# Special Mathematics Seminar

The aim of this seminar is to provide research level talks to a broad audience, including advanced students.

Accessibility is the keyword suggested to all speakers.

The time schedule is chosen for giving the opportunity to anyone to attend the seminar.
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Contact:

Serge Richard (richard@math.nagoya-u.ac.jp), Rm. 237 in Sci. Bldg. A

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Schedule : Tuesday December 17, 2019 (18:30 - 20.00) in room 207 of the Science Building A

Speaker : *Rafael Tiedra*
(Pontifical Catholic University of Chile)

Title: Spectral and scattering theory of one-dimensional coupled photonic crystals

**Abstract : **
We study the spectral and scattering theory of light transmission in a system consisting of two asymptotically periodic waveguides (one-dimensional photonic crystals) coupled by a junction. Using analyticity techniques and commutator methods in a two-Hilbert spaces setting, we determine the nature of spectrum and prove existence and completeness of the wave operators of the system. Joint work with G. De Nittis, M. Moscolari and S. Richard.

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Schedule : Thursday May 9, 2019 (18:30 - 20.00) in room 207 of the Science Building A

Speaker : *Max Lein*
(Advanced Institute of Materials Research, Tohoku University)

Title: Introduction to an analytic-algebraic approach to linear response theory

**Abstract : **
Linear response theory is a tool with which one can study systems that are driven out of equilibrium by external perturbations. It has been used to give a first-principles derivation of Ohm's law, where the current depends linearly on the applied external electric field. Simply put, the conductivity coefficients that quantify the response can be obtained from a "Taylor expansion".

Justifying this Taylor expansion of the current density in the electric field has been the subject of a lot of research over the last few decades, starting with the work of Green and Kubo. And making linear response theory rigorous has been a steady source of inspiration for mathematicians.

The approach to linear response theory I will discuss in this talk combines elements from functional analysis with insights from operator algebras. This is advantageous, because it is not tailored to a specific model, works for operators on the continuum or the discrete alike and can deal with disorder.

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Schedule : Thursday January 24, 2019 (18:30 - 20.00) in room 207 of the Science Building A

Speaker : *Rafael Tiedra*
(Pontifical Catholic University of Chile)

Title: Ruled strips with asymptotically diverging twisting

**Abstract : **
We consider the Dirichlet Laplacian in a two-dimensional strip composed of segments translated along a straight line with respect to a rotation angle with velocity diverging at infinity. We show that this model exhibits a "raise of dimension" at infinity leading to an essential spectrum determined by an asymptotic three-dimensional tube of annular cross section. If the cross section of the asymptotic tube is a disc, we also prove the existence of discrete eigenvalues below the essential spectrum. Joint work with David Krejcirik (Prague).

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Schedule : Thursday July 12, 2018 (18:30 - 20.00) in room 207 of the Science Building A

Speaker : *Francesco Buscemi* (Nagoya University)

Title: The Information-Disturbance Trade-Off in Quantum Theory

**Abstract : **
Heisenberg's "uncertainty relations" (HURs) are among those few ideas of modern science that found their way into common language. They are also among the most debated results since the inception of quantum theory. The reason for this is not to be found in their mathematical derivation, which is absolutely transparent, but rather in their interpretation. In this talk I will review HURs, the idea behind HURs, some problems with the original idea, and how a modern, information-theoretic approach can help in clarifying the whole matter. At the end of my talk, we will come to appreciate why uncertainty relations are so important to our understanding of nature and, at the same time, so problematic to be, 90 years since their first appearance, still at the center of heated scientific discussions.

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Schedule : Thursday June 28, 2018 (18:30 - 20.00) in room 207 of the Science Building A

Speaker : *Jeremy Faupin* (University of Lorraine, France)

Title: Scattering theory for dissipative quantum systems

**Abstract : **
In this talk, we will consider an abstract pseudo-hamiltonian given by a dissipative operator of the form H = H_V - i C*C, where H_V = H_0 + V is self-adjoint and C is a bounded operator. Such operators are frequently used to study scattering theory for dissipative quantum systems. We will recall conditions implying existence of the wave operators associated to H and H_0, and we will see that they are asymptotically complete if and only if H does not have spectral singularities on the real axis. For Schroedinger operators, the spectral singularities correspond to real resonances.
This is joint work with Juerg Froehlich.

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