Colloquia
Klaus Richter (University of Regensburg, Germany)
Title: Probing the Quantum Mechanics of Many-Body Chaos
Date: 5th December 2018, 4pm
The dynamics and spread of quantum information in complex many-body systems is presently attracting a lot of attention across various fields, ranging from cold atom physics via condensed quantum matter to high energy physics and quantum gravity. This includes questions of how a quantum system thermalises, phenomena like many-body localisation and non-classicality in many-body quantum physics. Here concepts such as the famous spin echo that are based on rewinding time provide a useful way to monitor complex quantum information dynamics. Central to these developments are so-called out-of-time-order correlators (OTOCs). They presently receive particular attention as sensitive probes for chaos and the temporal growth of complexity in interacting systems.
We will address such phenomena using semiclassical path intergral techniques based on interfering Feynman paths, thereby bridging the classical and quantum many-body world. This enables us to compute echoes and OTOCs in terms of coherent sums and interprete the results in terms of multi-particle interference, thereby including entanglement and correlation effects. Moreover, on the numerical side we devise a semiclassical method for Bose-Hubbard systems far-out-of equilibrium that allows us to calculate many-body quantum interference on time scales far beyond the famous scrambling/Ehrenfest time.
Yuji Kodama (The Ohio State University)
Title: KP Solitons in Shallow Water
Date: 9th October 2018, 4:00pm
The Kadomtsev–Petviashvili (KP) equation describes weakly dispersive and small amplitude waves propagating in a quasi-two-dimensional situation. Recently a large variety of exact soliton solutions of the KP equation has been found and classified. Those soliton solutions are localized along certain lines in a two-dimensional plane and decay exponentially everywhere else, and they are called line-soliton solutions. The classification is based on the far-field patterns of the solutions, which consist of a finite number of line-solitons. Each soliton solution is then defined by a point of the totally non-negative Grassmann variety, which can be parameterized by a unique derangement of the symmetric group of permutations.
In this talk, I will give a brief summary of our theory and discuss an application to the Mach reflection problem in shallow water, which has an important implication to tsunami amplification along the shore. The problem describes the resonant interaction of solitary waves appearing in the reflection of an obliquely incident wave onto a vertical wall. In particular, nonlinear interactions among small amplitude, obliquely propagating long waves on the surface of shallow water often generate web-like patterns. I discuss how line-soliton solutions of the KP equation can approximate such web-pattern in shallow water wave. The talk will be elementary and include many figures of the wave-patterns from real ocean.
Mark Hoefer (University of Colorado Boulder)
Talk: Dispersive hydrodynamics: the mathematics and physics of nonlinear waves in dispersive media
Date: 10th October 2018, 4pm
Dispersive hydrodynamics--modeled by hyperbolic conservation laws with dispersive perturbation--has emerged as a unified mathematical framework for the description of multiscale nonlinear wave phenomena in dispersive media and accurately describes a plethora of physical systems. This talk will be a tour through some recent mathematical and experimental results in this growing field of research. Parallels and analogies to classical hydrodynamics will be presented such as the generation of shock waves subject to appropriate regularization and their description in terms of characteristics. In contrast, from the existence of expansion shocks to the generation of dissipative shock waves in a conservative medium, dispersive regularization also leads to a number of counterintuitive, perhaps bizarre, effects, which will also be described. To keep you engaged, this tour will include lots of video and animations of in-house experiments and simulations.
Folkert Mueller-Hoissen (University of Goettingen)
Talk - Taking Solitons to the Limit: Recent insights into the fascinating world of the KP equation
Date - 7th March 2018
The KP equation generalizes the famous Korteweg-deVries (KdV) equation to a "completely integrable" PDE in three dimensions, which appears in different areas of physics and plays deep roles in some problems of mathematics. It possesses multi-soliton solutions forming network-like structures on a plane, evolving in time, modeling quite well certain wave patterns on a fluid surface. What are the rules underlying the evolution of these patterns? We approach this problem by idealizing the waves, at fixed time, to a piecewise linear structure, a graph, with a number (wave height) attached to each edge. In this talk, we focus on two insights from the analysis of this "tropical limit".
