Abstract: I will give a pedagogical introduction to toric geometry without requiring previous knowledge in algebraic geometry. The lecture series will be based on the toric geometry package in the open-source Sage (http://www.sagemath.org) mathematics software system. Various examples relevant to string theory are used to illustrate the techniques. Each lecture will contain exercises to be solved in the accompanying computer lab.

Abstract: I will give a pedagogical introduction to toric geometry without requiring previous knowledge in algebraic geometry. The lecture series will be based on the toric geometry package in the open-source Sage (http://www.sagemath.org) mathematics software system. Various examples relevant to string theory are used to illustrate the techniques. Each lecture will contain exercises to be solved in the accompanying computer lab.

Abstract: I will give a pedagogical introduction to toric geometry without requiring previous knowledge in algebraic geometry. The lecture series will be based on the toric geometry package in the open-source Sage (http://www.sagemath.org) mathematics software system. Various examples relevant to string theory are used to illustrate the techniques. Each lecture will contain exercises to be solved in the accompanying computer lab.

Professor Steinhardt is one of the most influential theoretical cosmologists for many years. He has recently argued that the high precision data favor a cyclic evolution of the universe rather than inflationary Big Bang cosmology that he himself had previously been a key contributor to.

This course will begin with a short summary of some aspects of the history of Quantum Mechanics, which will include Einstein’s photon hypothesis, his analysis of monatomic quantum gases (including Bose-Einstein condensation for ideal Bose gases), and a modern interpretation of Heisenberg’s discovery of Matrix Mechanics. A brief review of the “deformation point of view” will be given, emphasizing the fact that the atomistic nature of matter can be understood as arising from “quantization”, i.e., from a “deformation” of continuum theories of matter. Subsequently, some of the key features of Quantum Physics distinguishing it from Classical Physics - Entanglement, Kochen-Specker Theorem, violation of Bell Inequalities, etc. - and some of the puzzling features of Quantum Mechanics will be recalled. A short presentation of the theory of indirect (weak) measurements and observations, as pioneered by Kraus, and of the phenomenon of “purification” will follow next. This will prepare the ground for a discussion of a novel general approach to Quantum Mechanics that claims to solve the so-called “measurement problem” and eliminates an undue role of “observers” in the formulation of Quantum Mechanics. It will then be time to consider some concrete applications of Quantum Theory. Presumably, examples of irreversible behavior exhibited by open systems in a quantum-mechanical description - including a derivation of the (first and the second) fundamental laws of thermodynamics, a brief review of the derivation of Brownian motion from unitary quantum dynamics and possibly of some further dynamical phenomena - will be discussed at the beginning of this section of the course. Afterwards, the foundations of Equilibrium Quantum Statistical Mechanics, including the KMS condition and its derivation by Haag, Hugenholtz and Winnink, will be reviewed. This formalism will then be applied to studying some phase transitions in Quantum Statistical Mechanics, (using the method of “infrared bounds”). The course will end more or less where it started: Aspects of the theory of interacting Bose gases, including the discussion of various limiting regimes useful to understand, for example, Bose-Einstein condensation, will be discussed in some detail.

Abstract: I will give a pedagogical introduction to toric geometry without requiring previous knowledge in algebraic geometry. The lecture series will be based on the toric geometry package in the open-source Sage (http://www.sagemath.org) mathematics software system. Various examples relevant to string theory are used to illustrate the techniques. Each lecture will contain exercises to be solved in the accompanying computer lab.

This course will begin with a short summary of some aspects of the history of Quantum Mechanics, which will include Einstein’s photon hypothesis, his analysis of monatomic quantum gases (including Bose-Einstein condensation for ideal Bose gases), and a modern interpretation of Heisenberg’s discovery of Matrix Mechanics. A brief review of the “deformation point of view” will be given, emphasizing the fact that the atomistic nature of matter can be understood as arising from “quantization”, i.e., from a “deformation” of continuum theories of matter. Subsequently, some of the key features of Quantum Physics distinguishing it from Classical Physics - Entanglement, Kochen-Specker Theorem, violation of Bell Inequalities, etc. - and some of the puzzling features of Quantum Mechanics will be recalled. A short presentation of the theory of indirect (weak) measurements and observations, as pioneered by Kraus, and of the phenomenon of “purification” will follow next. This will prepare the ground for a discussion of a novel general approach to Quantum Mechanics that claims to solve the so-called “measurement problem” and eliminates an undue role of “observers” in the formulation of Quantum Mechanics. It will then be time to consider some concrete applications of Quantum Theory. Presumably, examples of irreversible behavior exhibited by open systems in a quantum-mechanical description - including a derivation of the (first and the second) fundamental laws of thermodynamics, a brief review of the derivation of Brownian motion from unitary quantum dynamics and possibly of some further dynamical phenomena - will be discussed at the beginning of this section of the course. Afterwards, the foundations of Equilibrium Quantum Statistical Mechanics, including the KMS condition and its derivation by Haag, Hugenholtz and Winnink, will be reviewed. This formalism will then be applied to studying some phase transitions in Quantum Statistical Mechanics, (using the method of “infrared bounds”). The course will end more or less where it started: Aspects of the theory of interacting Bose gases, including the discussion of various limiting regimes useful to understand, for example, Bose-Einstein condensation, will be discussed in some detail.

