Università di Trento
Title: New mathematical models and numerical algorithms for Newtonian and general relativistic continuum physics
Abstract: In the first part of the talk we present high order arbitrary high order accurate (ADER) finite volume and discontinuous Galerkin finite element schemes for the numerical solution of a new unified first order symmetric hyperbolic and thermodynamically compatible (SHTC) formulation of Newtonian continuum physics.
In the second part of the talk, we show a successful extension of the GPR model to general relativity, leading to a novel and unified first order hyperbolic formulation of general relativistic continuum mechanics.
In the last part of the talk we introduce a new, provably strongly hyperbolic first order reduction of the CCZ4 formalism of the Einstein field equations of general relativity and its solution with high order ADER discontinuous Galerkin finite element schemes.
Università di Firenze
Title: Francesco Maurolico, a Renaissance interpreter of Euclid
Abstract: Francesco Maurolico (1494-1575) has been one of the most interesting restorers of Greek mathematics during the Renaissance. His approach to the restoration of Classics was creative rather than philological, even in the case of Euclid’s Elements. Among his extant writings we find a quite faithful “reading” of some Books of the Elements (V, VII-X), but the most innovative work is a compendium of the Euclidean text…
University of Utah
Title: Boundedness of varieties of general type
Abstract: Complex projective varieties are subsets of complex projective space defined by a set of homogeneous polynomials. Varieties of general type are the higher dimensional analog of Riemann surfaces of genus g≥2. In this talk I will discuss recent progress on the classification of these varieties.
Title: Norms of random matrices with independent entries
Abstract: The spectral norm of any matrix is bigger than the largest Euclidean norm of its rows and columns. We show that for Gaussian matrices with independent entries this obvious bound may be reversed in average up to a universal constant. We will also discuss similar bounds for Schatten norms and other random matrices with independent entries.
The talk is based on a joint work with Ramon van Handel (Princeton) and Pierre Youssef (Paris).
University of Pittsburgh
Title: Models for Thin Prestrained Structures
Abstract: Variational methods have been extensively used in the past decades to rigorously derive nonlinear models in the description of thin elastic films. In this context, natural growth or differential swelling-shrinking lead to models where an elastic body aims at reaching a space-dependent metric. We will describe the effect of such, generically incompatible, prestrain metrics on the singular limits’ bidimensional models. We will discuss metrics that vary across the specimen in both the midplate and the thin (transversal) directions. We will also cover the case of the oscillatory prestrain, exhibit its relation to the non-oscillatory case via identifying the effective metrics, and discuss the role of the Riemann curvature tensor in the limiting models.
Title: Post-Turing tissue pattern formation: Insights from mathematical modelling
Abstract: Cells and tissue are objects of the physical world, and therefore they obey the laws of physics and chemistry, notwithstanding the molecular complexity of biological systems. What are the mathematical principles that are at play in generating such complex entities from simple laws? Understanding the role of mechanical and mechano-chemical interactions in cell processes, tissue development, regeneration and disease has become a rapidly expanding research field in the life sciences. To reveal the patterning potential of mechano-chemical interactions, we have developed two classes of mathematical models coupling dynamics of diffusing molecular signals with a model of tissue deformation. First, we derived a model based on energy minimization that leads to 4-th order partial differential equations of evolution of infinitely thin deforming tissue (pseudo-3D model) coupled with a surface reaction-diffusion equation. The second approach (full-3D model) consists of a continuous model of large tissue deformation coupled with a discrete description of spatial distribution of cells to account for active deformation of single cells. The models account for a range of mechano-chemical feedbacks, such as signalling-dependent strain, stress, or tissue compression. Numerical simulations based on the arbitrary Lagrangian-Eulerian and fully Eulerian formulations show ability of the proposed mechanisms to generate development of various spatio-temporal structures. In this study, we compare the resulting patterns of tissue invagination and evagination to those encountered in developmental biology. The new class of patterns is compared to the classical Turing patterns. Analytical and numerical challenges of the proposed models are discussed.
