Discrete harmonic analysis seminar - past schedule for the year 2007/2008:
Current page of the seminar is available here.
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Discrete harmonic analysis seminar - past schedule for the year 2007/2008:Current page of the seminar is available here. |
The first part exhibits some well known examples of intertwining of commutative Markov semigroups. The second part gives a group Theoretical point of view of interwining together with an illustration using the Heisenberg group.
Kerov introduced a class of polynomials which, when evaluated at free cumulants, yield (normalizations of) irreducible symmetric group characters. More recently, Corteel, Goupil, and Schaeffer have considered ćontent evaluation" of symmetric functions. For a distinguished class of symmetric functions, content evaluation also yields symmetric group characters. Thus there is an a priori connection between free cumulants and content evaluation. We will explore this connection. We also present connections with symmetric functions of Jucys-Murphy elements, and with the Laurent expansion of the unitary "Weingarten function" of Collins and Sniady. In particular we give an explicit expression for all coefficients in this Laurent expansion in terms of contents and characters.
The optimal control problem of the time evolution of quantum spin of Pauli two-level system subjected to an external field with the minimum energy function will be illustrated and formulated in terms of the quantum spin up and spin down states of the Pauli two-level system.
Financial mathematics may be regarded as the branch of the applied mathematics which are concerned with the financial markets. Generally, financial mathematics derives, and extends the mathematical models to describe the dynamics of stock prices in the financial market and uses stochastic calculus to obtain the fair price of the derivative of the stock. In terms of practice, financial mathematics also overlaps heavily with the field of financial engineering and computational finance. Arguably, all three are largely synonymous, although the latter two focus on applications, while the former focuses on modelling of stock prices and option pricing. In this talk, we will go through four subjects below which might have you an insight into the theory of stock price models and the theory of option pricing. We will first discuss some of stochastic processes which are suitable for an adequate description of the dynamics of underlying assets such as stocks. We will next discuss the concepts of arbitrage-free pricing and risk neutral valuation, which are indeed the fundamental concepts of the theory of option pricing. We finally introduce the MTS distributions and processes and use them to develop the GARCH option pricing model with the MTS innovations. The following is the structure of this talk : I. Stock-price models : 1. Binomial model : log returns can be modeled by a random walk - Cox-Ross- Rubinstein model 2. Diffusion models : log returns can be modeled by a Wiener process - Black- Scholes model 3. Pure jump models : log returns can be modeled by a Levy process - Levy models such as VG model , CGMY model II. Asset pricing models : 1. Absolute pricing models - Equilibrium pricing models - CAPM, 2. Relative pricing models - Arbitrage-free pricing - Black-Scholes model III. Option pricing: 1. No-arbitrage argument approach - Black-Scholes PDE 2. Risk-neutral valuation approach - Risk-neutral measure - Equivalent martingale measure IV. The MTS - GARCH option pricing model
Let G=(V,E) be a connected, locally finite graph with a fixed origin o\in V, and A the adjacency matrix. We consider continuous-time quantum walks on G, which are classical stochastic processes defined by P(X(t)=x)=|\langle \delta_x, e^{itA}\delta_o\rangle|^2, x\in V, P(Y(t)=n)=\sum_{x\in V_n}|\langle \delta_x, e^{itA}\delta_o\rangle|^2, \quad n=0,1,2,\dots, respectively, where V=\bigcup_{n=0}^\infty V_n is the stratification. It is interesting to study asymptotic behavior of Y(t) when the graph G grows. We derive a limit theorem by means of the asymptotic spectral analysis of growing graphs with quantum probabilistic techniques.
