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I would like to provide a short, possibly elementary, introduction to the problem of computing string amplitudes at higher genus for superstrings. Essentially, I will recall which is the mathematical problem in defining the path integral measure (which has a well defined algebraic geometry realization for bosonic strings) and the solution proposed by d~@~YHocker and Phong for the genus 2 case. Their main results are the chiral splitted form of the measure, and its explicit expression in genus two. They proposed the splitting form to work at any genus and assumed some restriction for the explicit form which however did not permitted them to find a solution for genera higher then 2. I will tell something about the technology which permitted us to find explicit solutions for genus 3 and four. Indeed, we showed that the restrictions imposed by d~@~YHocker and Phong have no solution whereas the most general form compatible with modular invariance and clustering provide a unique solution, at least for genus 3 and 4. I will try to be as less technical as possible.

In this talk I will give an introduction to the simulation of quantum many-body systems using the so-called tensor networks. After a brief historical review, I will introduce the basics on tensor network representations of quantum states, and will explain some recent developments. In particular, in the last part of my talk I will focus on recent results obtained in the simulation of 2-dimensional quantum lattice systems of infinite size.

There are a number of arguments in the philosophical, physical and cosmological literatures for the thesis that time is not fundamental to the description of nature. According to this view, time should be only an approximate notion which emerges from a more fundamental, timeless description only in certain limiting approximations. My first task is to review these arguments and explain why they fail. I will then examine the opposite view, which is that time and change are fundamental and, indeed, are perhaps the only aspects of reality that are not emergent from a more fundamental, microscopic description. The argument involves several aspects of contemporary physics and cosmology including 1) the problem of the landscape of string theory, 2) cosmological inflation and the problem of initial conditions, 3) the interpretation of the “wavefunction of the universe,” and the problem of what is an observable in classical and quantum general relativity. It also involves issues in the foundations of mathematics and the issue of the proper understanding of the role of mathematics in physics. The view that time is real and not emergent is, I will argue, supported by considerations arising from all these issues It leads finally to a need for a notion of law in cosmology which replaces the freedom to choose initial conditions with a notion of laws evolving in time. The arguments presented here have been developed in collaboration with Roberto Mangabeira Unger .

I discuss how we can give a satisfactory account of theory confirmation for theories with random data, such as Copenhagen quantum theory, despite the lack of a completely satisfactory definition of probabilistic theories of nature. I also explain why neither this nor any other proposed account of scientific confirmation works for many-worlds theories

I will identify six \'problems of time\' that arise in connection with quantum gravity and review the extent to which some of them can be regarded as solved, highlighting the very different aspects that they assume depending on one\'s starting point: Hamiltonian vs. path-integral, discrete vs continuous.

The evidence for the big bang is now overwhelming. However, the basic question of what caused the bang remains open. One possibility is that time somehow \'emerged,\' placing the universe in an inflationary state. Another, perhaps more conservative possibility, is that the big bang was a violent event in a pre-existing universe. I will describe model calculations employing the AdS/CFT correspondence which show how this is possible, and which point to a new explanation for the origin of large scale structure in the universe.

In recent work with Bob Coecke and others, we have developed a categorical axiomatization of quantum mechanics. This analyzes the main structural features of quantum mechanics into simple and general elements, which admit an elegant diagrammatic representation. This enables an illuminating and effective analysis of quantum information protocols and computational structures. One aspect which is brought to light in this analysis is that protocols such as teleportation use entanglement to achieve a logical information flow which has an apparent retro-causal (or even `backward-in-time\') component. However, there is also a physical or operational description of the same systems, based on an abstract structural characterization of quantum measurements, which is entirely causally consistent. The systematic relationship between these two descriptions suggests a novel perspective on the flow of time and information in the quantum realm.

In recent years there has been a growing awareness that studies on quantum foundations have close relationships with other fields such as probability and information theory. In this talk I give another example of how such interdisciplinary work can be fruitful, by applying some of the lessons from quantum mechanics, in particular from Bell\'s theorem, to a debate on the philosophical foundations of decision theory. I argue that the basic assumptions of the popular causal decision theory -- which was developed partly in response to a puzzle proposed by the physicist William Newcomb and published by the philosopher Robert Nozick -- are analogous to the basic assumptions of a local hidden-variables theory in the context of Bell\'s theorem. Both have too strong a prejudice about the causal structure of the world: there are possible games the world can pose such that an agent who operates by those theories is constrained to choose losing strategies no matter what evidence he or she acquires.