Title: Combinatorial topological dynamics
Abstract: Combinatorial topological dynamics is an emerging field with roots in the concept of combinatorial vector field introduced on the turn of the 20th and 21st century by American mathematician Robert Forman as a tool used by him to construct combinatorial Morse theory. I will recall the basic ideas of this theory. I will also present some recent results on combinatorial vector fields motivated by applications in sampled dynamics.
Università di Firenze
Title: The distance function from a real algebraic variety, old and new.
Abstract: The Euclidean distance function from a conic was computed by means of invariant theory in XIX century.
The distance function from the variety of corank one matrices was computed independently by Beltrami and Jordan a few years later and gave rise to the Singular Value Decomposition. Today this function is the core of engineering applications, like “offset surfaces”. More generally, the distance function from a real algebraic variety is a root of an algebraic function. Having in mind applications to the spectral theory of tensors, we show a duality property of this function and we describe its lowest and highest
We show how this fits in the ED phylosophy, where ED stands for “Euclidean Distance”.
Politecnico di Torino
Title: An overview of mathematical models for cell migration
Abstract: Cell-extracellular matrix interaction and the mechanical properties of cell nucleus have been demonstrated to play a fundamental role in cell movement across fibre networks and micro-channels. From the point of view of application understanding this process is important to describe on one hand the spread of cancer metastases and on the other hand to optimize medical scaffold that can be use to cure chronic wounds. From the point of view of mathematics, the problem can be addressed using different methods. In fact, in the talk, I will describe several mathematical models developed to deal with such a phenomenon, starting from modelling cell adhesion mechanics to the inclusion of influence of nucleus stiffness in the motion of cells, through continuum mechanics, kinetic models and individual cell-based models.
Title: New results in the topological classification of Gromov boundaries of hyperbolic groups
Abstract: The classification of finite simple groups is one of the greatest mathematical achievememnts of the 21’st century. In contrast with that, a classification of infinite finitely presented groups is an undecidable algorithmic problem, due to classical results from 1950’s. As a result, mathematicians study some special classes of infinite finitely presented groups, and try to classify them up to weaker equivalence relations than isomorphism.
Geometric group theory studies infinite groups by way of viewing them as certain geometric objects. In 1980’s M.Gromov proposed to study a vast class of the so called word-hyperbolic groups, whose behaviour as geometric objects resembles that of the hyperbolic plane. Instead of classifying them up to isomorphism, one may try to classify their “behaviour at infinity”, encoded in an object called Gromov boundary. Despite more than 30 years of efforts, there are still many quite basic open questions concerning the topological classification of Gromov boundaries of hyperbolic groups.
During the talk I will describe some recent developments concerning this problem. The first consists of a satisfactory description of the topology of the Gromov boundary of a free product of hyperbolic groups with amalgamation along finite groups. The other consists of showing that Gromov boundary of a hyperbolic group is a space belonging to some countable family of spaces called Markov compacta, which are describable in certain algorithmic way out of finite amount of initial data.
Università di Torino
Title: Spiralling and other solutions in limiting profiles of competition-diffusion systems
Abstract: Reaction-diffusion systems with strong interaction terms appear in many multi-species physical problems as well as in population dynamics. The qualitative properties of the solutions and their limiting profiles in different regimes have been at the center of the community’s attention in recent years.
A prototypical example is a system of equations which appears, for example, when looking for solitary wave solutions for Bose-Einstein condensates of two different hyperfine states which overlap in space. Phase separation in such systems has been described in the recent literature, both physical and mathematical. Relevant connections have been established with optimal partition problems involving spectral functionals. The classification of entire solutions and the geometric aspects of phase separation are of fundamental importance as well. We intend to focus on the most recent developments of the theory in connection with problems featuring anomalous diffusions, non-local and
non symmetric interactions.
GUEST PLENARY SPEAKER
Maria J. Esteban
CNRS & Université Paris-Dauphine
Title: Functional inequalities, flows, symmetry and spectral estimates
Abstract: In this talk I will review recent result about how the use of linear and nonlinear flows has been key to prove functional inequalities and qualitative properties for their extremal functions. I will also explain how from these inequalities and their best constants, optimal spectral estimates can be obtained for Schrödinger operators.
This is a topic which is at the crossroads of nonlinear analysis and probability, with implications in differential geometry and potential applications in modelling in physics and biology.