Seminars from until

It is well-known that considerations of symmetry, based on the construction of suitable Noether charges, lead to the definition of a host of conserved quantities (energy, linear momentum, center of mass, etc.) for an asymptotically flat initial data set and a great deal of progress in Mathematical Relativity in recent decades essentially amounts to establishing fundamental properties for such quantities (space-time positive mass theorems, Penrose inequalities, etc.). In this talk we first review this classical theory and then show how it can be extended to the setting in which the initial data set carries a non-compact boundary. This is based on joint work with S. Almaraz e L. Mari (arXiv:1907.02023).

Painlevé equations were discovered at the beginning of XX-th century by Paul Painlevé for purely mathematical reasons. Their solutions, the Painlevé transcendents, are, in general, "new transcendental functions" and, as the classical special functions, they appear in many problems of mathematics and physics. Applications "exploded" at the end of XX-th century: Einstein metrics, Frobenius manifolds, correlation function of the $2$-dimensional Ising model, reduction of integrable PDEs, reduction of self-dual Yang-Mills equations, random matrix theory, $2$-dimensional CFT (conformal blocks), non perturbative effects in strings theory ($2$d quantum gravity)...

First Lecture

Firstly, we will recall basics about Painlevé equations. Each Painlevé vector field initially defined on a trivial bundle of fibre $C^2$ (the naive phase space) can be extended to a fiber bundle (the Okamoto bundle) whose fiber is the Okamoto space of initial conditions: an affine rational surface endowed with a canonical symplectic structure. On this extension the Painlevé vector field is complete.

We will detail the "simplest case": Painlevé VI. The basis of the Okamoto bundle is $C\setminus \{0,1\}$ and we have a non-linear monodromy (with two generators) which induces a dynamics on each Okamoto space.

Painlevé VI can be interpreted as a traduction of the isomonodromic deformations of some linear second order equations (the linearized equations). The space of monodromy data of the linearized equations is an affine cubic surface: the character variety of Painlevé VI. It is endowed with a canonical symplectic structure.

For generic values of the parameters the character variety is non-singular and there exists an analytic (symplectic) diffeomorphism, the Riemann-Hilbert map (RH) between each Okamoto variety and the character variety. The dynamics on the Okamoto variety (non linear monodromy) is conjugated by RH to a dynamics on the character variety. An essential result is that this last dynamics is algebraic and can be explicitely calculated. As a consequence it is possible to prove that (generically) the dynamics of PVI is "rich" (chaotic...) and that the Galois-Malgrange differential groupoid is "as big as possible" (in particular PVI is not integrable!).

The main purpose of the minicourse is to describe a generalization of this picture to the others Painlevé equations (it is a work in progress...).

For PVI the linearized equation is Fuchsian. The character variety is the set of (classical) monodromy representations up to equivalence.

In the other cases the linearized equation has (as the Painlevé equation itself) irregular singularities.

There are Stokes phenomena and it is necessary to "add" in some sense these phenomena to the classical monodromy data. The corresponding character variety is the set of the extended (wild) monodromy data up to equivalence. It is also a cubic surface.

The non-linear monodromy remains but it is a "poor information". The induced dynamics on the Okamoto variety is "too small". Some years ago (2012), I proposed to define a "better dynamics", the wild dynamics. The idea is to generalize the wild dynamics I defined before in the linear case: this dynamics is generated by the classical dynamics, the Stokes phenomena and some continuous tori actions (exponential tori actions).

In the Painlevé case the definition of the non-linear Stokes phenomena and of the exponential tori actions are far to be trivial... In 2012 they were conjectural.The rigorous approach is due to A. Bittmann and quite recent: 2016.

The wild dynamics induces, via RH, a (symplectic) dynamics on the character variety. This dynamics is a priori local. I conjectured that it is in fact rational (2012).

In the second part of the first lecture I will detail the case of PII. I will give a simple and totally explicit description (partly conjectural...) of the wild dynamics on the character variety. An essential point is that it is rational.

The $9$ lines on the character variety of PII play an essential role in the wild dynamics and they are related to some special solutions (Boutroux truncated, tritrucated, bitruncated solutions). There are also related to the resurgence (in Ecalle spirit) of PII. An heuristic principle follows: the lines on the character varieties (affine cubic surfaces) play a central role in the theory of Painlevé equations.