There is a subclass of, at fixed time, tree-shaped KP soliton solutions. In the tropical limit, their evolutions correspond to sequences of rooted binary trees related by "rotation in trees", which is a realization of the associativity relation (a b) c --> a (b c). The solitons from the tree class then realize what is known as Tamari lattices in combinatorics. They form (the 1-skeleton of) polytopes, called associahedra or Stasheff polytopes. Further combinatorial structures (higher Tamari orders, "polygon equations") emerge from this setting.
The KdV and the Boussinesq equation are reductions of the KP equation to two dimensions. In this case the tropical limit of a soliton solution lives on a graph in (two-dimensional) space-time. This associates a particle picture with the wave solution and the graph displays an interaction (collisions). The interaction is richer if the particles carry internal degrees of freedom. Therefore we promote scalar PDEs to vector and matrix versions. The interaction of KdV solitons in the tropical limit is then ruled by a Yang-Baxter map (A. Veselov 2003), a solution of the famous (quantum) Yang-Baxter equation. The latter is a crucial structure underlying two-dimensional quantum integrable models and exactly solvable models of statistical mechanics. Whereas the associated particle model only exhibits elastic scattering, in case of the Boussinesq equation there is also inelastic scattering. Of course, these structures have counterparts in the KP case, and there is much more to be discovered from an analysis of matrix KP solitons in the tropical limit.
All this is based on joint work with Aristophanes Dimakis (University of the Aegean, Greece). The presentation will be elementary.
Richard Bertram (Florida State University)
Talk - How simple concepts from dynamics can drive biological experiment
Date - 13th September 2017
Sometimes knowing a few concepts from dynamics can take you a long way. In this presentation I will demonstrate this claim by describing some research on the origin of pulsatile insulin secretion from pancreatic islets of Langerhans. This research has largely been driven by mathematical insights, which include some well known and very useful concepts about dynamical systems. I hope to relay the message that mathematical theory, computer simulations, and experimental studies can be combined very effectively to increase our understanding of complex biological phenomena.
Alexander Veselov (Loughborough University)
Talk - Markov triples: arithmetic, geometry and dynamics
Date - 10th May 2017
Markov triples are the integer solutions of the Markov equation x^2+y^2+z^2=3xyz, which surprisingly appeared in many areas of mathematics: initially in classical number theory, but more recently in hyperbolic and algebraic geometry, the theory of Teichmueller spaces, Frobenius manifolds and Painleve equations.
I am planning to explain some of these magical relations and properties of Markov triples, including recent results about their growth found jointly with K. Spalding.
Jonathan Halliwell (Imperial College London)
Talk - How the Quantum Universe became Classical
Date - 17th May 2017
Quantum theory has been spectacularly successful in its description of the microscopic realm and there is not one shred of experimental evidence that suggests that its basic structure is incorrect in any way. Yet despite being the fundamental theory of matter, it is not the theory of the ordinary everyday world of our experience and indeed some of its features such as the existence of superposition states and entanglement defy intuition in a truly profound way. This is because the macroscopic world and our everyday experience is best described by the classical mechanics of Newton and the laws of thermodynamics, theories utterly different to quantum theory. This then leads to a very interesting question. If macroscopic objects are made of atoms, and atoms are described by quantum theory, how do large collections of atoms come to be described by classical physics? In brief, how does classical physics emerge from quantum theory at sufficiently large scales? In this talk I outline in simple terms how this transition from quantum to classical physics takes place, with an emphasis on simple physical ideas and pictures, and not elaborate mathematics. It includes a simple account of the decoherent histories approach to quantum theory which is particularly suited to this task.
Talk - How the Quantum Universe became Classical
Date - 17th May 2017
Quantum theory has been spectacularly successful in its description of the microscopic realm and there is not one shred of experimental evidence that suggests that its basic structure is incorrect in any way. Yet despite being the fundamental theory of matter, it is not the theory of the ordinary everyday world of our experience and indeed some of its features such as the existence of superposition states and entanglement defy intuition in a truly profound way. This is because the macroscopic world and our everyday experience is best described by the classical mechanics of Newton and the laws of thermodynamics, theories utterly different to quantum theory. This then leads to a very interesting question. If macroscopic objects are made of atoms, and atoms are described by quantum theory, how do large collections of atoms come to be described by classical physics? In brief, how does classical physics emerge from quantum theory at sufficiently large scales? In this talk I outline in simple terms how this transition from quantum to classical physics takes place, with an emphasis on simple physical ideas and pictures, and not elaborate mathematics. It includes a simple account of the decoherent histories approach to quantum theory which is particularly suited to this task.