RCS seminar by those interested

Date: 27 February 2025.

Location: Zoom meeting. Click here to enter.→Finished.

※You can enter the room from 15:00 on the day of the event (please do not enter before then).

※Please enter with your affiliation after your name (we recommend that you set up the display of your name in advance).

Content

15:30～16:30 (including Q&A session)

Presenter: Tsuyoshi Jomoto (Sophia University Graduate School of Science and Technology, Mathematics Division, Master student, 2nd year）

Title: Crystal bases on spin representation of type $B_n$

Abstract (translated):

A quantum group is an algebra defined on the ${\mathbb Q}(q)$. It is regarded as a $q$-analogue of the enveloping algebra of a Lie algebra, whose module is also defined on the ${\mathbb Q}(q)$. In a nutshell, a crystal basis is obtained as a ${\mathbb Q}$-basis of an module or subalgebra ‘specialised to $q=0$’. Setting $q=0$ simplifies the action of quantum groups, but much of the information on the representation of quantum groups can be attributed to combinatorics without being lost.

In this presentation, we will introduce some interesting aspects of the crystal basis using the example of $B$-type spin representations, which is also the topic of my master thesis.

Keywords (translated): 

Quantum groups, crystal basis, tensor product theorem, combinatorics, Young diagrams.

LIVE: 

Permitted.

Handout: 

Not to be disclosed (individual request to participants will be considered).

16:30～17:00

Breaks etc.

17:00～18:00 (including Q&A session)

Presenter: So Dainichi (Rikkyo University, College of Science, Department of Physics, 4th year)

Title: Primordial black holes and initial fluctuations 

Abstract (translated):

Primordial black holes (PBHs) are black holes that form in the early universe. The black hole binary collision detected by LIGO in 2015 is a candidate for a PBH because it is massive for a black hole formed from a star. In this presentation, we calculate the mass and abundance of PBHs from their formation process and derive the requirements of their abundance on the initial fluctuations of the Universe.

Keywords (translated): 

Black holes, early universe, big bang, dark matter.

LIVE: 

Permitted.

Handout: available on the page of this website.

Date and time: Tuesday, 5 March 2024, 14:00 - .

Location: Zoom meeting (Click here to enter.→Finished.)

※Please enter with your affiliation after your name.

Content

Presenter: Keigo Horikoshi（Rikkyo University Graduate School of Science, Physics major, Master student, 2nd year）

Title (translated): JT gravity and duality in the theory of random matrices

Abstract (translated): 

We focus on Jackiw-Teitelboim (JT) gravity and its duality in random matrix theory. This duality was proved by Saad, Shenker and Stanford (arXiv:1903.11115) in 2019, and the path integral, including even perturbation effects of JT gravity, can be computed using the theory of random matrices, in particular the resolvent recurrence formula. The aim of this talk is to understand the duality between these two theories.

Date: 6 Oct, 23 Oct, 1 Nov, 4 Nov, 7 Nov, 12 Nov 2022.

Location: Zoom meeting.

Content

Thu 6 Oct, 13:25 - (4½+ hrs).

Presenter: Yudai Ide（Rikkyo University, College of Science, Department of Physics, 3rd year）

Title (translated): Reconstruction of thermodynamics.

Abstract (translated):

First, a brief description of what the field of thermodynamics is trying to achieve is given. After the basics of isothermal and adiabatic operations, Helmholtz free energy and internal energy are introduced. Entropy is introduced by linking Helmholtz free energy and internal energy to Carnot's theorem. The physical meaning of entropy is also considered. The relationship between the thermodynamic variables introduced in this way is derived by partial differentiation. Some exercises are given.

Keywords (translated): 

Thermodynamics, Maximum work, Helmholtz free energy, Adiabatic work, Internal energy, Poisson's relation, Maximum absorbed heat, Entropy, Maxwell's relation.

Message (translated): 

In the first 30 minutes or so, I will explain why thermodynamics should not be regarded as low-level physics, with no equations at all. After that, the physical considerations will increase, but due to the structure of thermodynamics, difficult equations are unlikely to appear, so even first year students should be able to understand if they can do partial differentiation.

LIVE: Permitted.

Handout: available on the page of this website.