The moduli space of flat $C^\infty$ connections on rank $n$ vector bundles on a smooth complex algebraic curve $\Sigma$ has an explicit description as the space of representations of the fundamental group: $${\mathcal M}_B = {\rm Hom}(\pi_1(\Sigma),G)/G,\qquad G={\rm GL}_n({\mathbb C})$$ often called the character variety (or Betti moduli space). It has an algebraic Poisson structure with symplectic leaves ${\mathcal M}_B({\bf{\mathcal C}})\subset {\mathcal M}_B$ given by fixing the conjugacy classes of monodromy around each puncture. Choosing suitable generators of $\pi_1(\Sigma)$ this becomes the quotient by $G$ of a space of matrices satisfying a relation of the form $$[A_1,B_1]\cdots[A_g,B_g]M_1\cdots M_m=1$$ where $[a,b]=aba^{-1}b^{-1}$. Thus ${\mathcal M}_B$ “looks like” a multiplicative version of a symplectic (Marsden-Weinstein) quotient, with $G$-valued moment map $\mu$ given by the left-hand side of the relation, so that ${\mathcal M}_B=\mu^{-1}(1)/G$. This “quasi-Hamiltonian” theory was set-up by Alekseev-Malkin-Meinrenken, and leads to the construction of the symplectic manifolds ${\mathcal M}_B({\bf \mathcal C})$ as the multiplicative symplectic quotient of the fusion product of two basic types pieces (conjugacy classes ${\mathcal C}$ and doubles ${\rm \bf D}$).

However this story is just the tip of the iceberg: In algebraic geometry the space ${\mathcal M}_B$ just parametrises the special class of algebraic connections on rank $n$ algebraic vector bundles on $\Sigma$ with tame/regular singular behaviour at the punctures. In this course I’ll first review the topological description of the full category of algebraic connections in terms of Stokes local systems, and the resulting explicit presentations of the wild character varieties (many of which go back to Birkhoff 1913). In turn I’ll describe the extension of the above story, constructing the wild character varieties as algebraic symplectic/Poisson varieties, by fusing together and reducing some new basic pieces (the fission spaces ${\mathcal A}$ and ${\mathcal B}$).

If time permits I'll discuss other topics such as

1. the fact the Drinfeld-Jimbo quantum group quantises a quite simple wild character variety,
2. the notion of wild Riemann surface and the resulting wild mapping group action on the wild character varieties,
3. the upgrading of the symplectic structure to a hyperkahler structure, yielding the link to meromorphic Higgs bundles,
4. the quasi-Hamiltonian fusion approach to complex WKB.

Some references/sources:

Quasi-Hamiltonian geometry (for compact Lie groups) was defined in:

• Alekseev, Malkin, Meinrenken, Lie group valued moment maps, math.DG/9707021

and the extension of this formalism to complex groups that we use is in:

• Alekseev, Bursztyn, Meinrenken, Pure spinors on Lie groups, arXiv:0709.1452

The wild extension is in the sequence of papers:

1. Quasi-Hamiltonian Geometry of Meromorphic Connections, arXiv:math/0203161, published in Duke 2007 [[the generic case, for any reductive group $G$]].
2. Through the analytic halo: Fission via irregular singularities, arXiv:1305.6465, Ann. Inst. Fourier 59, 7 (2009) 2669-2684 [[the simplest non-generic case]
3. Geometry and braiding of Stokes data; Fission and wild character varieties, Annals of Math. 179 (2014) 301-365 (arXiv:1111.6228) [[the general untwisted case]]
4. P.Boalch and D. Yamakawa, Twisted wild character varieties, arXiv:1512.08091 [[the general case]]

An earlier analytic approach to these symplectic manifolds is in P.Boalch, Symplectic Manifolds and Isomonodromic Deformations, Adv. in Math. 163 (2001) 137-205, and the hyperkahler upgrade is in:

• P. Boalch and O. Biquard, Wild non-abelian Hodge theory on curves, Compos. Math. 140 (2004) no. 1, 179-204

A “simple as possible” description of several intrinsic approaches to Stokes data for general linear groups is in

• arXiv:1903.12612 (Topology of the Stokes phenomenon).

The introduction of this paper aims to give a good guide to the Stokes data literature.

Several explicit descriptions of some of the simplest examples is in arXiv:1501.00930 (Wild Character Varieties, points on the Riemann sphere and Calabi's examples), and reviews of several different directions related to this story are in arXiv:1305.6593, arXiv:1703.10376.