1. The existence of phase transitions in quantum crystals. A translation invariant system of interacting quantum anharmonic oscillators indexes by the elements of a simple cubic lattice Z^d is considered. For the translation and rotation invariant models on the lattice Z^d (d>=3) with the nearest neighbour interactions the existence of phase transitions has been proven. In the scalar case, for the translation invariant models with the nearest neighbour interactions on the lattice Z^d (d>=3) a weaker result - the existence of phase transitions of the first order (due to Landau classification) has been obtained. The proofs are based on the representation of local Gibbs states in terms of path measures and thereby on the use of the infrared estimates and the Garsia-Rodemich-Rumsey inequality. Keywords: phase transition, quantum anharmonic crystal, Euclidean approach. 2. Decay of correlations in quantum models. The decay of the two-point correlation function of N-component ferromagnetic quantum model on the lattice Z^d (with arbitrary d) is shown to be the same as the decay of the pair-interaction potential J_ll', if the temperature is above the transition point T_C(1) of the corresponding one-component model. Keywords: quantum model, decay of correlations.
Statistics of vertex sets on quasi bounded graphs is considered. It is expressed by sums of weight functions over compatible collections of connected sets called animals. We present conditions that have to be satisfied to obtain estimates and prove convergence for unbounded, irregular graphs. The idea of the proof was initiated by L. R. Dobrushin and A. Bovier.
It is well known that, among all Levy processes, essentially only Brownian motion and Poisson process have a chaos decomposition property, which leads to a unitary isomorphism between the L^2 space of the process and the symmetric Fock space. In the case of a general Levy process, one may either use multiple stochastic integrals in orthogonalized power jump processes, or use an expansion in orthogonal polynomials of white noise. We will compare both approaches and derive a special class of Levy processes for which the corresponding infinite-dimensional polynomials have a nice structure. This class will be an infinite-dimensional counterpart of the Meixner class of orthogonal polynomials on the real line. Furthermore, we will discuss an extension of these results to the case of free Levy processes.
It is well known that, among all Levy processes, essentially only Brownian motion and Poisson process have a chaos decomposition property, which leads to a unitary isomorphism between the L^2 space of the process and the symmetric Fock space. In the case of a general Levy process, one may either use multiple stochastic integrals in orthogonalized power jump processes, or use an expansion in orthogonal polynomials of white noise. We will compare both approaches and derive a special class of Levy processes for which the corresponding infinite-dimensional polynomials have a nice structure. This class will be an infinite-dimensional counterpart of the Meixner class of orthogonal polynomials on the real line. Furthermore, we will discuss an extension of these results to the case of free Levy processes.
We study derivatives of functions on the set of generalized Young diagrams with respect to various set of parameters (such as: shape of a Young diagram, free cumulants of a Young diagram, other kinds of cumulants). In this way we are able to find explicit formulas for characters expressed in terms of free cumulants (Kerov polynomials) and other kinds of cumulants. One of our main tools is the Stanley-F\'eray character formula.
Guest talk on Differential Geometry and Topology Seminar. Irreducible representations of the symmetric groups S_n are indexed by Young characters. There are many ways of parametrizing Young diagrams, but one of them surpasses all the other by its beauty: the one which uses "free cumulants" (also appearing in the random matrix theory and Voiculescu's free probability). Kerov character polynomials express the characters of the symmetric groups in terms of the free cumulants; they have a surprisingly rich structure and there are many open conjectures concerning them. During my talk I will concentrate on their conjectured connection with the dimensions of some (co)homologies of some mysterious objects (Schubert varieties?). The talk is intended to be non-technical.
The talk will concern structures related to the subordination property for free additive and multiplicative convolutions. This includes the s-free product of Hilbert spaces, s-free convolutions, s-free independence and the s-free product of graphs.
We shall discuss the notion of bm-independence and show the general construction of bm-product of algebras. Then we shall exhibit the related construction for graphs - bm-product of graphs - which generalizes the comb product and the markovian product of graphs. Next we will formulate the general form of the bm-Central limit Theorem associated with positive symmetric cones, and explain the proof of it. Finally, we shall present an example of the Donsker's invariance principle for bm-independence.
The main purpose of this talk is to derive a white noise calculus for the negative binomial process, studying in details an associated family of field operators. In particular, by using higher powers of negative binomial Jacobi fields, we give an identification of the square of white noise algebra and a representation of the Virasoro algebra.
We show how to extend the classical theorem of Pitman on Brownian motion to higher dimensions.
