Naotaka Kubo (Postdoctoral Researcher, Yukawa Institute for Theoretical Physics, Kyoto University)
M2-brane is an interesting object in M-theory and string theory. A three-dimensional đ©=6 super conformal Chern Simons theory with gauge group U(đ1)Ăđ(đ2), called ABJ theory, describes the low energy behavior of M2-brane On the one hand, it has been considered that when |đ1âđ2| is larger than the absolute value of Chern Simons level, the supersymmetry is broken. On the other hand, it was predicted that an interesting phenomenon called duality cascade occurs, and supersymmetry is not broken in some cases.
Motivated by this situation, we performed non-perturbative tests by focusing on the partitionfunction on đ3. The result strongly suggests that the duality cascade indeed occurs. We also proposed that the duality cascade occurs in theories with more general gauge groups and we performed non-perturbative tests in the same way. I will review and explain our physical prediction in the first half of my talk. In the second half of my talk , I will explain the non-perturbative tests . This part is mathematical because the partition function reduces to a matrix model by using the supersymmetric localization technique.
December. 14, 16:00-18:10, on Zoom
Hideki Inoue (Nagoya University)
Levinsonâs theorem is a surprising result in quantum scattering theory, which relates the number of bound states and the scattering part of the underlying quantum system. For the last about ten years, it has been proved for several models that once recast in an operator algebraic framework this relation can be understood as an index theorem for the MĂžller wave operators. Resulting index theorems are called topological version of Levinsonâs theorem or shortly topological Levinsonâs theorem. In this talk, we first review the background and the framework of our investigation. New analytical and topological results are provided for Schrödinger operators on the half-line. This talk is based on my Ph.D thesis.
December. 6, 16:00-18:10, on Zoom
Martin Skrodzki (Visiting Researcher, iTHEMS)
State-of-the-art acquisition devices produce surface representations of increasingly high resolution. While these detailed representations are important for production, they are problematic e.g. when exchanging drafts via the internet or when a quick rendering for comparison is necessary. In the first part of the talk, I will present results and further research questions from a paper I recently co-authored on 'Variational Shape Approximation'. This approach aims at linearizing the input surface and representing it via a set of localized planar segments. In the second part of the talk, I will present some ongoing research on surface representations via balls. This work started with constructions from spherical neodym magnets and provided a set of mathematical questions. These investigations are joint work with FU Berlin and OIST.
November. 30, 16:00-18:10, on Zoom
Takahiro Kitayama (The University of Tokyo)
In 3-dimensional topology the great progress during the last two decades revealed that various properties of 3-manifolds are well understood from their fundamental groups. I will give an introduction to the study of splittings of 3-manifolds along surfaces, with an emphasis on an application of group representations. A fundamental and difficult problem in general is to find surfaces essentially embedded in a given 3-manifold. I will explain how such surfaces are detected by deformations of representations of the fundamental group, and what information of detected surfaces is described in terms of topological invariants derived from representations.
November. 16, 16:00-18:10, on Zoom
Tianfeng Hou (Prediction Science Laboratory, RIKEN)
The use of numerical simulations for complex systems is common. However, significant uncertainties may exist for many of the involved variables, and in order to ensure the reliability of our simulation results and the safety of such complex systems, a stochastic approach providing statistics of the probability distribution of the results is of crucial importance. However, when a highly accurate result is required, the conventional Monte Carlo based probabilistic methodology inherently requires many repetitions of the deterministic analysis and in cases where that deterministic simulation is (relatively) time consuming, such probabilistic assessment can easily become computationally intractable. Hence, to reduce the computational expense of such probabilistic assessments as much as possible, the targets of this seminar are twofold: (1), to exploit an efficient sampling strategy to minimize the number of needed simulations of Monte Carlo based probabilistic analysis; (2), to investigate a surrogate model to reduce the computational expense of single deterministic simulation.
