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Information theory and computational complexity have emerged as central concepts in the study of biological and physical systems, in both the classical and quantum realm. The low-energy landscape of classical and quantum systems can be characterized in terms of constraint satisfaction problems, with the physical behavior of these theories – for example, glassy behavior – governed by the computational complexity of such problems. Information theory also provides a language for laying the foundations of nonequilibrium statistical mechanics, and plays a central role in understanding the emergence of a small number of useful parameters in models of physical and biological phenomena.This school, organized jointly by the ICTS and Brandeis University, will cover a variety of forefront research topics in physics and biology, in which information and computation play a central role:• Thermodynamics of information processing• Coding and computation in statistical mechanics• Quantum information i...

Information theory and computational complexity have emerged as central concepts in the study of biological and physical systems, in both the classical and quantum realm. The low-energy landscape of classical and quantum systems can be characterized in terms of constraint satisfaction problems, with the physical behavior of these theories – for example, glassy behavior – governed by the computational complexity of such problems. Information theory also provides a language for laying the foundations of nonequilibrium statistical mechanics, and plays a central role in understanding the emergence of a small number of useful parameters in models of physical and biological phenomena.This school, organized jointly by the ICTS and Brandeis University, will cover a variety of forefront research topics in physics and biology, in which information and computation play a central role:• Thermodynamics of information processing• Coding and computation in statistical mechanics• Quantum information i...

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 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...