Second Lecture

The main topic of this lecture is the heuristic principle stated at the end of the first lecture:

the lines on the character varieties (affine cubic surfaces) play a central role in the theory of Painlevé equations.

This principle works very well for PVI. Surprisingly it seems that this remains unnoticed until very recently.

In a first part (mainly in XIX-th algebraic geometry style), we will describe the lines on the cubic surfaces, in particular the 27 lines in the case of nonsingular (complete) cubic surfaces.

In the second part of the lecture I will explain some relations between the 24 lines on the character variety of PVI and some partial reducibility properties of the monodromy representations of the linearized equations.

Afterwards I will sketch a generalization for the other Painlevé equations.

The moduli space of flat $C^\infty$ connections on rank $n$ vector bundles on a smooth complex algebraic curve $\Sigma$ has an explicit description as the space of representations of the fundamental group: $${\mathcal M}_B = {\rm Hom}(\pi_1(\Sigma),G)/G,\qquad G={\rm GL}_n({\mathbb C})$$ often called the character variety (or Betti moduli space). It has an algebraic Poisson structure with symplectic leaves ${\mathcal M}_B({\bf{\mathcal C}})\subset {\mathcal M}_B$ given by fixing the conjugacy classes of monodromy around each puncture. Choosing suitable generators of $\pi_1(\Sigma)$ this becomes the quotient by $G$ of a space of matrices satisfying a relation of the form $$[A_1,B_1]\cdots[A_g,B_g]M_1\cdots M_m=1$$ where $[a,b]=aba^{-1}b^{-1}$. Thus ${\mathcal M}_B$ “looks like” a multiplicative version of a symplectic (Marsden-Weinstein) quotient, with $G$-valued moment map $\mu$ given by the left-hand side of the relation, so that ${\mathcal M}_B=\mu^{-1}(1)/G$. This “quasi-Hamiltonian” theory was set-up by Alekseev-Malkin-Meinrenken, and leads to the construction of the symplectic manifolds ${\mathcal M}_B({\bf \mathcal C})$ as the multiplicative symplectic quotient of the fusion product of two basic types pieces (conjugacy classes ${\mathcal C}$ and doubles ${\rm \bf D}$).

However this story is just the tip of the iceberg: In algebraic geometry the space ${\mathcal M}_B$ just parametrises the special class of algebraic connections on rank $n$ algebraic vector bundles on $\Sigma$ with tame/regular singular behaviour at the punctures. In this course I’ll first review the topological description of the full category of algebraic connections in terms of Stokes local systems, and the resulting explicit presentations of the wild character varieties (many of which go back to Birkhoff 1913). In turn I’ll describe the extension of the above story, constructing the wild character varieties as algebraic symplectic/Poisson varieties, by fusing together and reducing some new basic pieces (the fission spaces ${\mathcal A}$ and ${\mathcal B}$).

If time permits I'll discuss other topics such as

1. the fact the Drinfeld-Jimbo quantum group quantises a quite simple wild character variety,
2. the notion of wild Riemann surface and the resulting wild mapping group action on the wild character varieties,
3. the upgrading of the symplectic structure to a hyperkahler structure, yielding the link to meromorphic Higgs bundles,
4. the quasi-Hamiltonian fusion approach to complex WKB.

Some references/sources:

Quasi-Hamiltonian geometry (for compact Lie groups) was defined in:

• Alekseev, Malkin, Meinrenken, Lie group valued moment maps, math.DG/9707021

and the extension of this formalism to complex groups that we use is in:

• Alekseev, Bursztyn, Meinrenken, Pure spinors on Lie groups, arXiv:0709.1452

The wild extension is in the sequence of papers:

1. Quasi-Hamiltonian Geometry of Meromorphic Connections, arXiv:math/0203161, published in Duke 2007 [[the generic case, for any reductive group $G$]].
2. Through the analytic halo: Fission via irregular singularities, arXiv:1305.6465, Ann. Inst. Fourier 59, 7 (2009) 2669-2684 [[the simplest non-generic case]
3. Geometry and braiding of Stokes data; Fission and wild character varieties, Annals of Math. 179 (2014) 301-365 (arXiv:1111.6228) [[the general untwisted case]]
4. P.Boalch and D. Yamakawa, Twisted wild character varieties, arXiv:1512.08091 [[the general case]]

An earlier analytic approach to these symplectic manifolds is in P.Boalch, Symplectic Manifolds and Isomonodromic Deformations, Adv. in Math. 163 (2001) 137-205, and the hyperkahler upgrade is in:

• P. Boalch and O. Biquard, Wild non-abelian Hodge theory on curves, Compos. Math. 140 (2004) no. 1, 179-204

A “simple as possible” description of several intrinsic approaches to Stokes data for general linear groups is in

• arXiv:1903.12612 (Topology of the Stokes phenomenon).

