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Tbilisi Analysis & PDE Seminar

Past seminars

### Past Seminars

Seminar on June 6, 2022

Speaker: Prof. Björn Birnir, CNLS and the University of California at Santa Barbara; https://birnir.math.ucsb.edu/

The title of the lecture: “The statistical theory of Stochastic Nonlinear Partial Differential Equations with application to the angiogenesis equations”

Abstract: We develop the statistical theory for stochastic nonlinear PDEs with both additive and multiplicative noise. The canonical example is the stochastic Navier-Stokes equation. We solve the Kolmogorov-Hopf equation for the invariant measure determining the statistical quantities. Then the theory is applied to the stochastic angiogenesis equations describing how veins grow through the human body.

Seminar on May 23, 2022

Speaker: Prof. Oleksiy Karlovych, NOVA University Lisbon, Portugal; https://docentes.fct.unl.pt/oyk/

The title of the lecture: “Algebras of convolution type operators with continuous data do not always contain all rank one operators”

Abstract: See poster

Seminar on May 9, 2022

Speaker: Prof. Volker Mehrmann, TU Berlin, Germany; https://www.bimos.tu-berlin.de/menue/bimos_people/members/professors/volker_mehrmann/https://en.wikipedia.org/wiki/Volker_Mehrmann

The title of the lecture: “Modeling analysis and numerical simulation of multi-physical systems: A change of paradigm”

Abstract: Most real-world dynamical systems consist of subsystems from different physical domains, modeled by partial-differential equations, ordinary differential equations, and algebraic equations, combined with input and output connections. To deal with such complex systems, in recent years the class of dissipative port-Hamiltonian (pH) systems have emerged as a very efficient new modeling methodology. The main reasons are that the network-based interconnection of pH systems is again pH, Galerkin projection in PDE discretization and model reduction preserves the pH structure and the physical properties are encoded in the geometric properties of the flow as well as the algebraic properties of the equations. Furthermore, dissipative pH system form a very robust representation under structured perturbations and directly indicates Lyapunov functions for stability analysis. We discuss dissipative pH systems and describe, how many classical models can be formulated in this class. We illustrate some of the nice algebraic properties, including local canonical forms, the formulation of an associated Dirac structure, and the local invariance under space-time dependent diffeomorphisms. The results are illustrated with some real-world examples.

Seminar on April 25, 2022

Speaker: Prof. Natasha Samko, The Arctic University of Norway, Narvik, Norway; http://www.nsamko.com/

The title of the lecture: “Weighted boundedness of certain sublinear operators in generalized Morrey spaces on quasi-metric measure spaces under the growth condition”

Abstract: We prove the weighted boundedness of Calderón-Zygmund and maximal singular operators in generalized Morrey spaces on quasi-metric measure spaces, in general non-homogeneous, only under the growth condition on the measure, for a certain class of weights. The weights and characteristics of the spaces are independent of each other. The weighted boundedness of the maximal operator is also proved in the case when lower and upper Ahlfors exponents coincide with each other. Our approach is based on two important steps. The first is a certain transference theorem, where without using homogeneity of the space, we provide a condition which ensures that every sublinear operator with the size condition, bounded in Lebesgue space, is also bounded in generalized Morrey space. The second is a reduction theorem which reduces the weighted boundedness of the considered sublinear operators to that of weighted Hardy operators and the non-weighted boundedness of some special operators.

Seminar on April 11, 2022

Speaker: Professor Alexander Pushnitski, King’s College London, UK; https://www.kcl.ac.uk/people/alexander-pushnitski

The title of the lecture: “The spectra of some arithmetical matrices”

Abstract: I will discuss the spectral theory of a family of infinite arithmetical matrices, whose (n,m)-th entry involves the least common multiple of n and m, denoted LCM(n,m). The simplest example of such a matrix is {1/LCM(n,m)}, where n,m range over natural numbers. It turns out that an explicit formula for the asymptotics of eigenvalues of this matrix can be given. This is recent joint work with Titus Hilberdink (Reading).

