2015-08-03

IP1: The Euclidean Distance Degree of an Algebraic VarietyGiven a real n*n symmetric matrix A, let us consider the distance function from A to the variety X of rank one symmetric matrices. The critical points of this function correspond to the eigenvectors of A, in particular they are all real. We study this setting when X is any algebraic variety. The critical points satisfy a remarkable duality property and their number is called the Euclidean Distance Degree of X, which is an orthogonal invariant. Chern classes may be used to compute these invariants, and pose some computational challenges. Many examples in tensor spaces arise from applications in computer vision and other areas, with a common geometric flavour.
Other interesting examples and open problems arise from classical varieties, starting from conics.

MS01: Semidefinite Optimization: Geometry, Algebra and Applications 1

MS02: Coding Theory 1

MS03: Algorithms and Complexity in Polynomial Systems 1

MS04: Tropical Geometry 1

MS05: Geometry of Matrix Multiplication

MS06: Applications of Computational Algebraic Geometry to Theoretical Physics 1

MS07: Algebraic Structure in Graphical Models 1

MS08: Combinatorial Phylogenetics 1

IP02: Some Current Directions in Coding Theory

The theory of error-correcting codes was launched in Shannon's landmark 1948 paper, in which he proved that reliable information
transmission across a noisy channel was possible. The proof is non-constructive, and "Shannon's challenge", as it has come to be known, is to find efficient codes that can correct many errors and come equipped with efficient decoding algorithms. With the discovery of turbo codes, the rediscovery of low-density parity-check codes, and the introduction of polar codes, many argue that Shannon's challenge has been met. However, many challenges in coding theory remain, as new applications necessary for
our modern world continue to raise new questions. We will survey the field, placing particular emphasis on areas in which algebraic-geometric techniques have been, or can be, fruitfully applied.

CP01: Contributed Session I

MS09: Semidefinite Optimization: Geometry, Algebra and Applications 2

MS10: Coding Theory 2

MS11: Software and Applications in Numerical Algebraic Geometry 1

MS12: Tropical Geometry 2

MS13: Spectral Theory of Tensors and Tensor Rank 1

MS14: Algebraic Structures arising in Systems Biology 1

MS15: Markov Bases and their Applications in Statistics 1

2015-08-04

IP03: P–adic Integration and Number TheoryWe will give an introduction to the computation of line integrals in non-Archimedean geometry, together with applications to the solution of polynomial equations.

CP02: Contributed Session II

MS16: Nonnegative Rank

MS17: Coding Theory 3

MS18: Group Actions in Algebraic Geometry and Commutative Algebra

MS19: Tensor Decomposition: Ideals meet Applications 1

MS20: Applications of Computational Algebraic Geometry to Theoretical Physics 2

MS21: Maximum Likelihood Degrees and Critical Points 1

MS22: Pairings in Cryptography I

IP4: Applications of Numerical Algebraic Geometry

In broad terms, numerical algebraic geometry follows a computational geometric approach to solving systems of polynomial equations. This geometric approach has facilitated solving numerous problems in science and engineering throughout the past several decades. In this talk, I will survey some recent applications involving the use of numerical algebraic geometry including tensor decomposition, steady-state solutions to differential equations, and mechanism synthesis in kinematics.

MS23: Real Algebraic Geometry and Optimization 1

MS24: Coding Theory and Cryptography 1

MS25: Software and Applications in Numerical Algebraic Geometry 2

MS26: Tropical Geometry 3

MS27: Spectral Theory of Tensors and Tensor Rank 2

MS28: Algebraic Structures arising in Systems Biology 2

MS29: Algebraic Structure in Graphical Models 2

MS30: Class Groups of Global Fields

2015-08-05

IP05: Algebraic Codes and InvarianceAlgebraic error-correcting codes hold a central place in coding theory due to three, potentially unrelated, features. The most well-known feature is perhaps the combinatorial aspect: (1) Algebraic code pack Hamming space extremely densely, often outperform random error-correcting codes. Less well-known are the (2) Multiplicative property: Algebraic codes come endowed with a product operation where products of codewords remain far from each other and (3) Symmetries: Many algebraic codes are endowed with a rich group of symmetries that enable powerful uses of these codes.
In this talk I will briefly review algebraic codes and the first two properties, before turning to the third part and talk about recent works highlighting the role of symmetries in (mathematical) uses of error-correcting codes.

