Seiberg-Witten theory maps supersymmetric four-dimensional gauge theories with extended supersymmetry to algebraic completely integrable systems. For large class of such integrable systems the phase space is the moduli space of solutions of self-dual hyperKahler equations and their low-dimensional descendants. In particular, the list of such integrable systems includes Hitchin systems defined on Riemann surfaces with singularities at marked points (two-dimensional PDE), monopoles on circle bundles over surfaces (three-dimensional PDE or circle-valued Hitchin system) and instantons on torically fibered hyperKahler manifolds (four-dimensional PDE or elliptically valued Hitchin system). Deformations of four-dimensional gauge theory by curved backgrounds correspond to the quantization of the associated algebraic integrable systems. Quantization of Hitchin systems has relation to geometric Langlands correspondence and to the Toda two-dimensional conformal theory with Wg-algebra symmetry. Quantization of g-monopole and g-instanton moduli spaces relates to the representation theory of Drinfeld-Jimbo quantum affine algebras (and their rational and elliptic versions, Yangians and elliptic groups), associated respectively to g in the monopole case (circle-valued Hitchin) and to the central extension of the loop algebra of g in the instanton case (elliptically valued Hitchin). It is expected that there exists an analogue of geometric Langlands correspondence for quantization of the monopole and instanton algebraic integrable system (circle-valued and elliptically-valued Hitchin).

Seiberg-Witten theory maps supersymmetric four-dimensional gauge theories with extended supersymmetry to algebraic completely integrable systems. For large class of such integrable systems the phase space is the moduli space of solutions of self-dual hyperKahler equations and their low-dimensional descendants. In particular, the list of such integrable systems includes Hitchin systems defined on Riemann surfaces with singularities at marked points (two-dimensional PDE), monopoles on circle bundles over surfaces (three-dimensional PDE or circle-valued Hitchin system) and instantons on torically fibered hyperKahler manifolds (four-dimensional PDE or elliptically valued Hitchin system). Deformations of four-dimensional gauge theory by curved backgrounds correspond to the quantization of the associated algebraic integrable systems. Quantization of Hitchin systems has relation to geometric Langlands correspondence and to the Toda two-dimensional conformal theory with Wg-algebra symmetry. Quantization of g-monopole and g-instanton moduli spaces relates to the representation theory of Drinfeld-Jimbo quantum affine algebras (and their rational and elliptic versions, Yangians and elliptic groups), associated respectively to g in the monopole case (circle-valued Hitchin) and to the central extension of the loop algebra of g in the instanton case (elliptically valued Hitchin). It is expected that there exists an analogue of geometric Langlands correspondence for quantization of the monopole and instanton algebraic integrable system (circle-valued and elliptically-valued Hitchin).

I plan six lectures on possible directions of modification/generalization of the probability theory, both concerning mathematical foundations and applications within and without pure mathematics. Specifically, I will address two issues.

 

1. Enhancement of stochastic symmetry by linearization and Hilbertization of set-theoretic categories.

2. Non-symmetric probability theory in heterogeneous environments of molecular biology and of linguistics.

 
I will start with a category theoretic view on probability and entropy. This includes

representation of Lebesgue spaces as covariant functors from the category WF of weighted finite sets F into the category of set;

definition of the Boltzmann-Shannon entropy of an F as an image of F in the topological Grothendieck semigroup of WF.

(Much of this can be found in my article In a Search for a Structure, Part 1: On Entropy).

 
Next I will present several linearized measure-like structures, and associated entropies, such as the homology measures and their appearance in many particle systems.
(Some of it is presented in Singularities, expanders and topology of map. Part 2: From combinatorics to topology via algebraic isoperimetry. GAFA, Geom. func. anal., 20 (2010), 416-526, and in Geometry, Topology and Spectra of Non-Linear Spaces of Maps – Wolfgang Pauli Lectures, May 25, 2009.)

 
Also I say something about the von Neumann entropy.

 
Finally I will dicuss possible concepts of probability for heterogeneous systems, such as natural languages and their applications to automatic signals/texts analysis in the spirit of my article Ergostructures, Ergodic and the Universal Learning Problem: Chapters 1, 2.

 
(The above mentioned papers can be found on my web page at IHES in the section "recent").

I plan six lectures on possible directions of modification/generalization of the probability theory, both concerning mathematical foundations and applications within and without pure mathematics. Specifically, I will address two issues.

 

1. Enhancement of stochastic symmetry by linearization and Hilbertization of set-theoretic categories.

2. Non-symmetric probability theory in heterogeneous environments of molecular biology and of linguistics.

 
I will start with a category theoretic view on probability and entropy. This includes

representation of Lebesgue spaces as covariant functors from the category WF of weighted finite sets F into the category of set;

definition of the Boltzmann-Shannon entropy of an F as an image of F in the topological Grothendieck semigroup of WF.

(Much of this can be found in my article In a Search for a Structure, Part 1: On Entropy).

