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The Birch and Swinnerton-Dyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution. An elliptic curve, say E, can be represented by points on a cubic equation as below with certain A, B ∈ Q:y2 = x3 + Ax +BA Theorem of Mordell says that that E(Q), the set of rational points of E, is a finitely generated abelian group, and thus,E(Q) = Zr ⊕ T,for some non-negative integer r and a finite group T. Here, r is called the algebraic rank of E.The Birch and Swinnerton-Dyer conjecture relates the algebraic rank of E to the value of the L-function, L(E, s), attached to E at s = 1.Further theoretical understanding, corroborated by computations lead to a stronger version of the BSD conjecture. This refined version of the BSD conjecture provides a very precise formula for the leading term of L(E, s) at s = 1, the coefficient of (s − 1)r, in terms of various arithmetical data atta...

The School is focused on the basic physical and biological principles of the development of multicellular organisms.The purpose of this advanced School is to provide both an introduction to the basic physical and biological principles of development, and an overview of recent breakthroughs and promising research directions in the field. The selected topics will include morphogen signaling, somatogenesis, control of growth and division, tissue mechanics and regeneration, both in model organisms (drosophila, zebrafish, mouse) and in humans.Invited Lecturers include:James Briscoe (Crick Institute, London)Marie-Anne Félix (ENS, Paris)Mauro Giacca (ICGEB, Trieste)Thomas Gregor (Princeton University)Anna-Katerina Hadjantonakis (Memorial Sloan Kettering Cancer Center)Frank Jülicher (MPIPKS Dresden)Allon Klein (Harvard Medical School)Arthur Lander (UC Irvine)Andrew Oates (Crick Institute, London)Kenneth Poss (Duke University)Boris Shraiman (UC Santa Barbara)Aryeh Warmflash (Rice University)Eri...

The School is focused on the basic physical and biological principles of the development of multicellular organisms.The purpose of this advanced School is to provide both an introduction to the basic physical and biological principles of development, and an overview of recent breakthroughs and promising research directions in the field. The selected topics will include morphogen signaling, somatogenesis, control of growth and division, tissue mechanics and regeneration, both in model organisms (drosophila, zebrafish, mouse) and in humans.Invited Lecturers include:James Briscoe (Crick Institute, London)Marie-Anne Félix (ENS, Paris)Mauro Giacca (ICGEB, Trieste)Thomas Gregor (Princeton University)Anna-Katerina Hadjantonakis (Memorial Sloan Kettering Cancer Center)Frank Jülicher (MPIPKS Dresden)Allon Klein (Harvard Medical School)Arthur Lander (UC Irvine)Andrew Oates (Crick Institute, London)Kenneth Poss (Duke University)Boris Shraiman (UC Santa Barbara)Aryeh Warmflash (Rice University)Eri...

Quantum Theory has passed all experimental tests, with impressive accuracy. It applies to light and matter from the smallest scales so far explored, up to the mesoscopic scale. It is also a necessary ingredient for understanding the evolution of the universe.  It has given rise to an impressive number of new technologies. Yet it suffers from internal problems of consistency. On top of this, its unification with general relativity is still problematic, and no fully satisfactory quantum theory of gravity yet exists. This program will explore the current state of the art and future perspectives in the foundations of quantum theory, also in connection with its unification with general relativity. It consists of a two-week school, followed by a discussion meeting. The topics to be covered in the school include:School 1st Week - Theory: The measurement problem and its proposed solutions, Role of gravity in wave function collapse, Weak measurements, Quantum Measure and the Born rule, Trace Dy...

Quantum Theory has passed all experimental tests, with impressive accuracy. It applies to light and matter from the smallest scales so far explored, up to the mesoscopic scale. It is also a necessary ingredient for understanding the evolution of the universe.  It has given rise to an impressive number of new technologies. Yet it suffers from internal problems of consistency. On top of this, its unification with general relativity is still problematic, and no fully satisfactory quantum theory of gravity yet exists. This program will explore the current state of the art and future perspectives in the foundations of quantum theory, also in connection with its unification with general relativity. It consists of a two-week school, followed by a discussion meeting. The topics to be covered in the school include:School 1st Week - Theory: The measurement problem and its proposed solutions, Role of gravity in wave function collapse, Weak measurements, Quantum Measure and the Born rule, Trace Dy...

Computational techniques are of great help in dealing with substantial, otherwise intractable examples, possibly leading to further structural insights and the detection of patterns in many abstract mathematical fields like finite group theory, combinatorics, homological algebra etc. Computational mathematics has recently emerged as an important area of study and has established itself as a powerful tool, aiding quick maturing of intuition about concrete mathematical structures.The first part of this program, a 5-day instructional workshop during November 05 - 09, 2016, is an attempt to introduce computational techniques in group theory and related topics to graduate students and young researchers. This activity will consist of five mini-courses:Computational Homological AlgebraComputational Representation TheoryComputational aspects of finite p-groupsComputer algebra system GAPIntroduction to Monster simple group and MoonshineThe plan is to have five hours of teaching, tutorials and g...

Computational techniques are of great help in dealing with substantial, otherwise intractable examples, possibly leading to further structural insights and the detection of patterns in many abstract mathematical fields like finite group theory, combinatorics, homological algebra etc. Computational mathematics has recently emerged as an important area of study and has established itself as a powerful tool, aiding quick maturing of intuition about concrete mathematical structures.The first part of this program, a 5-day instructional workshop during November 05 - 09, 2016, is an attempt to introduce computational techniques in group theory and related topics to graduate students and young researchers. This activity will consist of five mini-courses:Computational Homological AlgebraComputational Representation TheoryComputational aspects of finite p-groupsComputer algebra system GAPIntroduction to Monster simple group and MoonshineThe plan is to have five hours of teaching, tutorials and g...