The introduction of this paper aims to give a good guide to the Stokes data literature.

Several explicit descriptions of some of the simplest examples is in arXiv:1501.00930 (Wild Character Varieties, points on the Riemann sphere and Calabi's examples), and reviews of several different directions related to this story are in arXiv:1305.6593, arXiv:1703.10376.

A modelação matemática é uma das formas mais eficientes de compreender fenómenos e prever acontecimentos futuros. Muitos desses modelos são formulados com recurso a Equações com Derivadas Parciais (EDPs), ferramentas matemáticas que capturam as alterações de certas quantidades sujeitas a leis como difusão e reação em função de variáveis contínuas (tempo, espaço, preço,…). Falaremos de passeio aleatório, transporte, opções, calor, superfícies mínimas, vibrações e tráfego automóvel, abordando e deduzindo algumas das EDPs mais famosas da Física, Biologia e Economia.

O mini-curso é acessível a alunos que tenham frequentado uma disciplina de cálculo ou análise em várias variáveis.

Third Lecture

Firstly I will discuss the problem of the definition of the wild monodromy for an arbitrary irregular singularity in the linear case, in relation with Stokes phenomena, $k$-summability, multisummability, Laplace transform and resurgence. The source of this topic is, a century ago, a R. Garnier paper 1919.

I will detail the basic example: the monodromy and Stokes phenomena in the case of Hypergeometic Equations (classical and confluent). I will explain the confluence of monodromy towards wild monodromy.

In the second part of the lecture I will describe some non-linear Stokes phenomena and the corresponding unfoldings: saddle-nodes, symplectic saddle-nodes.

I will end with the application of all the tools to the case of the confluence of PVI towards PV. As a byproduct, it is possible to get a proof of the rationality of the wild dynamics of PV (M. Klimes). It is extremely technical and I will give only the (simple!) basic ideas and the main lines.

It is a first step towards a proof of the following conjecture (Ramis 2012):

The (wild) dynamics on the (wild) character variety of each Painlevé equation is rational and explicitely computable.

Na era dos “Grandes Dados” a tendência é analisar toda a informação disponível. Será que é mesmo necessário? Será que o esforço e tempo despendido para analisar esses “Grandes Dados” conduz a uma informação significativamente maior do que usar técnicas de amostragem?

Neste minicurso discute-se esta questão e introduzem-se conceitos básicos de amostragem em populações finitas.

The moduli space of flat $C^\infty$ connections on rank $n$ vector bundles on a smooth complex algebraic curve $\Sigma$ has an explicit description as the space of representations of the fundamental group: $${\mathcal M}_B = {\rm Hom}(\pi_1(\Sigma),G)/G,\qquad G={\rm GL}_n({\mathbb C})$$ often called the character variety (or Betti moduli space). It has an algebraic Poisson structure with symplectic leaves ${\mathcal M}_B({\bf{\mathcal C}})\subset {\mathcal M}_B$ given by fixing the conjugacy classes of monodromy around each puncture. Choosing suitable generators of $\pi_1(\Sigma)$ this becomes the quotient by $G$ of a space of matrices satisfying a relation of the form $$[A_1,B_1]\cdots[A_g,B_g]M_1\cdots M_m=1$$ where $[a,b]=aba^{-1}b^{-1}$. Thus ${\mathcal M}_B$ “looks like” a multiplicative version of a symplectic (Marsden-Weinstein) quotient, with $G$-valued moment map $\mu$ given by the left-hand side of the relation, so that ${\mathcal M}_B=\mu^{-1}(1)/G$. This “quasi-Hamiltonian” theory was set-up by Alekseev-Malkin-Meinrenken, and leads to the construction of the symplectic manifolds ${\mathcal M}_B({\bf \mathcal C})$ as the multiplicative symplectic quotient of the fusion product of two basic types pieces (conjugacy classes ${\mathcal C}$ and doubles ${\rm \bf D}$).