We introduce Kerov polynomials and describe some of their properties.
In the case of weights (infinite states) GNS construction is a little more complicated. For normalised states we deal with map A\ni a\mapsto a\Omega\in H, where \Omega is a cyclic vector. For weights this mapping is replaced by so called GNS map that should be treated as unbounded densely defined closed operator acting from the algebra into Hilbert space. The application of this concept to Tomita-Takesaki theory and to locally compact quantum groups will be discussed.
We present a class of stationary quantum stochastic processes with (free, conditionally free, monotonically independent) increments, whose conditional moments satisfy of Laha-Lukacs formulas. They all have classical version Markov processes. The results are obtained together with W. Bryc and N. Demni.
In 2000 Carnovale and Koornwinder [G. Carnovale, T.H. Koornwinder, A q-analogue of convolution on the line, Methods Appl. Anal. 7 (2000), 705-726] defined a q-convolution. They proved that for some classes of measures it is associative and commutative. No positivity-preserving properties were disscuced. That question was posed by M. Bozejko on the conference in Bedlewo 2007. We anwser it partially. The notions of q-positivity and q-moments were studied in [A. Kula, A q-analogue of complete monotonicity, Colloq. Math. 111 (2008), 169-181]. In the talk we start from the algebraic interpretation of q-positivity and this leads us to a definition of (p,q)-convolution. It has a form similar to the q-convolution of Carnovale and Koornwinder. As for the properties, it is associative, commutative and symmetric with p and p^{-1}. Moreover, a (kind of) positivity-preserving property follows immediately form the algebraic construction. For the new convolution we find an appropriate analogue of Fourier transform and also present a central limt theorem. The above is a joint work with Eric Ricard (UFR, Besancon).
We present the contruction of posiive definite functions on some class of discrete groups (Coxeter, free product groups), which is clasical Abelian groups is Riesz product function. We also present applications to non-commutative Khinchine inequality and Boolean and Coxeter probability . Connections with completely bounded multipliers (Schur multipiers) on groups also will done.
The talk gives short survey of some recent results connecting random matrices, non-colliding processes and queues in series. In particular, our main goal will be to prove the following result discovered by Baryshinkov (2001) and Gravner, Tracy and Widom (2001). Let B(t)=(B_1(t),..., B_n(t)) be a standard n-dimensional Brownian motion. Then the random variable: M_n=sup_{0 < s_1 < ... < s_{n-1} < 1} sum_{i=0}^{n-1} (B_i(s_{i+1})-B_i(s_i)) has the same law as the largest eigenvalue of an n-dimensional GUE random matrix. What is even more surprising another approach is through a celebrated Burke theorem, which will be also given. Finally, I hope to mention few very new results and related open problems.
We exhibit some properties of matrix-valued stochastic processes and their corresponding eigenvalues. A deep insight to the latter makes use of the so-called radial Dunkl processes with classical reduced root systems. This allows to recover and answer probabilistic questions using determinantal representation of some special functions on some complex Lie algebras. The last part will be concerned with construction via matrix theory of some free processes.
The discrete method for constructing the copulas in R^n will be presented. Next the non-discrete meaning for the previous "deus ex machina" will be dredged up.
We show the method for constructing positive definite functions on 2-partitions using positive defined functions on S_{\infty}.
We show the method for constructing positive defined functions on 2-partitions using positive defined functions on S_{\infty}.
The aim of this talk is to show how to carry out the join construction of compact quantum groups avoiding braiding and replacing the unit interval by an arbitrary unital C*-algebra (noncommutative compact Hausdorff space). This is done in terms of equivariantly projective Hopf-Galois extensions of C*-algebras. The completion of the extended algebra is a natural candidate for a non-crossed product example of a principal extension of C*-algebras in the sense of Ellwood (non-trivial noncommutative principal bundle). The main point is a general and explicit formula for a strong connection, which puts us directly into the framework of the index pairing between K-theory and K-homology. (Based on a joint work with L. Dąbrowski and T. Hadfield.)
This is an introductory talk on history and current state of the theory of approximation.