This seminar contains two parts and will be accompanied by a set of illustrative building physical case studies (analysis of the heat and moisture transfer through building components). The first part of this seminar focusses on the use of quasi-Monte Carlo based probabilistic assessment for building performance, since it has the potential to outperform the standard Monte Carlo method. More specifically, the quasi-Monte Carlo sampling strategies and related error estimation techniques will be introduced in detail. In addition, questions on under which conditions the quasi-Monte Carlo can outperform the standard Monte Carlo method will be answered by a set of analyses. The second part of this seminar targets the investigation of using model order reduction methods for optimizing the deterministic simulation, given that it generally allows a (large) reduction of the simulation time without losing the dynamic behavior of the conventional models (such as the transient finite element analysis). Particularly, the fundamental concepts of one common model order reduction method â proper orthogonal decomposition (POD) will be provided, and its potential use for simulating (building physical) problems with different levels of non-linearity and complexity will be illustrated.
November. 13, 16:00-18:10, on Zoom
Naoto Shiraishi (Gakushuin University)
If an isolated macroscopic quantum system is left at a nonequilibrium state, then this system will relax to the unique equilibrium state, which is called thermalization. Most of quantum many-body systems thermalize, while some many-body systems including integrable systems do not thermalize. What determines the presence/absence of thermalization and how to understand thermalization from microscopic quantum mechanics are profound long-standing problems.
In the first part of my talk, I briefly review some established results of quantum thermalization. I first clarify the problem of thermalization in a mathematical manner, and then introduce several important results and insights: typicality of equilibrium states [1], relaxation caused by large effective dimension [2], and eigenstate thermalization hypothesis (ETH) [3,4] and weak-ETH [5].
In the second part of my talk, I explain some of my results. First, I introduce a model which is non-integrable and thermalizes but does not satisfy the ETH [6,7]. This finding disproves the conjectures that all nonintegrable systems satisfy the ETH and that the ETH is a necessary condition for thermalization. I also discuss the hardness of the problem of thermalization from the viewpoint of computational science [8]. Then, I move to an analytical approach to a concrete model, and prove that S=1/2 XYZ chain with a magnetic field is nonintegrable [9]. This is the first example of proof of nonintegrability in a concrete quantum many-body system, which will help a mathematical approach to thermalization.
References:
[1] S. Popescu, A. Short, A. Winter, Nat. Phys. 2, 754 (2006)
[2] P. Reimann, Phys. Rev. Lett, 101, 190403 (2008)
[3] M. Srednicki, PRE 50, 888 (1994)
[4] M. Rigol, V. Dunjko & M. Olshanii, Nature 452, 854 (2008)
[5] T. Mori, arXiv:1609.09776 (2016)
[6] N. Shiraishi and T. Mori, Phys. Rev. Lett. 119, 030601 (2017)
[7] T. Mori and N. Shiraishi, Phys. Rev. E 96, 022153 (2017)
[8] N. Shiraishi and K. Matsumoto, in preparation
[9] N. Shiraishi, Europhys. Lett. 128, 17002 (2019)
November. 10, 16:00-18:10, on Zoom
Kosuke Suzuki (Hiroshima University)
In this talk, I will introduce mathematical aspects of quasi-Monte Carlo (QMC) integration. We aim to approximate the integral of a function on the d-dimensional hypercube [0,1]^d. A useful approach is Monte-Carlo (MC) integration, which uses randomly chosen samples. A drawback of MC is the rate of convergence; the standard deviation of the estimator converges as 1/sqrt(n) asymptotically in n. To have a better rate of convergence as O(log^d N/N) or more, QMC uses deterministic, uniformly distributed points.
In the first part, I will give an overview of QMC, such as star-discrepancy, Koksma-Hlawka inequality, and some explicit constructions as lattices and digital nets.
In the second part, I will show that QMC using lattices and digital nets can achieve a higher rate of convergence for smooth integrands.
November. 5, 16:00-18:10, on Zoom
Toshihiro Ota (Osaka University, iTHEMS)
- 1st part (math): In physics literature, âlattice modelsâ appear quite often as mathematical models of physical systems, e.g. Ising model, vertex models, lattice gauge theory. The aim of the 1st part is to introduce âwhat is (T)QFT,â âwhat is lattice model,â and âwhat does integrability meanâ in the language of mathematics. In turn, they will play a crucial role in the 2nd part of my talk. I also hope that this will lead to a good exchange among us, especially between physicists and mathematicians.