However this story is just the tip of the iceberg: In algebraic geometry the space ${\mathcal M}_B$ just parametrises the special class of algebraic connections on rank $n$ algebraic vector bundles on $\Sigma$ with tame/regular singular behaviour at the punctures. In this course I’ll first review the topological description of the full category of algebraic connections in terms of Stokes local systems, and the resulting explicit presentations of the wild character varieties (many of which go back to Birkhoff 1913). In turn I’ll describe the extension of the above story, constructing the wild character varieties as algebraic symplectic/Poisson varieties, by fusing together and reducing some new basic pieces (the fission spaces ${\mathcal A}$ and ${\mathcal B}$).

If time permits I'll discuss other topics such as

1. the fact the Drinfeld-Jimbo quantum group quantises a quite simple wild character variety,
2. the notion of wild Riemann surface and the resulting wild mapping group action on the wild character varieties,
3. the upgrading of the symplectic structure to a hyperkahler structure, yielding the link to meromorphic Higgs bundles,
4. the quasi-Hamiltonian fusion approach to complex WKB.

Some references/sources:

Quasi-Hamiltonian geometry (for compact Lie groups) was defined in:

• Alekseev, Malkin, Meinrenken, Lie group valued moment maps, math.DG/9707021

and the extension of this formalism to complex groups that we use is in:

• Alekseev, Bursztyn, Meinrenken, Pure spinors on Lie groups, arXiv:0709.1452

The wild extension is in the sequence of papers:

1. Quasi-Hamiltonian Geometry of Meromorphic Connections, arXiv:math/0203161, published in Duke 2007 [[the generic case, for any reductive group $G$]].
2. Through the analytic halo: Fission via irregular singularities, arXiv:1305.6465, Ann. Inst. Fourier 59, 7 (2009) 2669-2684 [[the simplest non-generic case]
3. Geometry and braiding of Stokes data; Fission and wild character varieties, Annals of Math. 179 (2014) 301-365 (arXiv:1111.6228) [[the general untwisted case]]
4. P.Boalch and D. Yamakawa, Twisted wild character varieties, arXiv:1512.08091 [[the general case]]

An earlier analytic approach to these symplectic manifolds is in P.Boalch, Symplectic Manifolds and Isomonodromic Deformations, Adv. in Math. 163 (2001) 137-205, and the hyperkahler upgrade is in:

• P. Boalch and O. Biquard, Wild non-abelian Hodge theory on curves, Compos. Math. 140 (2004) no. 1, 179-204

A “simple as possible” description of several intrinsic approaches to Stokes data for general linear groups is in

• arXiv:1903.12612 (Topology of the Stokes phenomenon).

The introduction of this paper aims to give a good guide to the Stokes data literature.

Several explicit descriptions of some of the simplest examples is in arXiv:1501.00930 (Wild Character Varieties, points on the Riemann sphere and Calabi's examples), and reviews of several different directions related to this story are in arXiv:1305.6593, arXiv:1703.10376.

A lógica moderna nasce da tentativa de formalizar toda a matemática. O estudo das limitações apontadas pelos teoremas de incompletude de Gödel foram um grande catalisador da área, tendo influenciado muitos novos ramos de estudo, nos quais podemos incluir a própria ciência da computação. O desenvolvimento da computação e inteligência artificial, e suas aplicações em áreas críticas, exigem novos métodos formais (leia-se lógicas) capazes de lidar com o raciocínio rigoroso necessário à sua análise. Para dar resposta às necessidades práticas é fundamental encontrar equilíbrios entre expressividade das lógicas consideradas e a complexidade de decisão dos problemas associados às mesmas.

Na primeira parte deste mini-curso darei uma perspectiva geral sobre lógica focando na variedade de escolhas possíveis relativamente a três aspectos fundamentais: sintaxe (linguagem), cálculos (sistemas dedutivos) e semântica. Na segunda parte apresentarei exemplos de lógicas modais, e terminarei com alguns resultados de combinação de lógicas, cujo objectivo último é compreender e controlar, se possível, os mecanismos por detrás da emergência da complexidade computacional em tarefas de raciocínio formal/simbólico.

Com o uso generalizado de dispositivos inteligentes e ligados “na nuvem” e a cada vez maior preocupação com segurança é cada vez mais necessário protocolos criptográficos capazes de correr em qualquer tipo de hardware. A criptografia baseada em curvas elípticas, ao apresentar chaves mais pequenas para o mesmo nível de segurança, torna-se especialmente apetecível para o uso em que a memória ou a transferência de dados são comodidades preciosas.