23 Oct (Sun), 15:00 - (1½ to 3 hrs).

Presenter: Hideto Komiya（Secondary school teacher）

Title (translated): Semantics -basic semantic theory-

Abstract (translated):

In this talk, I will take up the issue of 'word polysemy', which is one of the debates in semantics in linguistics, and introduce a theory, the 'basic semantic theory', with a detailed explanation. Concrete examples (mainly in Japanese and English) will also be discussed and explained in the lecture.

First, the view of the linguistic signifier in Saussure's structuralist linguistics, on which the basic semantic theory is based, will be described in detail, and then the question of why word polysemy is a problem in semantics will be explained.

Next, an outline of basic semantic theory as an extension of the aforementioned structuralist theory is presented, and how it provides an answer to the problem of word polysemy. E. Koseliu's theory of norma is also introduced here to address the problem of linguistic fluidity, i.e. the change of meaning of words.

Finally, I will discuss criticisms and explanations of the theory of basic meaning, issues related to the theory of basic meaning, and the affinity between the theory of basic meaning and cognitive linguistics, which has become mainstream in recent years.

Keywords (translated): 

Basic semantic theory, polysemy, structuralism, semantic change.

Message (translated): 

Anyone interested in linguistics and the meaning of words (with or without previous knowledge) is welcome.

LIVE: Permitted.

Handout: available only to those who have registered for the 8th Joint Study Group on Mathematics and Physics.

1 Nov (Tue), 20:00 - (1.5-3 hrs).

Presenter: So Dainichi（Rikkyo University, College of Science, Department of Physics, 2nd year）

Title (translated): On the business model of the rhythm game

Abstract (translated):

The roots of rhythm games go back to Parappa Rapper, released by KONAMI in 1996. The following year, in 1997, KONAMI released Beatmania, also from KONAMI, which became the pioneer of arcade rhythm games. The game's gameplay, in which the player controls the game to the beat of the music or in such a way as to reproduce the music, has won over many fans over the years and has evolved with the development of technology. Now, thanks to improvements in the performance of devices, mobile terminals have an operability comparable to that of arcade games.

In this age of sound game wars, we look at which rhythm games have lasted the longest and whether the up-and-coming sound games can survive, as well as considering the future of the sound game industry.

Keywords: 

(None.)

Message (translated): 

As this is a joint study group, I hope there will be more casual topics like this. It costs money, so it is not advisable to get too involved, but please give it a try.

LIVE: Permitted.

Handout: available on the page of this website.

Fri 4 Nov, 11:20 - (1½ to 3 hrs).

Presenter: Keigo Horikoshi（Rikkyo University Graduate School of Science, Physics major, Master student, 1st year）

Title (translated): Let's use analytical mechanics to understand special relativity

Abstract (translated):

Based on the principle of least action in analytical mechanics and the requirements of special relativity, the relativistic action of free particles and even the action of the electromagnetic field itself are shown. The Maxwell equations for electromagnetic fields are also derived. It is recommended that you have some elementary knowledge of analytical mechanics and electromagnetism, but I will be supplementing your knowledge of these subjects, so if you are interested, I would be happy to have you listen to my talk. I will probably talk about the topics that are of interest to those who have studied Analytical Mechanics according to the normal curriculum (?).

Keywords (translated): 

Analytical mechanics, special relativity, electrodynamics, least action principle, Lagrangian, Hamiltonian, Euler-Lagrange equation, free particle action, Lorentz transformations, Lorentz invariance, electromagnetic field tensor, Maxwell equations, classical field theory.

Message (translated): 

People who enjoy this kind of discussion are suited to theoretical physics research teams.

LIVE: Permitted.

Handout: available on the page of this website.

7 Nov (Mon), 13:30 - (1 to 1.5 hours *may be extended).

Presenter: Toshiki Takadera（Rikkyo University Graduate School of Science, Physics major, Master student, 2nd year）

Title (translated): The Dark Universe Explored by the Theory of Modified Gravity

Abstract (translated):

One of the great challenges of modern cosmology is the existence of dark energy. This is the cause of the current late-accelerating expansion of the Universe, but its true nature is still not well understood. As an approach to explaining this dark energy, modified gravity theory has been actively studied in recent years. Modified gravity is an extension of the standard theory of gravity, general relativity, and various theories have been proposed so far. In this talk, after an overview of general relativity, two modified gravity theories, scalar tensor theory and massive gravity, will be presented.

