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Highprobability grants
According to our matching algorithm, Gil Kalai is the likely recipient of the following grants.
Years 
Recipients 
Code 
Title / Keywords 
Matching score 
2004 — 2008 
Kannan, Ravindran [⬀] Kalai, Gil 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Three Topics in Combinatorics With Relations to Theoretical Computer Science
We intend to study three topics in combinatorics which are related to theoretical computer science. We will study the diameter problem for graphs of polytopes aiming at a polynomial upper bound, and also try to find versions of the simplex algorithm with subexponential worst case behavior. We will study Boolean functions, their Fourier transform and how it relates to questions in probability and complexity. We will also try to find general methods to relate the solution of a positive integer programming problem and its linear programming relaxation.
Combinatorics have now become the central mathematical discipline in theoretical computer science (and various applied areas of computer science as well). The role of combinatorics in computer science today is quite similar to the role of logic in the early days of computation. Problems from theoretical computer science enriched and enforced combinatorial thinking and areas which were regarded as having clear intellectual merit have gained surprising reallife applications. Linear programming and the simplex algorithm are among the most important applications of computers and in our earlier work as well as the planned research. We intend to study fundamental questions concerning linear programming. Another area of our research, the Fourier analysis of Boolean functions is a relatively new area which we helped to develop in the past. In view of the importance of Fourier analysis in other areas we should not have been surprised to see its recent applications in combinatorics and complexity theory and we intend to explore further connections.

1 
2007 — 2011 
Kalai, Gil 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Probabilistic and Geometric Combinatorics
The proposer proposes to study several topics in probabilistic and geometric combinatorics. These topics are related to theoretical computer science and other academic disciplines. The proposal is divided into two areas.
Part I concerns the study of influences, threshold behaviors and Fourier Analysis. Influence of variables on Boolean functions and the related notion of pivotality are of central importance in combinatorics, various other areas of mathematics, computer science and other areas as game theory, reliability theory and statistical physics. What is the correct notion of power or influence in complicated social situations? or in complex computer networks? How does noise effect such systems? and what is the correct way of modeling noise? It turns out that these questions are related to a recent mathematical theory which involve combinatorics, probability and ``Fourier analysis''. The proposer will study some basic open problems in this area. This is closely related to the study of Fourier analysis of Boolean functions, and to noise.
Part II concerns Helly type theorems and convex polytopes: One important theme in combinatorial geometry is to try to extract the combinatorial and topological content of various geometric theorems. The study of topological Helly type theorems is one example. Extending results from polytopes to simplicial and polyhedral spheres is another. Some of these problems are surprisingly easy to state and still they were not solved for many decades and any progress is expected to involve some deep new insights. These problems are related to the area of ``optimization'' and especially to ``linear programming''. Optimization  trying to find the best solution in a complicated situation with various constraints  is one of the most important applications of mathematics. The proposer presented some problems in this direction that he intends to study.

1 
2013 — 2017 
Kalai, Gil 
N/AActivity Code Description: No activity code was retrieved: click on the grant title for more information 
Problems in Probabilistic and Geometric Combinatorics
Combinatorics is an area centered around problems and applications. Some of these problems arise in the field itself and others come from other mathematical areas and from applications in other scientific fields. We present problems which in the interface between combinatorics, geometry and probability with much input from theoretical computer science and mathematical programming. We will study, also using discrete harmonic analysis, phase transition phenomena for combinatorial stochastic models and problems related to the combinatorics of simplicial and polyhedral complexes arising in geometric problems about convex sets.
The research topics were selected to have potential impact in other areas of mathematics, theoretical computer science, mathematical programming, and questions regarding social choice. What causes phase transition for stochastic combinatorial models, and what is the nature of this phase transition? How to analyze the noise sensitivity of Boolean functions and how it is related to models of computations on the one hand, and to voting rules on the other? What can we infer from combinatorics of polyhedra to the performance of optimization algorithms. These are areas where theory and practice are surprisingly interlaced, and information as well as inspiration go both ways.

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