Seminar on March 28, 2022

Speaker: Doctoral student Duvan Cardona Sanchez, Ghent University; https://sites.google.com/site/duvancardonas/

The title of the lecture: “Oscillating Fourier multipliers theory: geometric aspects and the role of the symmetries”

Abstract: Oscillating Fourier multipliers on the torus and on Rⁿ play a fundamental role in analysis and PDE and in the setting of Lie groups are still a subject of intensive research. The classical results by Fefferman and Stein in this direction (published between 1970 and 1972 in Acta Math.) have consolidated a fundamental theory for the harmonic analysis of these operators, even, in a more general setting that contains Calderón-Zygmund singular integrals of convolution type. In this talk, we present some recent results that extend the Fefferman and Stein theory of oscillating Fourier multipliers to arbitrary Lie groups of polynomial growth.

Seminar on March 14, 2022

Speaker: Prof. Hans Georg Feichtinger, University of Vienna;

https://www.univie.ac.at/nuhag-php/home/fei.php

The title of the lecture: “The Banach Gelfand Triple and its role in Classical Fourier Analysis and Operator Theory”

Abstract: The Banach Gelfand Triple (S0, L2, S0*)(Rd) (which arose in the context of Time-Frequency Analysis) is a simple and useful tool, both for the derivation of mathematically valid theorems AND for teaching relevant concepts to engineers and physicists (and of course mathematicians, interested in applications!).

In this context, the basic terms of an introductory course on Linear System’s Theory can be explained properly: Translation invariant systems are viewed as linear operators, which can be described as convolution operator by some impulse response, whose Fourier transform is well defined (and is called transfer function), and there is a kernel theorem:

Operators T : S0(Rd) to S0*(Rd) have a "matrix representation" using some sigma in S0*(R2d). Most importantly, dual space S0 ∗ (Rd), the space of so-called mild distributions, contains all kinds of objects relevant for signal processing: periodic signals, discrete signals, and of course discrete and periodic signals. One can show that the generalized Fourier transform for such functions works well and reduced to the DFT/FFT (Fast Fourier Transform).

An important tool is the STFT (Short-Time Fourier Transform). Mild distributions are exactly those tempered distributions which have a bounded short-time Fourier transform, and the w∗-convergence just corresponds to uniform convergence of the STFT over compact subsets of the time-frequency plane.

Slides: Click to see

Seminar on February 28, 2022

Speaker: Prof. George Tephnadze, The University of Georgia;

The title of the lecture: “Almost everywhere convergence of partial sums of trigonometric and Vilenkin systems and certain summability methods”

Abstract: The classical theory of the Fourier series deals with the decomposition of a function into sinusoidal waves. Unlike these continuous waves, the Vilenkin (Walsh) functions are rectangular waves. There are many similarities between these theories, but there exist differences also. Much of these can be explained by modern abstract harmonic analysis, combined with martingale theory.

This talk is devoted to investigating tools that are used to study almost everywhere convergence of the partial sums of trigonometric and Vilenkin systems. In particular, these methods combined with martingale theory helps to give a simpler proof of an analogy of the famous Carleson-Hunt theorem for Fourier series with respect to the Vilenkin system. We also define an analogy of Lebesgue points for integrable functions and we will describe which certain summability methods are convergent in these points.

Seminar on February 14, 2022

Speaker: Prof. Paata Ivanisvili, University of California, USA;

The title of the lecture: “Convex hull of a space curve”

Abstract: Finding a simple description of a convex hull of a set K in n-dimensional Euclidean space is a basic problem in mathematics. When K has some additional geometric structures one may hope to give an explicit construction of its convex hull. A good starting point is when K is a space curve. In this talk I will describe convex hulls of space curves which have a "very" positive torsion. In particular, we obtain parametric representation of the boundary of the convex hull, different formulas for their Euclidean volumes of the convex hull, the area of its boundary, and the solution to a general moment problem corresponding to such curves.

Seminar on January 31, 2022

Speaker: Prof. Heiko Gimperlein, University of Innsbruck and University of Parma;

https://www.macs.hw.ac.uk/~hg94/

The title of the lecture: “Boundary regularity for fractional Laplacians: a geometric approach”

Abstract: We consider the sharp boundary regularity of solutions to the Dirichlet problem for the fractional Laplacian on a smoothly bounded domain in Euclidean space. The fractional Laplacian is defined via the extension method, as a Dirichlet-to-Neumann operator for a degenerate elliptic problem in a half-space of one higher dimension. We use techniques from geometric microlocal analysis to analyse the regularity of solutions, with particular emphasis on asymptotic expansions and Hölder continuous data. Detailed and sharp results about this problem have been obtained by Gerd Grubb, and we present a complementary approach. Extensions to polygonal domains are mentioned.