Keywords (translated): 

Theory of Modified Gravity, Cosmology, Dark Energy, General Relativity.

Message (translated): 

I will try to explain it as clearly as possible, as it may be difficult to understand. It is not assumed that the presenter is familiar with general relativity, so we hope that those who are not familiar with it will be able to listen with ease.

LIVE: Permitted.

Handout: available only to participants on the day of the presentation.

Sat 12 Nov, 13:30 - (3-4½ hrs).

Presenter: Tomoaki Murata（Rikkyo University Graduate School of Science, Physics major, Doctoral student）

Title (translated): Introduction to early universe cosmology

Abstract (translated):

The basic content of cosmology will be reviewed. This time, special attention will be given to the inflationary theory of the early universe, with the aim of understanding the phenomena of that time using mathematical formulae. Current research in inflationary theory will also be presented.

Keywords (translated): 

Early Universe, Inflation, Classical Field Theory.

Message (translated): 

In the first hour I will introduce myself and talk a little about the ecology of doctoral studies. I would be happy if people who are wondering whether to do a doctorate, or what a doctorate is, could come and listen to my talk. I would be happy if people who are wondering whether to enter a doctoral programme, or what a doctoral programme is at all, could listen to my talk.

LIVE: Permitted.

Handout: it will be made available to the members only (on the mailing list).

Tuesday 15 March, 10:30-16:30 (including lunch break)

Speaker: Keigo Horikoshi（Rikkyo University, College of Science, Department of Physics, 4th year）

Title (translated): Invitation to string theory（*Files are available here.）

Abstract (translated): 

String theory, in which the fundamental unit is not a particle but a string. According to this theory, the quantisation of strings gives rise to photons and gravitons. This talk starts with the Lagrangian of classical strings and describes the quantisation of strings and photon states. Using a covariant description, I will also talk about the well-known properties of the Maxwell field from the point of view of string theory.

Message from speaker (translated):

It's probably difficult to understand in one sitting, so I hope you'll listen to it as if to say, 'I didn't know there was such a thing'. If you're interested, let's try physics.

Wednesday 16 March, 15:20-18:50 (+ Q&A, etc.)

Speaker: Hideto Komiya（Rikkyo University Graduate School of Intercultural Communication, Master student, 1st year）

Title (translated): What do words mean? An invitation to semantics

Abstract (translated):

In this talk, I will review various theories such as structuralist linguistics, philosophy of language, and cognitive linguistics, focusing on topics in the semantics of linguistics. Specifically, the meaning of 'word' (lexical semantics) will be discussed with several concrete examples, and related linguistic theories will be introduced at the same time. Communicating with language inevitably implies that we already understand the meaning of words, but in fact we are unconscious about the meaning of words. Here we hope to convey the interest of thinking about language by 'shedding light' on such aspects of language to which we usually do not pay attention.

Message from speaker (translated):

Regardless of previous knowledge, anyone with even a passing interest in the meaning of words and linguistics in general is welcome to attend.

Saturday 19 March, 13:25-17:00

Speaker: Takumi Teramoto（Rikkyo University, College of Science, Department of Mathematics, 4th year）

Title (translated): Super-introduction to algebraic geometry

Abstract (translated):

Starting with a review of commutative ring theory and topological space theory, the talk will cover the rudiments of algebraic geometry, focusing on affine algebraic varieties and affine schemes. We will try to convey the atmosphere of algebraic geometry to those who are not familiar with it.

Message from speaker (translated):

As I am not fully versed in algebraic geometry myself, I do not intend to cover much applied material. The aim of this talk is to make you feel that you understand the correspondence between rings and figures. Depending on the number of participants, the time allocation between the review and the algebraic geometry content may be readjusted.

21 Mar (Mon), 15:20-17:00

Speaker: Takumi Teramoto（Rikkyo University, College of Science, Department of Mathematics, 4th year）

Title (translated): Introduction to music theory for science students

Abstract (translated):

A collection of topics in music theory, especially those that science students might enjoy. Why do the three notes Do, Mi and So sound so beautifully resonant? Why is an octave divided into 12 tones?' and 'Why is an octave divided into 12 tones?' from a mathematical and physical point of view.