- 2nd part (physics): In the 2nd part, I would like to explain where an integrable lattice model may come from, especially for people in the physics background. I will show a certain class of integrable lattice models is realized by Wilson-ât Hooft lines in 4d quiver gauge theories. I will also explain a bit how these gauge theories are constructed from brane configurations in string theory. String dualities allow us to relate the original 4d setups to 4d partially topological Chern-Simons theory, which is a partial TQFT and generates integrable lattice models.
October. 2, 16:00-18:10, on Zoom
Kenta Sato (iTHEMS)
The four color theorem states that, given any separation of a plane into contiguous regions, no more than four colors are required to color the regions. Although this theorem was already proved about 40 years ago, another proof without using a computer is not found still now.
In this talk, I will introduce an algebraic approach to this theorem, which states that a conjecture about singularities of algebraic varieties implies the four color theorem. In particular, I would like to focus on the connection of three different fields in mathematics: graph theory, convex geometry and algebraic geometry.
September. 24, 17:10-18:10, on Zoom
Genki Ouchi (iTHEMS)
In this talk, I would like to talk about symmetries of algebraic varieties, especially cubic fourfolds and K3 surfaces. It is known that symmetries of cubic fourfolds and K3 surfaces are related to sporadic finite groups as Mathieu groups and Conway groups in both algebraic geometry and string theory. Relations between cubic fourfolds and K3 surfaces are studied in the context of derived categories, Hodge theory and so on. I would like to explain the direct relation among symmetries of cubic fourfolds and K3 surfaces via their derived categories.
September. 24, 16:00-17:00, on Zoom
Ken Furukawa (Prediction Science Laboratory, RIKEN)
The theory of maximal regularity is a powerful tool to get solutions having the best regularity to linear partial differential equations (PDEs) of parabolic type. The theory is also applicable to show well-posedness of various non-linear PDEs.
In the first part, We introduce the history of the development of the theory of maximal regularity and the way to apply non-linear PDEs.
In the second part, We give some applications to PDEs, e. g. the primitive equations, the Navier-Stokes equations, and elliptic equations with dynamic boundary conditions.
September. 8, 16:00-18:10, on Zoom
[DAY 1: Aug 31] Makiko Sasada (University of Tokyo)
[DAY 2: Sept 1] Kenichi Bannai (Keio University)
This is a series of lectures on "Geometric Perspectives for Fluid Dynamic Limit Theory" and this series is a Math-Phys joint seminar.
One of the fundamental problems in the natural and social sciences is to explain macroscopic phenomena that we can observe from the rules governing the microscopic system giving rise to the phenomena. Hydrodynamic limit provides a rigorous mathematical method to derive the deterministic partial differential equations describing the time evolution of macroscopic parameters, from the stochastic dynamics of a microscopic large scale interacting system.
In the article "Topological Structures of Large Scale Interacting Systems via Uniform Locality" joint with Yukio Kametani, we introduce a general framework encompassing a wide variety of interacting systems in order to systematically investigate various microscopic stochastic large scale interacting systems in a unified fashion. In particular, we introduced a new cohomology theory called the uniformly local cohomology to investigate the underlying geometry of the interacting system. Our theory gives a new interpretation of the macroscopic parameters, the role played by the group action on the microscopic system, and the origin of the diffusion matrix associated to the macroscopic deterministic partial differential equation obtained via the space-time scaling limit of the microscopic system.
The purpose of the series of lectures is to introduce to the audience the theory of hydrodynamic limits, especially the relation between the macroscopic observables and the microscopic interacting system. We then explain our new perspective of how geometry comes into play in investigating the interacting system, and introduce the ideas and results of our article.
August. 31 and September. 1, 13:00-15:00, on Zoom
Tadahiro Miyao (Hokkaido University)
1. Stability of ferromagnetism in many-electron systems
Abstract: I construct a model-independent framework describing stabilities of ferromagnetism in strongly correlated electron systems. Within the new framework, I reinterpret the Marshall-Lieb-Mattis theorem and Liebâs theorem; in addition, from the new perspective, I prove that Liebâs theorem still holds true even if the electron-phonon and electron-photon interactions are taken into account. I also examine the NagaokaThouless theorem and its stability. These examples verify the effectiveness of the new viewpoint.