 
Next I will present several linearized measure-like structures, and associated entropies, such as the homology measures and their appearance in many particle systems.
(Some of it is presented in Singularities, expanders and topology of map. Part 2: From combinatorics to topology via algebraic isoperimetry. GAFA, Geom. func. anal., 20 (2010), 416-526, and in Geometry, Topology and Spectra of Non-Linear Spaces of Maps – Wolfgang Pauli Lectures, May 25, 2009.)

 
Also I say something about the von Neumann entropy.

 
Finally I will dicuss possible concepts of probability for heterogeneous systems, such as natural languages and their applications to automatic signals/texts analysis in the spirit of my article Ergostructures, Ergodic and the Universal Learning Problem: Chapters 1, 2.

 
(The above mentioned papers can be found on my web page at IHES in the section "recent").

I plan six lectures on possible directions of modification/generalization of the probability theory, both concerning mathematical foundations and applications within and without pure mathematics. Specifically, I will address two issues.

 

1. Enhancement of stochastic symmetry by linearization and Hilbertization of set-theoretic categories.

2. Non-symmetric probability theory in heterogeneous environments of molecular biology and of linguistics.

 
I will start with a category theoretic view on probability and entropy. This includes

representation of Lebesgue spaces as covariant functors from the category WF of weighted finite sets F into the category of set;

definition of the Boltzmann-Shannon entropy of an F as an image of F in the topological Grothendieck semigroup of WF.

(Much of this can be found in my article In a Search for a Structure, Part 1: On Entropy).

 
Next I will present several linearized measure-like structures, and associated entropies, such as the homology measures and their appearance in many particle systems.
(Some of it is presented in Singularities, expanders and topology of map. Part 2: From combinatorics to topology via algebraic isoperimetry. GAFA, Geom. func. anal., 20 (2010), 416-526, and in Geometry, Topology and Spectra of Non-Linear Spaces of Maps – Wolfgang Pauli Lectures, May 25, 2009.)

 
Also I say something about the von Neumann entropy.

 
Finally I will dicuss possible concepts of probability for heterogeneous systems, such as natural languages and their applications to automatic signals/texts analysis in the spirit of my article Ergostructures, Ergodic and the Universal Learning Problem: Chapters 1, 2.

 
(The above mentioned papers can be found on my web page at IHES in the section "recent").

I plan six lectures on possible directions of modification/generalization of the probability theory, both concerning mathematical foundations and applications within and without pure mathematics. Specifically, I will address two issues.

 

1. Enhancement of stochastic symmetry by linearization and Hilbertization of set-theoretic categories.

2. Non-symmetric probability theory in heterogeneous environments of molecular biology and of linguistics.

 
I will start with a category theoretic view on probability and entropy. This includes

representation of Lebesgue spaces as covariant functors from the category WF of weighted finite sets F into the category of set;

definition of the Boltzmann-Shannon entropy of an F as an image of F in the topological Grothendieck semigroup of WF.

(Much of this can be found in my article In a Search for a Structure, Part 1: On Entropy).

 
Next I will present several linearized measure-like structures, and associated entropies, such as the homology measures and their appearance in many particle systems.
(Some of it is presented in Singularities, expanders and topology of map. Part 2: From combinatorics to topology via algebraic isoperimetry. GAFA, Geom. func. anal., 20 (2010), 416-526, and in Geometry, Topology and Spectra of Non-Linear Spaces of Maps – Wolfgang Pauli Lectures, May 25, 2009.)

 
Also I say something about the von Neumann entropy.

 
Finally I will dicuss possible concepts of probability for heterogeneous systems, such as natural languages and their applications to automatic signals/texts analysis in the spirit of my article Ergostructures, Ergodic and the Universal Learning Problem: Chapters 1, 2.

 
(The above mentioned papers can be found on my web page at IHES in the section "recent").

I plan six lectures on possible directions of modification/generalization of the probability theory, both concerning mathematical foundations and applications within and without pure mathematics. Specifically, I will address two issues.

 

1. Enhancement of stochastic symmetry by linearization and Hilbertization of set-theoretic categories.

2. Non-symmetric probability theory in heterogeneous environments of molecular biology and of linguistics.

 
I will start with a category theoretic view on probability and entropy. This includes

representation of Lebesgue spaces as covariant functors from the category WF of weighted finite sets F into the category of set;

definition of the Boltzmann-Shannon entropy of an F as an image of F in the topological Grothendieck semigroup of WF.

(Much of this can be found in my article In a Search for a Structure, Part 1: On Entropy).

 
Next I will present several linearized measure-like structures, and associated entropies, such as the homology measures and their appearance in many particle systems.
(Some of it is presented in Singularities, expanders and topology of map. Part 2: From combinatorics to topology via algebraic isoperimetry. GAFA, Geom. func. anal., 20 (2010), 416-526, and in Geometry, Topology and Spectra of Non-Linear Spaces of Maps – Wolfgang Pauli Lectures, May 25, 2009.)