Fields Medalist Alain Connes (Collège de France, IHES and Ohio State University) presents a set of three lectures at the Arnold Sommerfeld Center in the period March 1-3, 2016.

The speakers will not only present the latest findings about the universe and its possible beginnings but also discuss if there are limits of what can be theorized about, and how far the scientific method can be taken. Does it reach to the boundaries of the universe or is there a fundamental limit of what can be known.

Fields Medalist Alain Connes (Collège de France, IHES and Ohio State University) presents a set of three lectures at the Arnold Sommerfeld Center in the period March 1-3, 2016.

Fields Medalist Alain Connes (Collège de France, IHES and Ohio State University) presents a set of three lectures at the Arnold Sommerfeld Center in the period March 1-3, 2016.

Abstract: I will give a pedagogical introduction to toric geometry without requiring previous knowledge in algebraic geometry. The lecture series will be based on the toric geometry package in the open-source Sage (http://www.sagemath.org) mathematics software system. Various examples relevant to string theory are used to illustrate the techniques. Each lecture will contain exercises to be solved in the accompanying computer lab.

Abstract: I will give a pedagogical introduction to toric geometry without requiring previous knowledge in algebraic geometry. The lecture series will be based on the toric geometry package in the open-source Sage (http://www.sagemath.org) mathematics software system. Various examples relevant to string theory are used to illustrate the techniques. Each lecture will contain exercises to be solved in the accompanying computer lab.

This course will begin with a short summary of some aspects of the history of Quantum Mechanics, which will include Einstein’s photon hypothesis, his analysis of monatomic quantum gases (including Bose-Einstein condensation for ideal Bose gases), and a modern interpretation of Heisenberg’s discovery of Matrix Mechanics. A brief review of the “deformation point of view” will be given, emphasizing the fact that the atomistic nature of matter can be understood as arising from “quantization”, i.e., from a “deformation” of continuum theories of matter. Subsequently, some of the key features of Quantum Physics distinguishing it from Classical Physics - Entanglement, Kochen-Specker Theorem, violation of Bell Inequalities, etc. - and some of the puzzling features of Quantum Mechanics will be recalled. A short presentation of the theory of indirect (weak) measurements and observations, as pioneered by Kraus, and of the phenomenon of “purification” will follow next. This will prepare the ground for a discussion of a novel general approach to Quantum Mechanics that claims to solve the so-called “measurement problem” and eliminates an undue role of “observers” in the formulation of Quantum Mechanics. It will then be time to consider some concrete applications of Quantum Theory. Presumably, examples of irreversible behavior exhibited by open systems in a quantum-mechanical description - including a derivation of the (first and the second) fundamental laws of thermodynamics, a brief review of the derivation of Brownian motion from unitary quantum dynamics and possibly of some further dynamical phenomena - will be discussed at the beginning of this section of the course. Afterwards, the foundations of Equilibrium Quantum Statistical Mechanics, including the KMS condition and its derivation by Haag, Hugenholtz and Winnink, will be reviewed. This formalism will then be applied to studying some phase transitions in Quantum Statistical Mechanics, (using the method of “infrared bounds”). The course will end more or less where it started: Aspects of the theory of interacting Bose gases, including the discussion of various limiting regimes useful to understand, for example, Bose-Einstein condensation, will be discussed in some detail.