Seminar on January 17, 2022

Speaker: Nikolai L. Vasilevski, Department of Mathematics, CINVESTAV, Mexico City, Mexico;

https://www.math.cinvestav.mx/~nvasilev/

The title of the lecture: “On analytic type function spaces and direct sum decomposition of $L_2(D, d\nu)$”

Seminar on December 20

Speaker: Prof. Dr. Alexander Mielke, Weierstrass Institute for Applied Analysis and Stochastics and Humboldt-Universität zu Berlin;

https://www.wias-berlin.de/people/mielke/

The title of the lecture: “On a rigorous derivation of a wave equation with fractional damping from a system with fluid-structure interaction”

Abstract: We consider a linear system that consists of a linear wave equation on a horizontal hypersurface and a parabolic equation in the half-space below. The model describes longitudinal elastic waves in organic monolayers at the water-air interface, which is an experimental setup that is relevant for understanding wave propagation in biological membranes. We study the scaling regime where the relevant horizontal length scale is much larger than the vertical length scale and provide a rigorous limit leading to a fractionally damped wave equation for the membrane. We provide the associated existence results via linear semigroup theory and show convergence of the solutions in the scaling limit. Moreover, based on the energy–dissipation structure for the full model, we derive natural energy and a natural dissipation function for the fractionally damped wave equation with a time derivative of order 3/2.

Seminar on December 06

Speaker: Sergey MIKHAILOV, Professor of Computational and Applied Mathematics, Dept. of Mathematical Sciences, Brunel University London, UK,

http://people.brunel.ac.uk/~mastssm/

The title of the lecture: “Volume and Layer Potentials for the Stokes System with Non-smooth Anisotropic Viscosity Tensor and Some Applications”

Seminar on November 22

Speaker: Prof. Vladimir Rabinovich, National Polytechnic Institute of Mexico, ESIME Zacatenco,

The title of the lecture: “Interaction problems for the Dirac operators on Rn

Recorded talk: N/A

Seminar on November 8

Speaker: Prof. Eugine Shargorodsky, King’s College London, UK,

https://www.kcl.ac.uk/people/eugene-shargorodsky

The title of the lecture: “Negative eigenvalues of two-dimensional Schrödinger operators”

Abstract: According to the celebrated Cwikel-Lieb-Rozenblum inequality, the number of negative eigenvalues of the Schrödinger operator , $-\Delta-V, V>=0$ on $L_2((R^d), d>=3$, is estimated above by $$const\int_{\R^d}V(x)^{d/2}dx$$ It is well known that this estimate does not hold for d=2. I will present estimates for the number of negative eigenvalues of a two-dimensional Schrödinger operator in terms of weighted $L_1$-norms and $LlogL$ type Orlicz norms of the potential obtained over the last decade and discuss related open problems.

Recorded talk: N/A

Seminar on October 25, 2021

Speaker: Prof. Grigori Rozenblioum, the Chalmers University of Technology and University of Gothenburg, Sweden; St. Petersburg State University and Leonhard Euler International Mathematical Institute in Saint Petersburg,

https://www.chalmers.se/en/staff/Pages/grigori-rozenblioum.aspx

The title of the lecture: “Spectral properties of the Neumann-Poincare operator for the elasticity system and related questions about zero-order pseudodifferential operators

Abstract: The Neumann-Poincare operator is the double layer potential. Unlike the well-studied electrostatic problem where this operator is compact, the elasticity system is not compact. For the 3D homogeneous isotropic body with smooth boundary, this operator has an essential spectrum consisting of 3 points determined by Lame parameters. The eigenvalues of this operator may converge only to these points. We developed the machinery of spectral analysis of polynomially compact pseudodifferential operators and use it to find the asymptotics of these eigenvalues.

Recorded talk N/A

Seminar on October 11, 2021

Dorina Mitrea, Professor and Chair Department of Mathematics,

https://www.baylor.edu/math/index.php?id=962935

The title of the lecture: “On boundedness of Singular Integral Operators on Holder Spaces”

Abstract:A central question in Calderon-Zygmund Theory is that of the L2-boundedness of Singular Integral Operators. An equally relevant issue is that of the boundedness of Singular Integral Operators on the scale of Holder spaces. In this talk I will present results in this regard which are applicable to large classes of Singular Integral Operators on general geometric settings.