2. Order preserving operator inequalities in many-electron systems
Abstract: In this talk, I will introduce order preserving operator inequalities and explain how these inequalities are applied to the mathematical study of ferromagnetism. As examples of applications, Lieb's theorem of the Hubbard model and its stabilities will be discussed in terms of the inequalities.
July. 31, 16:00-18:10, on Zoom
Yuichi Ike (Fujitsu)
1. Topological data analysis and its applications
In this talk, I will explain some methods in topological data analysis (TDA) and their applications. First I recall persistent homology, which is a central tool to analyze the "shape" of a point cloud set. Then I show several applications to material science and time-series analysis. I also talk about our collaborative research with Inria on noise-robust persistent homology and an automated vectorization method of persistence diagrams.
2. Persistence-like distance on sheaf category and displacement energy
In this talk, I will talk about relation among sheaf theory, persistence modules, and symplectic geometry. We introduce a persistence-like distance on Tamarkin sheaf category and prove a stability result with respect to Hamiltonian deformation of sheaves. Based on this result, we propose a new sheaf-theoretic method to give a lower bound of the displacement energy of compact subsets of a cotangent bundle.
This is a joint work with Tomohiro Asano.
July. 15, 16:00-18:10, on Zoom
Ryusuke Hamazaki (Riken iTHEMS)
In quantum mechanics, the existence of large energy gaps allows us to trace out the degrees of freedom of irrelevant energy scale. Consequently, we can treat a system within a constrained subspace obtained by the projection of the total Hilbert space. While this statement has widely been used to approximate quantum dynamics in various contexts, a general and quantitative justification stays lacking.
In this talk, we show a universal and rigorous error bound for such a constrained-dynamics approximation in generic gapped quantum systems [1,2]. This universal bound is a linear function of time that only involves the energy gap and coupling strength, provided that the latter is much smaller than the former. If time allows, I will briefly talk about generalizations of our result to e.g., quantum many-body systems and open quantum systems.
[1]Z. Gong, N. Yoshioka, N. Shibata, and RH, Phys. Rev. Lett. 124, 210606 (2020).
[2]Z. Gong, N. Yoshioka, N. Shibata, and RH, Phys. Rev. A 101, 052122 (2020).
June. 24, 16:00-18:10, on Zoom
Tasuku Soma (The University of Tokyo)
Matrix scaling is a classical problem with a wide range of applications. It is known that the Sinkhorn algorithm for matrix scaling is interpreted as alternating e-projections from the viewpoint of classical information geometry.
Recently, a generalization of matrix scaling to completely positive maps called operator scaling has been found to appear in various fields of mathematics and computer science, and the Sinkhorn algorithm has been extended to operator scaling.
In this study, the operator Sinkhorn algorithm is studied from the viewpoint of quantum information geometry through the Choi representation of completely positive maps. The operator Sinkhorn algorithm is shown to coincide with alternating e-projections with respect to the symmetric logarithmic derivative metric, which is a Riemannian metric on the space of quantum states relevant to quantum estimation theory. This talk is based on joint work with Takeru Matsuda.
June. 17, 16:00-18:10, on Zoom
Masaki Taniguchi (Riken iTHEMS)
The classification problem of knots is one of the central topics in a study of topology.
In the first part, we review classical knot theory and theory of 2-dimensional knots in the 4-dimensional space.
In the second part, we focus on a problem considered in differential topology. In the studies of differential topology, people are interested in the difference between continuous and smooth. As the main result of this talk, we introduce a theorem that tells us the difference between continuous and smooth 2-dimensional knots. The proof uses Yang-Mills gauge theory for 4-manifolds obtained by the surgery of 2-knots.
May. 27, 16:00-18:10, on Zoom
Yukimi Goto (Riken iTHEMS)
In this talk, I will introduce the mathematical studies on the ionization problem. Some experimental & numerical evidences say that any doubly charged atomic ion X^{2-} is not stable. This 'fact' is called the ionization conjecture in mathematical physics literatures. My hope is to illustrates the interplay between mathematical and physical ideas. The talk is directed towards researchers on various aspects of quantum mechanics. In the first part, we will discuss the many-body aspects of quantum mechanics and introduce some basic notions. The second part will deal with the mathematical results in some approximation theories.