This course will begin with a short summary of some aspects of the history of Quantum Mechanics, which will include Einstein’s photon hypothesis, his analysis of monatomic quantum gases (including Bose-Einstein condensation for ideal Bose gases), and a modern interpretation of Heisenberg’s discovery of Matrix Mechanics. A brief review of the “deformation point of view” will be given, emphasizing the fact that the atomistic nature of matter can be understood as arising from “quantization”, i.e., from a “deformation” of continuum theories of matter. Subsequently, some of the key features of Quantum Physics distinguishing it from Classical Physics - Entanglement, Kochen-Specker Theorem, violation of Bell Inequalities, etc. - and some of the puzzling features of Quantum Mechanics will be recalled. A short presentation of the theory of indirect (weak) measurements and observations, as pioneered by Kraus, and of the phenomenon of “purification” will follow next. This will prepare the ground for a discussion of a novel general approach to Quantum Mechanics that claims to solve the so-called “measurement problem” and eliminates an undue role of “observers” in the formulation of Quantum Mechanics. It will then be time to consider some concrete applications of Quantum Theory. Presumably, examples of irreversible behavior exhibited by open systems in a quantum-mechanical description - including a derivation of the (first and the second) fundamental laws of thermodynamics, a brief review of the derivation of Brownian motion from unitary quantum dynamics and possibly of some further dynamical phenomena - will be discussed at the beginning of this section of the course. Afterwards, the foundations of Equilibrium Quantum Statistical Mechanics, including the KMS condition and its derivation by Haag, Hugenholtz and Winnink, will be reviewed. This formalism will then be applied to studying some phase transitions in Quantum Statistical Mechanics, (using the method of “infrared bounds”). The course will end more or less where it started: Aspects of the theory of interacting Bose gases, including the discussion of various limiting regimes useful to understand, for example, Bose-Einstein condensation, will be discussed in some detail.

This course will begin with a short summary of some aspects of the history of Quantum Mechanics, which will include Einstein’s photon hypothesis, his analysis of monatomic quantum gases (including Bose-Einstein condensation for ideal Bose gases), and a modern interpretation of Heisenberg’s discovery of Matrix Mechanics. A brief review of the “deformation point of view” will be given, emphasizing the fact that the atomistic nature of matter can be understood as arising from “quantization”, i.e., from a “deformation” of continuum theories of matter. Subsequently, some of the key features of Quantum Physics distinguishing it from Classical Physics - Entanglement, Kochen-Specker Theorem, violation of Bell Inequalities, etc. - and some of the puzzling features of Quantum Mechanics will be recalled. A short presentation of the theory of indirect (weak) measurements and observations, as pioneered by Kraus, and of the phenomenon of “purification” will follow next. This will prepare the ground for a discussion of a novel general approach to Quantum Mechanics that claims to solve the so-called “measurement problem” and eliminates an undue role of “observers” in the formulation of Quantum Mechanics. It will then be time to consider some concrete applications of Quantum Theory. Presumably, examples of irreversible behavior exhibited by open systems in a quantum-mechanical description - including a derivation of the (first and the second) fundamental laws of thermodynamics, a brief review of the derivation of Brownian motion from unitary quantum dynamics and possibly of some further dynamical phenomena - will be discussed at the beginning of this section of the course. Afterwards, the foundations of Equilibrium Quantum Statistical Mechanics, including the KMS condition and its derivation by Haag, Hugenholtz and Winnink, will be reviewed. This formalism will then be applied to studying some phase transitions in Quantum Statistical Mechanics, (using the method of “infrared bounds”). The course will end more or less where it started: Aspects of the theory of interacting Bose gases, including the discussion of various limiting regimes useful to understand, for example, Bose-Einstein condensation, will be discussed in some detail.

This course will begin with a short summary of some aspects of the history of Quantum Mechanics, which will include Einstein’s photon hypothesis, his analysis of monatomic quantum gases (including Bose-Einstein condensation for ideal Bose gases), and a modern interpretation of Heisenberg’s discovery of Matrix Mechanics. A brief review of the “deformation point of view” will be given, emphasizing the fact that the atomistic nature of matter can be understood as arising from “quantization”, i.e., from a “deformation” of continuum theories of matter. Subsequently, some of the key features of Quantum Physics distinguishing it from Classical Physics - Entanglement, Kochen-Specker Theorem, violation of Bell Inequalities, etc. - and some of the puzzling features of Quantum Mechanics will be recalled. A short presentation of the theory of indirect (weak) measurements and observations, as pioneered by Kraus, and of the phenomenon of “purification” will follow next. This will prepare the ground for a discussion of a novel general approach to Quantum Mechanics that claims to solve the so-called “measurement problem” and eliminates an undue role of “observers” in the formulation of Quantum Mechanics. It will then be time to consider some concrete applications of Quantum Theory. Presumably, examples of irreversible behavior exhibited by open systems in a quantum-mechanical description - including a derivation of the (first and the second) fundamental laws of thermodynamics, a brief review of the derivation of Brownian motion from unitary quantum dynamics and possibly of some further dynamical phenomena - will be discussed at the beginning of this section of the course. Afterwards, the foundations of Equilibrium Quantum Statistical Mechanics, including the KMS condition and its derivation by Haag, Hugenholtz and Winnink, will be reviewed. This formalism will then be applied to studying some phase transitions in Quantum Statistical Mechanics, (using the method of “infrared bounds”). The course will end more or less where it started: Aspects of the theory of interacting Bose gases, including the discussion of various limiting regimes useful to understand, for example, Bose-Einstein condensation, will be discussed in some detail.