Nesta palestra procura-se então dar uma ideia de alguns protocolos baseados em curvas elípticas e das estruturas subjacentes, começando pelo clássico Diffie-Helman sobre curvas elípticas até ao Sike, o protocolo proposto ao NIST no recente concurso para escolha da próxima geração de protocolos.

Na era dos “Grandes Dados” a tendência é analisar toda a informação disponível. Será que é mesmo necessário? Será que o esforço e tempo despendido para analisar esses “Grandes Dados” conduz a uma informação significativamente maior do que usar técnicas de amostragem?

Neste minicurso discute-se esta questão e introduzem-se conceitos básicos de amostragem em populações finitas.

As entidades gestoras de serviços de água necessitam de reinvestir massivamente na reabilitação das suas infraestruturas para garantirem o melhor serviço aos seus clientes, de forma permanente, sustentável e segura.

Extrair informação relevante dos dados disponíveis, adquiridos em contínuo, permite direcionar as principais decisões a curto e longo prazo e identificar riscos e vulnerabilidades para os poder solucionar antes que os clientes sejam impactados.

A lógica moderna nasce da tentativa de formalizar toda a matemática. O estudo das limitações apontadas pelos teoremas de incompletude de Gödel foram um grande catalisador da área, tendo influenciado muitos novos ramos de estudo, nos quais podemos incluir a própria ciência da computação. O desenvolvimento da computação e inteligência artificial, e suas aplicações em áreas críticas, exigem novos métodos formais (leia-se lógicas) capazes de lidar com o raciocínio rigoroso necessário à sua análise. Para dar resposta às necessidades práticas é fundamental encontrar equilíbrios entre expressividade das lógicas consideradas e a complexidade de decisão dos problemas associados às mesmas.

Na primeira parte deste mini-curso darei uma perspectiva geral sobre lógica focando na variedade de escolhas possíveis relativamente a três aspectos fundamentais: sintaxe (linguagem), cálculos (sistemas dedutivos) e semântica. Na segunda parte apresentarei exemplos de lógicas modais, e terminarei com alguns resultados de combinação de lógicas, cujo objectivo último é compreender e controlar, se possível, os mecanismos por detrás da emergência da complexidade computacional em tarefas de raciocínio formal/simbólico.

Um dos muitos casos de estudo na Mecânica dos Fluidos é o movimento da água quando parte do seu contorno está em contacto com a atmosfera, como no caso dos rios. Este é um problema complexo, mas as características da água e certas hipóteses sobre o seu movimento permitem simplificá-lo.

Falarei sobre como foi construído um modelo para o problema, como o analisei dum ponto de vista teórico, provando a existência de solução, e como efetuei aproximações numéricas do mesmo.

A modelação matemática é uma das formas mais eficientes de compreender fenómenos e prever acontecimentos futuros. Muitos desses modelos são formulados com recurso a Equações com Derivadas Parciais (EDPs), ferramentas matemáticas que capturam as alterações de certas quantidades sujeitas a leis como difusão e reação em função de variáveis contínuas (tempo, espaço, preço,…). Falaremos de passeio aleatório, transporte, opções, calor, superfícies mínimas, vibrações e tráfego automóvel, abordando e deduzindo algumas das EDPs mais famosas da Física, Biologia e Economia.

O mini-curso é acessível a alunos que tenham frequentado uma disciplina de cálculo ou análise em várias variáveis.

Dynamical systems are often subjected to noise perturbations either from external sources or from their own intrinsic uncertainties. While it is well believed that noises can have dramatic effects on the stability of a deterministic system at both local and global levels, mechanisms behind noise surviving or robust dynamics have not been well understood especially from distribution perspectives. This talk attempts to outline a mathematical theory for making a fundamental understanding of these mechanisms in white noise perturbed systems of ordinary differential equations, based on the study of stationary measures of the corresponding Fokker-Planck equations. New existence and non-existence results of stationary measures will be presented by relaxing the notion of Lyapunov functions. Limiting behaviors of stationary measures as noises vanish will be discussed in connection to important issues such as stochastic stability and bifurcations.

Two-dimensional CFTs have an infinite set of commuting conserved charges, known as the quantum KdV charges, built out of the stress tensor. We compute the thermal correlation functions of the these KdV charges on a circle. We show that these correlation functions are given by quasi-modular differential operators acting on the torus partition function.