If you are interested in the experimental or numerical results, you can refer to following.
References
T. Andersen, H. K. Haugen and H. Hotop, "Binding Energies in Atomic Negative Ions: III," J. Phys. Chem. Ref. Data 28 1511-1533 (1999).
H. Hogreve, "On the maximal electronic charge bound by atomic nuclei," J. Phys. B 31 L439-446 (1998).
A V. Sergeev, and S. Kais, "Critical nuclear charges for N-electron atoms," Int. J. Quant. Chem. 75 533-542 (1999).
May. 13, 16:00-18:10, on Zoom
Keita Mikami (Riken iTHEMS)
Resonance is one of the most studied object in mathematical study of Schrödinger operators. One possible reason is that resonance is appeared in many other fields like arithmetic, physics, and topography. This series of talks target both mathematicians and researchers in other fields. The goal of the talk is to introduce the study of resonances for two body Schrödinger operators.
In the first part, we briefly review spectral theory and how we use it in the study of Schrödinger operators. The aim of this part is to introduce the audience some basic notions used in the study of Schrödinger operators.
In the second part, we give brief introduction of resonances and its application to both mathematicians and researchers in other fields. We start from mathematical definition of resonances to its applications in the other fields.
May. 1, 16:00-18:10, on Zoom
Mikio Furuta (University of Tokyo)
The Wilson-Dirac operator is a discrete version of Dirac operator defined on regular lattices. When the discrete version is a fine approximation of the Dirac operator on a Z/2-graded Clifford module on a torus, it is known that
(1) an integer-valued index is defined for the Wilson-Dirac operator, and
(2) the index is equal to the Atiyah-Singer index of the Dirac operator on the torus.
These have been well established up to around 2000. The strategy of all the previous works is to make use of the discrete version of the heat kernel for Neuberger's overlap Dirac operator. Therefore the strategy cannot be generalized to mod 2 index nor family version of index.
In this talk I would like to explain a new approach to the index of Wilson-Dirac operator which can be immediately generalized to these various cases.
Joint work with H. Fukaya, S. Matsuo, T. Onogi, S. Yamaguchi and M. Yamashita.
Feb. 25, 16:00-18:10, Main Research Building Room 160
Martin Skrodzki (iTHEMS)
In 2009, Joselli et al. introduced the neighborhood grid data structure for fast computation of neighborhood estimates for point clouds in arbitrary dimensions. Even though the data structure has been used in several applications and was shown to be practically relevant, it is theoretically not yet well understood even in the two-dimensional case. The purpose of this talk is to present the data structure, give a time-optimal building algorithm, and motivate several associated questions from enumerative combinatorics as well as low-dimensional (probabilistic) geometry. In case of questions that have been solved in the past, corresponding proofs will be provided. For the open question, the talk will list them as an outlook to possible future collaboration.
Feb. 7, 16:00-18:10, Main Research Building Room 160
Shu Nakamura (Gakushuin University)
Plan of the seminar: we separate each talk into two. In the first 60 minutes the speaker gives an introductory talk for non-mathematicians. After a short break, the second 60 minutes is spent for a bit more detailed talk for mathematicians (working in other areas). We welcome you joining both parts of the seminar or only the first/second half.
Abstracts:
Talk 1: Semiclassical analysis, microlocal analysis and scattering theory.
I plan to talk about overview on the semiclassical analysis and related topics, especially its intrinsic relationship with microlocal analysis and (microlocal) scattering theory. Roughly speaking, the microlocal analysis is an application of semiclassical idea to the analysis of singularities, and its analogue in momentum space is the microlocal scattering theory. We discuss basic notions of these, and mention several recent results.
Talk 2: Microlocal structure of the scattering matrix with long-range perturbations.
As an example of topics discussed in Talk 1, we discuss recent results on the scattering matrix with long-range perturbations. In particular, we show that the scattering matrix is expressed as a Fourier integral operator, and in some cases we can decide its spectral properties. Our approach is fairly geometric and abstract, and thus applies not only to usual Schrödinger operators but also to higher order operators and discrete Schrödinger operators.
Jan. 23, 16:00-18:10, Main Research Building Room 435-437