Message from speaker (translated):

This time I would like to touch on many of the musical concepts related to pitch and chords. The main content of the lecture will be to explain the reasons for musical phenomena, rather than the practical content of music, such as the types of chords. Although the lecture is aimed at science-minded people, it will also be of interest to non-science-minded people, but there are some parts of the mathematics that require knowledge of High School Mathematics II+B.

Content

Friday 8 October, 5th period

Speaker: Keigo Horikoshi（Rikkyo University, College of Science, Department of Physics, 4th year）

Title (translated): Japanese dialects

Detail (translated): 

The existence of a single language in Japan - dialects of the Japanese language, their characteristics, transmission and relationship to society are introduced using general rather than specific words. The content is introductory and elementary rather than specialised, so no prior knowledge is required, but there may be some ambiguous expressions for those with knowledge, so discussion is encouraged where appropriate. Anyone interested should read the books listed in the bibliography.

Friday 8 October, 18:50-19:20

Time for questions and answers, chatting, etc. (Free to disperse.)

Friday 29 October, 4th period + half of 5th period

Speaker: Ryo Suzuki（Rikkyo University, College of Science, Department of Mathematics, 1st year）

Title (translated): Let's pretend to understand Atiyah-MacDonald

Detail (translated): 

In this talk I will try to explain the contents of Atiyah-MacDonald's Introduction to Commutative Algebra in my own way. In particular, I hope to convey the geometric viewpoint leading to the theory of schemes and the rudiments of the theory of Noetherian ring.

After the end:

Time for questions and answers, chatting, etc. (Free to disperse.)

Content

Tuesday 30 March, 13:00-15:00 (with a break in between)

Speaker: Yudai Ide（Rikkyo University, College of Science, Department of Physics, 1st year）

Title (translated): Applications of linear algebra in physics

Keywords (translated): Linear algebra, matrices, eigenvalue problems, equations of motion for restoring forces, coordinate transformations, Lorentz transformations, quantum mechanics, Schrodinger equation, well potentials.

Detail (translated): 

The talk will show how linear algebra is used in physics. In particular, coupled oscillations in classical mechanics, coordinate transformations in special relativity and applications to quantum mechanics will be covered.

1. Applications to classical mechanics

2. Applications to special relativity

3. Applications to quantum mechanics

Presentation of pdfs with mathematical formulae and explanations, with screen sharing.

Message from speaker (translated):

The content should be understandable if you can do the basic linear algebra calculations taught in first year undergraduate courses. I have not taken any lectures on quantum mechanics, so I may say something strange, but please correct me at the end if you notice anything. I am looking for deformations of the equations, but some of them are not correct. If you find any deformations or miscalculations, please correct them during the break or after the presentation.

I am sure that there are many people who know more about the content of this presentation than I do. I would like to ask them to actively point out the problems. Thank you for your cooperation.

Tuesday, 30 March, 15:00 -

Time for questions and answers, chatting, etc. (Free to disperse.)

Wednesday 31 March, 2nd, 3rd and 4th periods

Speaker: Keigo Horikoshi（Rikkyo University, College of Science, Department of Physics, 3rd year） 

Title (translated): Analytical mechanics, and on to quantum mechanics

Keywords (translated): Analytical mechanics, principle of least action, Lagrangian, Hamiltonian, symmetry, conserved quantities, Neter's theorem, canonical transformations, Poisson brackets, Hamilton-Jacobi equation, quantum mechanics, quantum theory, Schrodinger equation.

Detail (translated): 

The aim is to 'get there' from analytical mechanics to quantum mechanics. I will not go into too much detail, so if you are interested, please read the references I will introduce later. From the principle of least action, Lagrangian is introduced, leading first to analytical mechanics in Lagrangian form. Then, after a review of symmetry and conservation laws, we will talk about Hamiltonian forms, which are also related to quantum theory. After discussing canonical transformations and the Hamilton-Jacobi equation, we will move on to quantum mechanics, specifically the Schrodinger equation. If time permits, we will also cover topics related to the principle of least action and exercises in analytical mechanics.

1st session (2nd period): Analytical mechanics in Lagrangian form

From the discussion of the principle of least action, I will bring in analytical mechanics in Lagrangian form, with general coordinates and general velocities as dynamical variables. I also hope to talk about the somewhat descent transition from the Newtonian form. The first session ends with a discussion of symmetry and the corresponding conserved quantities and Noether's theorem.