This course will begin with a short summary of some aspects of the history of Quantum Mechanics, which will include Einstein’s photon hypothesis, his analysis of monatomic quantum gases (including Bose-Einstein condensation for ideal Bose gases), and a modern interpretation of Heisenberg’s discovery of Matrix Mechanics. A brief review of the “deformation point of view” will be given, emphasizing the fact that the atomistic nature of matter can be understood as arising from “quantization”, i.e., from a “deformation” of continuum theories of matter. Subsequently, some of the key features of Quantum Physics distinguishing it from Classical Physics - Entanglement, Kochen-Specker Theorem, violation of Bell Inequalities, etc. - and some of the puzzling features of Quantum Mechanics will be recalled. A short presentation of the theory of indirect (weak) measurements and observations, as pioneered by Kraus, and of the phenomenon of “purification” will follow next. This will prepare the ground for a discussion of a novel general approach to Quantum Mechanics that claims to solve the so-called “measurement problem” and eliminates an undue role of “observers” in the formulation of Quantum Mechanics. It will then be time to consider some concrete applications of Quantum Theory. Presumably, examples of irreversible behavior exhibited by open systems in a quantum-mechanical description - including a derivation of the (first and the second) fundamental laws of thermodynamics, a brief review of the derivation of Brownian motion from unitary quantum dynamics and possibly of some further dynamical phenomena - will be discussed at the beginning of this section of the course. Afterwards, the foundations of Equilibrium Quantum Statistical Mechanics, including the KMS condition and its derivation by Haag, Hugenholtz and Winnink, will be reviewed. This formalism will then be applied to studying some phase transitions in Quantum Statistical Mechanics, (using the method of “infrared bounds”). The course will end more or less where it started: Aspects of the theory of interacting Bose gases, including the discussion of various limiting regimes useful to understand, for example, Bose-Einstein condensation, will be discussed in some detail.

This course will begin with a short summary of some aspects of the history of Quantum Mechanics, which will include Einstein’s photon hypothesis, his analysis of monatomic quantum gases (including Bose-Einstein condensation for ideal Bose gases), and a modern interpretation of Heisenberg’s discovery of Matrix Mechanics. A brief review of the “deformation point of view” will be given, emphasizing the fact that the atomistic nature of matter can be understood as arising from “quantization”, i.e., from a “deformation” of continuum theories of matter. Subsequently, some of the key features of Quantum Physics distinguishing it from Classical Physics - Entanglement, Kochen-Specker Theorem, violation of Bell Inequalities, etc. - and some of the puzzling features of Quantum Mechanics will be recalled. A short presentation of the theory of indirect (weak) measurements and observations, as pioneered by Kraus, and of the phenomenon of “purification” will follow next. This will prepare the ground for a discussion of a novel general approach to Quantum Mechanics that claims to solve the so-called “measurement problem” and eliminates an undue role of “observers” in the formulation of Quantum Mechanics. It will then be time to consider some concrete applications of Quantum Theory. Presumably, examples of irreversible behavior exhibited by open systems in a quantum-mechanical description - including a derivation of the (first and the second) fundamental laws of thermodynamics, a brief review of the derivation of Brownian motion from unitary quantum dynamics and possibly of some further dynamical phenomena - will be discussed at the beginning of this section of the course. Afterwards, the foundations of Equilibrium Quantum Statistical Mechanics, including the KMS condition and its derivation by Haag, Hugenholtz and Winnink, will be reviewed. This formalism will then be applied to studying some phase transitions in Quantum Statistical Mechanics, (using the method of “infrared bounds”). The course will end more or less where it started: Aspects of the theory of interacting Bose gases, including the discussion of various limiting regimes useful to understand, for example, Bose-Einstein condensation, will be discussed in some detail.

Abstract: I will give a pedagogical introduction to toric geometry without requiring previous knowledge in algebraic geometry. The lecture series will be based on the toric geometry package in the open-source Sage (http://www.sagemath.org) mathematics software system. Various examples relevant to string theory are used to illustrate the techniques. Each lecture will contain exercises to be solved in the accompanying computer lab.