Recorded talk N/A

Seminar on June 21

Dr. Marius Mitrea, professor of mathematics and chair of the mathematics department at Baylor University, Waco, Texas, United States;

https://www.baylor.edu/math/index.php?id=962939

The title of the lecture: “Singular Integrals, Geometry of Sets, and Boundary Problems”

Abstract:Presently, it is well understood what geometric features are necessary and sufficient to guarantee the boundedness of convolution-type singular integral operators on Lebesgue spaces. This being said, dealing with other function spaces where membership entails more than a mere size condition (like Sobolev spaces, Hardy spaces, or the John-Nirenberg space BMO) requires new techniques. In this talk I will explore recent progress in this regard, and follow up the implications of such advances into the realm of boundary value problems.

Recorded talk N/A

Seminar on June 7

Prof. Valery Smyshlyaev, University College London, UK;

The title of the lecture: “High-frequency scattering of whispering gallery waves by boundary inflection: asymptotics and boundary integral equations”

Abstract:The talk is on a long-standing problem of scattering of a high-frequency whispering gallery wave by boundary inflection. Like Airy ODE and associated Airy function are fundamental for describing transition from oscillatory to exponentially decaying asymptotic behaviors, the boundary inflection problem leads to an arguably equally fundamental canonical inner boundary-value problem for a special PDE describing transition from a modal'' to a scattered'' high-frequency asymptotic regimes. An additional recent motivation comes from the problem seemingly holding the keys for numerical analysis of Galerkin-type methods for boundary integral equations (BIE) in high-frequency scattering by smooth non-convex obstacles. The talk first reviews the background, on asymptotically reducing a problem described by Hemlholtz equation to the inner problem. The latter is a Schr\"odinger equation on a half-line with a potential linear in both space and time, and was first formulated and analysed by M.M. Popov starting from 1970-s, and has been intensively studied since then (see [1] for a review and some further references). The associated solutions have asymptotic behaviors with a discrete spectrum at one end and with a continuous spectrum at the other end, and of central interest is to find the map connecting the above two asymptotic regimes. We report recent result in [1] proving that the solution past the inflection point has a searchlight'' asymptotics corresponding to a beam concentrated near the limit ray. This is achieved by a non-standard perturbation analysis at the continuous spectrum end, and the result allows interpretations in terms of a unitary scattering operator connecting the modal and the scattered asymptotic regimes.We also review some most recent progress on a reducing the inner problem to one-dimensional boundary integral equations and their further analysis. The integral equations are of improper weakly singular Volterra type of both first and second kinds (with appropriate jump conditions for the latter) and can be shown to be well-posed. Their subsequent regularization allows to express the solution in term of limit of uniformly convergent Neumann series with anticipated further benefits for the problem's asymptotic and possibly numerical analyses. Some parts of the work are joint with Ilia Kamotski, and with Shiza Naqvi.

Recorded talk N/A

Seminar on May 24

Prof. Simon Chandler-Wilde, University of Reading, UK;

The title of the lecture: “Do Galerkin methods converge for the classical 2nd kind boundary integral equations in polyhedra and Lipschitz domains? ”

Abstract:The boundary integral equation method is a popular method for solving elliptic PDEs with constant coefficients, and systems of such PDEs, in bounded and unbounded domains. An attraction of the method is that it reduces solution of the PDE in the domain to solution of a boundary integral equation on the boundary of the domain, reducing the dimensionality of the problem. Second kind integral equations, featuring the double-layer potential operator, have a long history in analysis and numerical analysis. They provided, through C. Neumann, the first existence proof to the Laplace Dirichlet problem in 3D, have been an important analysis tool for PDEs through the 20th century, and are popular computationally because of their excellent conditioning and convergence properties for large classes of domains. A standard numerical method, in particular for boundary integral equations, is the Galerkin method, and the standard convergence analysis starts with a proof that the relevant operator is coercive, or a compact perturbation of a coercive operator, in the relevant function space. A long-standing open problem is whether this property holds for classical second kind boundary integral equations on general non-smooth domains. In this talk we give an overview of the various concepts and methods involved, reformulating the problem as a question about numerical ranges. We solve this open problem through counterexamples, presenting examples of 2D Lipschitz domains and 3D Lipschitz polyhedra for which coercivity does not hold. This is joint work with Prof Euan Spence, Bath.

