Page 49 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 49
PLENARY SPEAKERS
Escaping the curse of dimensionality in combinatorics
Janos Pach, pach@cims.nyu.edu
Alfréd Rényi Mathematical Institute, Budapest, Hungary, and
MIPT, Moscow, Russian Federation
We discuss some notoriously hard combinatorial problems for large classes of graphs and hy-
pergraphs arising in geometric, algebraic, and practical applications. These structures escape
the “curse of dimensionality”: they can be embedded in a bounded-dimensional space, or they
have small VC-dimension or a short algebraic description. What are the advantages of low
dimensionality?
1. With the help of suitable topological and algebraic separator theorems, large families of
geometric objects embedded in a fixed-dimensional space can be split into subfamilies of
roughly the same size, and then the smaller families can be analyzed recursively.
2. Geometric objects in space can be compared by a number of naturally defined partial
orders, so that one can utilize the theory of partially ordered sets.
3. Graphs and hypergraphs of bounded VC-dimension admit very small epsilon-nets and
can be particularly well approximated by random sampling.
4. If the description complexity of a family of geometric objects is small, then the “com-
binatorial complexity” (number of various dimensional faces) of their arrangements is
also small. This typically guarantees the existence of efficient algorithms to visualize,
describe, control, and manipulate these arrangements.
The Beat of Math
Alfio Quarteroni, alfio.quarteroni@polimi.it
Politecnico di Milano and EPFL, Lausanne, Italy
Mathematical models based on first principles are devised for the description of the blood mo-
tion in the human circulatory system, as well as for the simulation of the interaction between
electrical, mechanical, and fluid-dynamical processes occurring in the heart. This is a classical
environment where multi-physics and multi-scale processes have to be addressed.
Appropriate systems of nonlinear differential equations (either ordinary and partial) and effi-
cient numerical strategies must be devised to allow for the analysis of both heart function and
dysfunction, and the simulation, control, and optimization of therapy and surgery.
This presentation will address some of these issues and a few representative applications of
clinical interest.
Acknowledgment: The work presented in this talk is part of the project iHEART that has re-
ceived funding from the European Research Council (ERC) under the European Union’s Hori-
zon 2020 research and innovation program (grant agreement No 740132)
47
Escaping the curse of dimensionality in combinatorics
Janos Pach, pach@cims.nyu.edu
Alfréd Rényi Mathematical Institute, Budapest, Hungary, and
MIPT, Moscow, Russian Federation
We discuss some notoriously hard combinatorial problems for large classes of graphs and hy-
pergraphs arising in geometric, algebraic, and practical applications. These structures escape
the “curse of dimensionality”: they can be embedded in a bounded-dimensional space, or they
have small VC-dimension or a short algebraic description. What are the advantages of low
dimensionality?
1. With the help of suitable topological and algebraic separator theorems, large families of
geometric objects embedded in a fixed-dimensional space can be split into subfamilies of
roughly the same size, and then the smaller families can be analyzed recursively.
2. Geometric objects in space can be compared by a number of naturally defined partial
orders, so that one can utilize the theory of partially ordered sets.
3. Graphs and hypergraphs of bounded VC-dimension admit very small epsilon-nets and
can be particularly well approximated by random sampling.
4. If the description complexity of a family of geometric objects is small, then the “com-
binatorial complexity” (number of various dimensional faces) of their arrangements is
also small. This typically guarantees the existence of efficient algorithms to visualize,
describe, control, and manipulate these arrangements.
The Beat of Math
Alfio Quarteroni, alfio.quarteroni@polimi.it
Politecnico di Milano and EPFL, Lausanne, Italy
Mathematical models based on first principles are devised for the description of the blood mo-
tion in the human circulatory system, as well as for the simulation of the interaction between
electrical, mechanical, and fluid-dynamical processes occurring in the heart. This is a classical
environment where multi-physics and multi-scale processes have to be addressed.
Appropriate systems of nonlinear differential equations (either ordinary and partial) and effi-
cient numerical strategies must be devised to allow for the analysis of both heart function and
dysfunction, and the simulation, control, and optimization of therapy and surgery.
This presentation will address some of these issues and a few representative applications of
clinical interest.
Acknowledgment: The work presented in this talk is part of the project iHEART that has re-
ceived funding from the European Research Council (ERC) under the European Union’s Hori-
zon 2020 research and innovation program (grant agreement No 740132)
47