2nd session (3rd period): Analytical mechanics in Hamiltonian form

The Lagrangian is transformed into a LeJandre transform to define the Hamiltonian, and then we move on to the Hamiltonian form where the general coordinates and the general momentum are the dynamical variables. After that, we talk in particular about canonical transformations and the Hamilton-Jacobi equation.

3rd session (4th period): From analytical mechanics to quantum mechanics

An introduction to quantum mechanics is considered. The most important Schrodinger equation in quantum mechanics will be explained in the language of particle-wave duality and analytical mechanics. We will also derive the Schrodinger equation from the Hamilton-Jacobi equation.

If there is not enough time, the topic will be moved to the next session.

Message from speaker (translated):

I can't guarantee the rigour of a maths department, so please keep an open mind. No prior knowledge of physics is required, but it is recommended that you have some knowledge of Newtonian mechanics at the level of high school physics. Basically I would like to use PowerPoint slides, but I may use OneNote or something similar for some parts.

Wednesday 31 March, 5th period (or at the end of the presentation) onwards

Time for questions and answers, chatting, etc. (Free to disperse.)

Content

Presenter: Shinichiro Kakuta（Rikkyo University, College of Science, Department of Mathematics, 4th year）

13：25 - 14：25: Part 1 'From the concrete to the abstract'

14：25 - 14：35: Break

14：35 - 15：35: Part 2 'What can we learn if we understand general topology?'

15：35 -   　　     : Small talk (ending at an appropriate time)

Content

27 Feb (Thursday)

Speaker: Shinichiro Kakuta（Rikkyo University, College of Science, Department of Mathematics, 3rd year）

Title (translated): From function sequences to variational problems

Keywords (translated): (ε, N)-definition, (ε, δ)-definition, limit superior, limit inferior, completeness, continuity and semi-continuity, uniform continuity, absolute convergence, conditional convergence, Bolzano-Weierstrass Theorem, pointwise convergence, uniform convergence, uniform convergence on compact sets, uniform absolute-convergence, Ascoli–Arzelà theorem, variational problems, curve families, functional, Euler equation, etc.

Detail (translated): 

Starting from the limits of sequences, which is the basic content of first year undergraduate calculus, we will discuss in detail the continuity of real one-variable real-valued functions, convergence and divergence of infinite series, the whole of the real numbers as a continuum, function sequences and function term series, and so on. And this is not just a review (or preparation) of the differential and integral calculus, but as an application of it we will consider variational problems involving the maximum and minimum (or maxima and minima) of some 'quantity'. If there is space, I would like to present a counter-example to the existence of solutions to variational problems, i.e. variational problems without solutions, which were thought to be obvious until the middle of the 19th century.

Friday 28 February

Speaker: Shinichiro Kakuta（Rikkyo University, College of Science, Department of Mathematics, 3rd year）

Title (translated): Theory and applications of Fourier analysis.

Keywords (translated): Linear operators, Periodic conditions, Hermite inner product, L2-norm, Fourier expansion, Fourier series, Fejér series, Fejér kernel functions, Fejér's theorem, The Stone–Weierstrass theorem, Parseval's identity, uniform norm, Basel problem, Kirchhoff's circuit laws, Rapidly decreasing functions, Schwartz space, Fourier and inverse Fourier transforms, Fourier inversion theorem, Plancherel formula, Function folding, Computed Tomography, distribution, Digital Sampling, etc. 

Detail (translated): 

Assuming only the discussion of number vector spaces in first year undergraduate linear algebra, students are introduced to the concept of Fourier transforms through linear ordinary differential equations, and from there the theory of Fourier analysis is introduced. The emphasis this time is on intuitive understanding and exposure to various applications rather than on the rigour of the theory. Examples of applications include the proof of the Basel problem, the solution of linear ordinary differential equations for RLC circuits, the principle of CT scanning, digital sampling, etc.

Tuesday 17 March

Speaker: Shinichiro Kakuta（Rikkyo University, College of Science, Department of Mathematics, 3rd year）

Title (translated): Quantum theory of path integrals

Keywords (translated): Double-slit experiments, Born rule, path integrals, wave functions, Schrödinger equation, Green function, Lagrangian, Hamiltonian, operators, coherent states, eigenstates, well potentials, harmonic oscillators, etc.