Recorded talk N/A

Seminar on April 19

Prof. Gerd Grubb, Department of Mathematical Sciences, University of Copenhagen (UCPH);

https://www.math.ku.dk/english/staff/?pure=en/persons/64670

The title of the lecture: “Sharp regularity results for electromagnetic fields in Lipschitz domainsBoundary problems for fractional-order operators”

Abstract:There has recently been an upsurge of interest in the fractional Laplacian ${\left( -\Delta \right)}^{a}$ $\left( 0<a<1 \right)$ and other fractional-order pseudodifferential operators $P$, because of applications in financial theory and probability (and also in differential geometry and mathematical physics). The boundary problems for $P$ on subsets $\Omega$ of ${R}^{n}$ are challenging since $P$ is nonlocal; here there have mostly been used real methods from potential theory and integral operator theory, or probabilistic methods. As we know, the pseudodifferential point of view should be useful too, in view of Vishik and Eskin's work in the sixties, and many later works. I shall tell about a circle of results developed in the last 8 years, based on the mu-transmission condition introduced by Hörmander (in his book '85 and in a lecture note '66), telling also how they differ from the results in Eskin's book '81. The pseudodifferential methods have not been popular in the applications community, partly because Fourier transform techniques (and complex functions) do not seem to be part of the toolbox, partly because the ps.d.o. methods originally have

Recorded talk N/A

Seminar on March 22

Prof. Dr. Martin Costabel, IRMAR, Institut Mathématique Université de Rennes 1;;

https://perso.univ-rennes1.fr/martin.costabel/

The title of the lecture: “Sharp regularity results for electromagnetic fields in Lipschitz domains”

Abstract: The standard energy spaces for the time-harmonic Maxwell equations in a bounded 3-dimensional domain are the Hilbert spaces of square-integrable vector fields whose divergence and curl are also square-integrable and whose tangential or normal components are zero on the boundary. Elements of these spaces are known to have additional regularity. The classical Gaffney inequality (1951) states that their gradients are square-integrable if the domain is smooth, and it is clear that there is less regularity if the boundary has corners and edges. For Lipschitz domains, regularity in the Sobolev space of order 1/2 has been known for 30 years, and recently a domain has been constructed that shows that in the scale of Sobolev spaces this is sharp. By the principle that Maxwell singularities are carried by gradients, this regularity result can also be equivalently formulated as a regularity result for the classical Dirichlet and Neumann problems of the Laplacian on bounded Lipschitz domains. I will describe the context and motivation for the result and explain the construction of the domain and some of its additional properties.

Recorded talk N/A

Seminar on March 8

Speaker: Prof. Dr. Pavel Exner,Doppler Institute for Mathematical Physics and Applied Mathematics;

https://gemma.ujf.cas.cz/~exner/personal.html

The title of the lecture: “Product formulae related to Zeno quantum dynamics”

Abstract: We present a new class of product formulae which involve a unitary group generated by a positive self-adjoint operator and a continuous projection-valued function. The problem is motivated by quantum description of decaying systems, in particular, the Zeno effect coming from frequently repeated measurements. Applied to it, the formula expresses the dynamics of such a system. An example of a permanent position ascertaining leading to the effective Dirichlet condition is given.

Recorded talk N/A

Seminar on February 22

Speaker: Prof. Paata Ivanishvili,North Carolina State University (NC), USA;

The title of the lecture: “Rademacher type and Enflo Type Coincide”

Abstract: Pick any finite number of points in a Hilbert space. If they coincide with vertices of a parallelepiped then the sum of the squares of the lengths of its sides equals the sum of the squares of the lengths of the diagonals (parallelogram law). If the points are in a general position then we can define sides and diagonals by labeling these points via vertices of the discrete cube {0,1}n. In this case the sum of the squares of diagonals is bounded by the sum of the squares of its sides no matter how you label the points and what n you choose. In a general Banach space we do not have parallelogram law. Back in 1978 Enflo asked: in an arbitrary Banach space if the sum of the squares of diagonals is bounded by the sum of the squares of its sides for all parallelepipeds (up to a universal constant), does the same estimate hold for any finite number of points (not necessarily vertices of the parallelepiped)? In the joint work with Ramon van Handel and Sasha Volberg we positively resolve Enflo's problem. Banach spaces satisfying the inequality with parallelepipeds are called of type 2 (Rademacher type 2), and Banach spaces satisfying the inequality for all points are called of Enflo type 2. In particular, we show that Rademacher type and enflo type coincide.

