Pavel Bleher, IUPUI
Mini-course. Renormalization Group and Critical Phenomena in Hierarchical Models of Statistical Physics

In the series of four lectures we will discuss various results on renormalization group (RG) and critical phenomena in hierarchical models. We will consider two different types of hierarchical models:
1. Dyson's hierarchical models,
2. Migdal-Kadanoff's spin models on hierarchical lattices.
The great importance of hierarchical models is that they allow investigation of various phenomena unavailable in other models. We will discuss evaluation of critical exponents, phase transitions in models with discrete and continuous symmetry, Fisher's and Lee-Yang's zeros, and others. For Dyson's hierarchical models the RG action is described in terms of a nonlinear integral operators, and we will explain a technique to investigate this nonlinear dynamics. For Migdal-Kadanoff's spin models on hierarchical lattices the RG action is described in terms of rational mappings in one and two dimensions, and we will find their attractors and the limiting Lee-Yang measures in dimension 1 and the limiting Fisher-Lee-Yang currents in dimension 2.
Part I of the lectures will be based on the works of Bleher-Sinai and Bleher-Major, and Part II on the works of Bleher-Lyubich-Roeder.

Title:   Mini-course. Renormalization Group and Critical Phenomena in Hierarchical Models of Statistical Physics
Speaker:   Roland Roeder [IUPUI]

Abstract:   In the series of four lectures we will discuss various results on renormalization group (RG) and critical phenomena in hierarchical models. We will consider two different types of hierarchical models:
1. Dyson's hierarchical models,
2. Migdal-Kadanoff's spin models on hierarchical lattices.
The great importance of hierarchical models is that they allow investigation of various phenomena unavailable in other models. We will discuss evaluation of critical exponents, phase transitions in models with discrete and continuous symmetry, Fisher's and Lee-Yang's zeros, and others. For Dyson's hierarchical models the RG action is described in terms of a nonlinear integral operators, and we will explain a technique to investigate this nonlinear dynamics. For Migdal-Kadanoff's spin models on hierarchical lattices the RG action is described in terms of rational mappings in one and two dimensions, and we will find their attractors and the limiting Lee-Yang measures in dimension 1 and the limiting Fisher-Lee-Yang currents in dimension 2.
Part I of the lectures will be based on the works of Bleher-Sinai and Bleher-Major, and Part II on the works of Bleher-Lyubich-Roeder.
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Increase to 19 or 23 credits in accordance to the Course Load Policy. $20 fee automatically billed for students enrolled in more than 19 credits.
12:00 Noon

Pavel Bleher, IUPUI
Mini-course. Renormalization Group and Critical Phenomena in Hierarchical Models of Statistical Physics

In the series of four lectures we will discuss various results on renormalization group (RG) and critical phenomena in hierarchical models. We will consider two different types of hierarchical models:
1. Dyson's hierarchical models,
2. Migdal-Kadanoff's spin models on hierarchical lattices.
The great importance of hierarchical models is that they allow investigation of various phenomena unavailable in other models. We will discuss evaluation of critical exponents, phase transitions in models with discrete and continuous symmetry, Fisher's and Lee-Yang's zeros, and others. For Dyson's hierarchical models the RG action is described in terms of a nonlinear integral operators, and we will explain a technique to investigate this nonlinear dynamics. For Migdal-Kadanoff's spin models on hierarchical lattices the RG action is described in terms of rational mappings in one and two dimensions, and we will find their attractors and the limiting Lee-Yang measures in dimension 1 and the limiting Fisher-Lee-Yang currents in dimension 2.
Part I of the lectures will be based on the works of Bleher-Sinai and Bleher-Major, and Part II on the works of Bleher-Lyubich-Roeder.

Title:   Mini-course. Renormalization Group and Critical Phenomena in Hierarchical Models of Statistical Physics
Speaker:   Roland Roeder [IUPUI]

Abstract:   In the series of four lectures we will discuss various results on renormalization group (RG) and critical phenomena in hierarchical models. We will consider two different types of hierarchical models:
1. Dyson's hierarchical models,
2. Migdal-Kadanoff's spin models on hierarchical lattices.
The great importance of hierarchical models is that they allow investigation of various phenomena unavailable in other models. We will discuss evaluation of critical exponents, phase transitions in models with discrete and continuous symmetry, Fisher's and Lee-Yang's zeros, and others. For Dyson's hierarchical models the RG action is described in terms of a nonlinear integral operators, and we will explain a technique to investigate this nonlinear dynamics. For Migdal-Kadanoff's spin models on hierarchical lattices the RG action is described in terms of rational mappings in one and two dimensions, and we will find their attractors and the limiting Lee-Yang measures in dimension 1 and the limiting Fisher-Lee-Yang currents in dimension 2.
Part I of the lectures will be based on the works of Bleher-Sinai and Bleher-Major, and Part II on the works of Bleher-Lyubich-Roeder.
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Theodore D. Drivas, Stony Brook University
Mini-course. Mathematical aspects of turbulence.

In Lecture 1 & 2, we will discuss some foundational aspects of three-dimensional incompressible fluid turbulence, including guiding experimental observations, Kolmogorov's 1941 theory on the structure of a turbulent flow, Onsager's 1949 conjecture on anomalous dissipation and weak Euler solutions, and Landau’s Kazan remark concerning intermittency. Mathematical examples and constructions that exhibit features of turbulent behavior will be discussed. In Lecture 3 & 4, we will discuss the formation of small and large scales in two-dimensional fluids (both viscous and inviscid). In the inviscid setting, we will discuss the mixing process which creates infinitely fine scales of motion at long times and serves as the dynamical mechanism for the direct enstrophy cascade. Rigorous statements can be made in this setting near steady states (Nadirashvili, Koch). For viscous fluids, we will discuss the stability and instability of Kolmogorov flow on two-dimensional flat tori (Meshalkin-Sinai) and a related example of non-uniqueness of smooth steady states of the Navier-Stokes equations (Yudovich). Destabilization of this laminar regime relates to the transition to turbulence.

Title:   Mini-course. Mathematical aspects of turbulence.
Speaker:   Theodore D. Drivas [Stony Brook University]

Abstract:   In Lecture 1 & 2, we will discuss some foundational aspects of three-dimensional incompressible fluid turbulence, including guiding experimental observations, Kolmogorov's 1941 theory on the structure of a turbulent flow, Onsager's 1949 conjecture on anomalous dissipation and weak Euler solutions, and Landau’s Kazan remark concerning intermittency. Mathematical examples and constructions that exhibit features of turbulent behavior will be discussed. In Lecture 3 & 4, we will discuss the formation of small and large scales in two-dimensional fluids (both viscous and inviscid). In the inviscid setting, we will discuss the mixing process which creates infinitely fine scales of motion at long times and serves as the dynamical mechanism for the direct enstrophy cascade. Rigorous statements can be made in this setting near steady states (Nadirashvili, Koch). For viscous fluids, we will discuss the stability and instability of Kolmogorov flow on two-dimensional flat tori (Meshalkin-Sinai) and a related example of non-uniqueness of smooth steady states of the Navier-Stokes equations (Yudovich). Destabilization of this laminar regime relates to the transition to turbulence.
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