MS31: Real Algebraic Geometry and Optimization 2

MS32: Coding Theory 4

MS33: Algorithms and Complexity in Polynomial Systems 2

MS34: On the Geometry and Topology of (Co)Ame`bas and Beyond

MS35: Tensor Decomposition : Ideals meet Applications 2

MS36: Algebraic Structures arising in Systems Biology 3

MS37: Markov Bases and their Applications in Statistics 2

MS38: Pairings in Cryptography 2

2015-08-06

IP06: Algebraic VisionReconstructing a 3-dimensional scene from 2-dimensional images of it is a fundamental problem in computer vision. Real solutions to systems
of polynomials play a key role in solving this problem. In this talk I will explain recent work that attempts to bring in tools from algebraic geometry and polynomial optimization to some of the foundational problems in 3D computer vision with a particular focus on the most basic problem of reconstruction from two images.

MS39: Polynomial Optimization and Moments 1

MS40: Coding Theory and Cryptography 2

MS41: Software and Applications in Numerical Algebraic Geometry 3

MS42: Combinatorial Methods in Algebraic Geometry 1

MS43: Symbolic Combinatorics 1

MS44: Algebraic Vision 1

MS45: Algebraic Structure in Graphical Models 3

MS46: Combinatorial Phylogenetics 2

IP07: Challenges in the Development of Open Source Computer Algebra Systems

Computer algebra is facing new challenges as mathematicians are inventing new and more abstract tools to answer difficult problems and connect apparently
remote fields of mathematics. On the mathematical side, while we wish to provide cutting-edge techniques for application areas such as commutative algebra, algebraic
geometry, arithmetic algebraic geometry, singularity theory, and many more, the implementation of an advanced and more abstract computational machinery often
depends on a long chain of more specialized algorithms and efficient data structures at various levels. On the software development side, for cross-border
approaches to solving mathematical problems, the efficient interaction of systems specializing in different areas is indispensable; handling complex examples or large
classes of examples often requires a considerably enhanced performance. Whereas the interaction of systems is based on a systematic software modularization and the design of mutual interfaces, a new level of computational performance is reached via parallelization, which opens up the full power of multi-core computers, or clusters of computers.
In my talk, I will report on the ongoing collaboration of groups of developers of several well-known open source computer algebra systems, including ({\sc{GAP}},
which pays particular emphasis to group theory, {\sc{Singular}}, a system for applications in algebraic geometry and singularity theory, and {\sc{Polymake}},
a software for polyhedral geometry. In presenting computational tools relying on this collaboration, and some of the mathematical challenges which lead us to develop such tools, I will in particular highlight the {\sc{Homalg}} project which provides an abstract structure and algorithms for abelian categories, aiming at concrete applications ranging from linear control theory to commutative algebra and algebraic geometry.
I will also comment on progress in the design of parallel algorithms for basic tasks in commutative algebra and algebraic geometry such as primary decomposition, normalization, finding adjoint curves, or parametrizing rational curves.

MS47: Polynomial Optimization and Moments 2

MS48: Verified Solutions of Algebraic Systems

MS49: Computational Approach to GIT and Moduli Theory 1

MS50: Symbolic Combinatorics 2

MS51: Algebraic Vision 2

MS52: Maximum Likelihood Degrees and Critical Points 2

MS53: Core Algorithms in Algebraic Geometry 1

2015-08-07

IP08: Progress Report on Geometric Complexity TheoryGeometric complexity theory is an approach to the fundamental lower bound problems of complexity theory via algebraic geometry and representation theory. This talk will give an overview of some progress in this field.

CP03: Contributed Session III

MS54: Polynomial Optimization and Moments 3

MS55: Applications of Polynomial Systems Solving in Cryptology 1

MS56: Core Algorithms in Algebraic Geometry 2

MS57: Combinatorial Methods in Algebraic Geometry 2

MS58: Geometric Complexity Theory

MS59: Aspects of Grassmann Manifolds With a view towards Applications

MS60: Maximum Likelihood Degrees and Critical Points 3

MS61: Computational Approach to GIT and Moduli Theory 2

MS62: Parametrization of Rational Curves in Projective Space

MS63: Applications of Polynomial Systems Solving in Cryptology 2

MS64: Recent developments in Geometric and algebraic Methods in Economics

MS65: Holonomic Functions and Holonomic Gradient Method

IP09: Hodge Theory and Combinatorics

A conjecture of Read predicts that the coefficients of the chromatic polynomial of any graph form a log-concave sequence. A related conjecture of Welsh predicts that the number of linearly independent subsets of varying sizes form a log-concave sequence for any configuration of vectors in a vector space. All known proofs use Hodge theory for projective varieties, and the more general conjecture of Rota for possibly ‘nonrealizable’ configurations is still open. In this talk, I will argue that two main results of Hodge theory, the Hard Lefschetz theorem and the Hodge-Riemann relations, continue to hold in a realm that goes far beyond that of Kahler geometry. This cohomology theory gives strong restrictions on numerical invariants of tropical varieties, and in particular those of graphs and matroids. Joint work with Karim Adiprasito and Eric Katz.