 
Also I say something about the von Neumann entropy.

 
Finally I will dicuss possible concepts of probability for heterogeneous systems, such as natural languages and their applications to automatic signals/texts analysis in the spirit of my article Ergostructures, Ergodic and the Universal Learning Problem: Chapters 1, 2.

 
(The above mentioned papers can be found on my web page at IHES in the section "recent").

I plan six lectures on possible directions of modification/generalization of the probability theory, both concerning mathematical foundations and applications within and without pure mathematics. Specifically, I will address two issues.

 

1. Enhancement of stochastic symmetry by linearization and Hilbertization of set-theoretic categories.

2. Non-symmetric probability theory in heterogeneous environments of molecular biology and of linguistics.

 
I will start with a category theoretic view on probability and entropy. This includes

representation of Lebesgue spaces as covariant functors from the category WF of weighted finite sets F into the category of set;

definition of the Boltzmann-Shannon entropy of an F as an image of F in the topological Grothendieck semigroup of WF.

(Much of this can be found in my article In a Search for a Structure, Part 1: On Entropy).

 
Next I will present several linearized measure-like structures, and associated entropies, such as the homology measures and their appearance in many particle systems.
(Some of it is presented in Singularities, expanders and topology of map. Part 2: From combinatorics to topology via algebraic isoperimetry. GAFA, Geom. func. anal., 20 (2010), 416-526, and in Geometry, Topology and Spectra of Non-Linear Spaces of Maps – Wolfgang Pauli Lectures, May 25, 2009.)

 
Also I say something about the von Neumann entropy.

 
Finally I will dicuss possible concepts of probability for heterogeneous systems, such as natural languages and their applications to automatic signals/texts analysis in the spirit of my article Ergostructures, Ergodic and the Universal Learning Problem: Chapters 1, 2.

 
(The above mentioned papers can be found on my web page at IHES in the section "recent").

I review the relationship between supersymmetric gauge theories and quantum integrable systems. From the quantum integrability side this relation includes various spin chains, as well as many well-known quantum many body systems like elliptic Calogero-Moser system and generalisations. From the gauge theory side one has supersymmetric gauge theories with four (and eight) supercharges in various space-time dimensions (compactified to two-dimensions, or in Omega-background). Gauge theory perspective provides the exact energy spectrum of corresponding quantum integrable system. Key elements, usually appearing in the topic of quantum integrability, such as Baxter equation, Yang-Yang function, Bethe equation, spectral curve, Yangian, quantum affine algebra, quantum elliptic algebra – all acquire meaning in the supersymmetric gauge theory.

I review the relationship between supersymmetric gauge theories and quantum integrable systems. From the quantum integrability side this relation includes various spin chains, as well as many well-known quantum many body systems like elliptic Calogero-Moser system and generalisations. From the gauge theory side one has supersymmetric gauge theories with four (and eight) supercharges in various space-time dimensions (compactified to two-dimensions, or in Omega-background). Gauge theory perspective provides the exact energy spectrum of corresponding quantum integrable system. Key elements, usually appearing in the topic of quantum integrability, such as Baxter equation, Yang-Yang function, Bethe equation, spectral curve, Yangian, quantum affine algebra, quantum elliptic algebra – all acquire meaning in the supersymmetric gauge theory.

I review the relationship between supersymmetric gauge theories and quantum integrable systems. From the quantum integrability side this relation includes various spin chains, as well as many well-known quantum many body systems like elliptic Calogero-Moser system and generalisations. From the gauge theory side one has supersymmetric gauge theories with four (and eight) supercharges in various space-time dimensions (compactified to two-dimensions, or in Omega-background). Gauge theory perspective provides the exact energy spectrum of corresponding quantum integrable system. Key elements, usually appearing in the topic of quantum integrability, such as Baxter equation, Yang-Yang function, Bethe equation, spectral curve, Yangian, quantum affine algebra, quantum elliptic algebra – all acquire meaning in the supersymmetric gauge theory.

I will start with an aspect of mathematics that is well understood that is the Mendelian dynamics in the spaces of alleles. (This is described in Mendelian Dynamics and Sturtevant’s Paradigm in the « recent » section on my website.)
Also I touch upon in this context on the categorical view on the entropy in dynamics as in In a Search for a Structure, Part 1: On Entropy, also in the « recent » section).
(2-3 lectures)

Then I will elaborate on the Poincaré-Sturtevant idea of describing geometries of spaces X by samples of probability measures on the set subsets of X, where Poincaré had in mind the reconstruction of the Euclidean geometry by the Brain and Sturtevant used it to make a genomic map of a chromosome of drosophila.
(1 lecture)

Also I dedicate a lecture to mathematical problems related to the structure and functions of proteins.
I conclude by speculations on further possible mathematical « unfoldings » of messages conveyed by molecular ­