Central objects of statistical learning theory are statistical models. The project is based on our results obtained jointly with N. Ay and J. Jost and covers the following topics: geometry of efficient estimations, geometry of natural gradient flows and properties of Kullback-Leibler divergence on statistical models, in particular graphical models, hidden Markov models, Boltzmann machine, multilayer perceptrons and infinite dimensional exponential models.
Grant: GA18-07776S
from 01/01/2018
to 31/12/2020
Grantor: Czech Science Foundation
Higher structures in algebra, geometry and mathematical physics
Objectives:
It has been gradually realized that various deep problems in algebra, geometry and mathematical physics, particularly in string theory, involve previously unknown higher structures. Over the years, this originally esoteric concept has become widely recognized, with parallel breakthroughs in the foundations of derived algebraic geometry and topology, category theory, representation theory and other seemingly unrelated fields. Our project aims to increase
the understanding of the topics mentioned above, by combining the expertise of the team members in different but tightly interlaced areas of mathematics and mathematical physics. More specifically, the project aims at topics such as the terminality conjecture for spaces relevant for string field theory, higher Lie algebras and gauge theory, M-branes, Penrose-Ward transform, Adams-Novikov spectral sequence, Riemann surfaces, and related issues. The common background of these themes are operads, higher category theory and homological algebra, combined with the standard methods of differential and algebraic geometry.
Grant: GA18-05974S
from 01/01/2018
to 31/12/2020
Grantor: Czech Science Foundation
Oscillations and concentrations versus stability in the equations of mathematical fluid dynamics
Objectives:
The project focuses on questions of possible singularities in the equations of mathematical fluid dynamics and their adequate description by means of weak and measure valued solutions. The main topics include:(i) dissipative solutions, (ii) admissibility criteria, (iii) equations with stochastic terms, (iv) applications in the numerical analysis.
The goal is to develop a consistent mathematical theory of fluids in motion in the framework of weak and measure valued solutions, developing the concept of dissipative solution, obtaining new admissibility criteria, solving problems with stochastic terms, analyzing the underlying numerical schemes.
Grant: GA18-09628S
from 01/01/2018
to 31/12/2020
Grantor: Czech Science Foundation
Advanced flow-field analysis
Objectives:
The research deals with an advanced flow-field analysis, particularly for transitional and turbulent flows. Local vortex identification and more general classification of flow regions based on the velocity gradient are investigated. The velocity gradient is usually decomposed in strain-rate tensor and vorticity tensor, consequently the identification and classification criteria are determined by the inner velocity-gradient configuration. The impact of configuration is usually significant though hidden, and will be, including representative data sets, analyzed and described. Large-scale 3D numerical experiments based on the solution of the Navier-Stokes equations (NSE) will be performed with the help of new effective methods (parallel domain decomposition (DD) with adaptive mesh refinement) while these methods will be further developed. Some suitable qualitative properties of the solutions to the NSE will be studied and described in detail, focusing on the regularity criteria with only one or two velocity components and one or several velocity-gradient entries.
Grant: GA18-00496S
from 01/01/2018
to 31/12/2020
Grantor: Czech Science Foundation
Singular spaces from special holonomy and foliations
Objectives:
Spaces with singularities naturally appear in differential geometry and mathematical physics. The project is based on the development of our results covering the following topics: the holonomy groups of cones over pseudo-Riemannian manifolds, their relations to Lorentzian manifold admitting imaginary Killing spinors, Sasakian and other special geometries, constructions of new examples of complete G2 and Spin(7)-holonomy metrics and study of their deformations, constructions of invariant Kaehler-Einstein and Einstein-Sasakian metrics on cohomogeneity one manifolds. We also plan to investigate geometry of the leaf space of foliations; in particular, to develop Losik's approach to these spaces and to study their characteristic classes.
Grant: GA18-00580S
from 01/01/2018
to 31/12/2020
Grantor: Czech Science Foundation
Function Spaces and Approximation
Objectives:
We shall study important properties of various function spaces and operators acting on them. We shall focus on optimality of the obtained results. We shall develop new sampling algorithms that will have important applications in theory of approximation. We shall concentrate on applications of results obtained in other fields of mathematics.