Recorded talk N/A

Seminar on February 8

Speaker: Prof. Michael Ruzhansky,Queen Mary University of London;

https://ruzhansky.org/

The title of the lecture: “Nonharmonic operator analysis”

Abstract: In this talk we will give a survey of our recent works on developing the nonharmonic symbolic calculus. This has applications to various questions for non-self-adjoint operators as well as for operators on manifolds with (or without) boundaries.

Recorded talk N/A

Seminar on January 25

Speaker: Prof. Leonid Parnovski,Department of Mathematics, University College London, UK

http://www.homepages.ucl.ac.uk/~ucahlep/

The title of the lecture: “Floating mats and sloping beaches: spectral asymptotics of the Steklov problem on polygons”

Abstract: I will discuss asymptotic behaviour of the eigenvalues of the Steklov problem (aka Dirichlet-to-Neumann operator) on curvilinear polygons. The answer is completely unexpected and depends on the arithmetic properties of the angles of the polygon.

Recorded talk N/A

Seminar on January 11

Speaker: Prof. Maria J. ESTEBAN, CEREMADE (CEntre de REcherche en MAthématiques de la DÉcision, French for Research Centre in Mathematics of Decision), Paris Dauphine University,

The title of the lecture: “Magnetic interpolations inequalities in dimensions 2 and 3”

Abstract: In this talk I will present some results concerning magnetic inequalities, similar to Gagliardo-Nirenberg inequalities, but involving magnetic operators. We will first consider the case of a general magnetic field where general results will be proved, but without much concrete information. Then, in the particular cases of constant or Aharonov-Bohm magnetic fields, we will be able to make those results more precise and get better estimates, or even complete information, about the best constants in the inequalities, or about the optimal extremals.

Recorded talk N/A

Seminar on December 14

Speaker: Prof. Ari Laptev, Department of Mathematics, Imperial College London, http://wwwf.imperial.ac.uk/~alaptev/

The title of the lecture: “Magnetic rings”

Abstract: We study functional and spectral properties of perturbations of a magnetic second-order differential operator on a circle. This operator appears when considering the restriction to the unit circle of a two dimensional Schrödinger operator with the Bohm-Aharonov vector potential. We prove some Hardy-type inequalities and sharp Keller-Lieb-Thirring inequalities.

Recorded talk: https://youtu.be/DEs7IWzOyKs

Seminar on November 30

Speaker: Prof. Mikhail Sodin, Raymond & Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, https://en-exact-sciences.tau.ac.il/profile/sodin

The title of the lecture: “Fourier uniqueness and non-uniqueness pairs”

Abstract: Motivated by a remarkable discovery by Radchenko and Viazovska and by a recent work by Ramos and Sousa, we find conditions sufficient for a pair of discrete subsets of the real axis to be a uniqueness or a non-uniqueness pair for the Fourier transform. These conditions are not too far from each other. The uniqueness theorem can be upgraded to the frame bound and an interpolation formula, which in turn produce an abundance of Poisson-like formulas. This is a report on a joint work in progress with Aleksei Kulikov and Fedor Nazarov.

Recorded talk: https://youtu.be/D9G3Sp8CkLQ

Seminar on November 16

Speaker: Prof. Kristian Seip, Norwegian University of Science and Technology, https://www.ntnu.edu/employees/kristian.seip

The title of the lecture: “Fourier interpolation with the zeros of the Riemann zeta function”

Abstract: Originating in work of Radchenko and Viazovska, a new kind of Fourier analytic duality, known as Fourier interpolation, has recently been developed. I will discuss the underlying general duality principle and present a new construction associated with the non-trivial zeros of the Riemann zeta function, obtained in joint work Andriy Bondarenko and Danylo Radchenko. I will emphasize how the latter construction fits into the theory of the Riemann zeta function.

Recorded talk: Play recording (58 mins)

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ჩვენი სოციალური ქსელი
UG
კონტაქტი
მისამართი: 77ა, მ. კოსტავას ქუჩა, თბილისი, 0171, საქართველო ტელ: 2 55 22 22; info@ug.edu.ge, ug@ug.edu.ge
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