Detail (translated): 

The fundamental equation of quantum mechanics is the Schrödinger equation. The path integral is a rewrite of the Schrödinger equation as an integral equation. This time, instead of trying to derive the Schrödinger equation from the quantisation of classical mechanics, we take the 'quantum mechanical particle image' from the double-slit experiment as our starting point and try to arrive at the idea of the path integral before the Schrödinger equation. The path integral is a 'quantum particle picture' imagined from the double-slit experiment.

In this presentation, the emphasis is on knowledge of different topics rather than understanding advanced theories and arguments. The prerequisite knowledge should be about what is common knowledge for high school students entering university science departments.

Tuesday 24 March

Speaker: Shinichiro Kakuta（Rikkyo University, College of Science, Department of Mathematics, 3rd year）

Title (translated): From vector calculus to modern geometry.

Keywords (translated): Gradients, Euler numbers, vector fields, divergences, rotations, potentials, Poincaré lemma, differential forms, Divergence Theorem, Gauss curvature, mean curvature, Levi-Civita connection, geodesic curvature, Gauss-Bonnet theorem, Euler's polyhedral formula, Möbius band, Klein's bottle, projective plane, Euler-Poincaré's theorem, etc.

Detail (translated): 

The basic concepts of vector analysis are introduced and the divergence theorem in vector analysis is considered. From there, curvature, differential forms, Euler numbers, etc. are touched upon, and the beautiful results that arise in modern geometry are seen. If there is room, we will also introduce more advanced topics in modern differential geometry and topology.

Content

Thursday 22 August, 2nd period

Self-introductions, preparation for presentation, small talk, etc.

Thursday 22 August, 3rd (, 4th ) period

Speaker: Rihito Kojima（Rikkyo University, College of Science, Department of Mathematics, 2nd year）

Title (translated): Foundations of artificial intelligence and deep learning.

Keywords (translated): Deep learning, gradient descent, error back propagation, CNN, Python.

Detail (translated):  

The basic structure and theory of deep learning will be explained, and as an application, the CNN method, which is a major method in image recognition, will be explained and its interpretation will be explained. In addition, depending on the time and the wishes of the participants, we may not go through it, but we will give examples of simple model implementations in Python and how they are actually used in Python, mainly by the library keras, which can be easily used.

Message from speaker (translated):

The presentation is basically a slide show. The chapters will be organised as follows:

1. What deep learning is in the first place

2. The structure and computational theory of neural nets

3. Explanation of some of the techniques that make deep learning realistic (gradient descent, error back propagation, etc.)

4. Applications of deep learning, explanation of CNN

5. Conclusion

I would like to be able to include programmatic implementations in each chapter.

I would also like to make it possible for the participants to present a problem (something they can think about a little) in the slideshow, for example, the linear inseparability problem in neural nets. However, we do not plan to take the time to solve the problems that much at the moment.

Thursday 12 September, 2nd period

Free time.

Thursday 12 September, 3rd, 4th and 5th periods

Speaker: Kisui Tazawa （Rikkyo University, College of Science, Department of Mathematics, 3rd year）

Title (translated): Actuarial mathematics

Keywords (translated): Normal, exponential, and beta distributions.

Detail (translated):

This course aims to deepen participants' understanding of continuous probability distributions, which are often used in the mathematics of actuarial exams, by dealing with questions similar to those in the exams.

Message from speaker (translated): 

First, the problem to be solved is presented. After explaining the knowledge required to solve the problem, the audience is invited to solve it.

Friday 13 September, 2nd, 3rd, 4th, 5th and 6th periods

Speaker: Shinichiro Kakuta（Rikkyo University, College of Science, Department of Mathematics, 3rd year）

Title (translated): 'Quantum mechanics' -- let us get started with linear algebra of matrices.

Keywords (translated): Complex Euclidean spaces, normal operators, self-adjoint operators, unitary operators, tensor products, spin, hydrogen atoms, momentum, angular momentum, particles, waves, fermions and bosons.

Detail (translated):

Starting from first-year linear algebra (in number vector space), the nth-order square matrix with complex components is considered as a linear operator $\mathbb{C}^n \to \mathbb{C}^n$, and some special linear operators are focused on to prepare for quantum mechanics. Then, without using any 'hard tools', the topics of quantum mechanics are explained as in the keywords.

Message from speaker (translated): 

Since linear algebra has applications in many different fields, we think that the study of linear algebra will be somewhat more fulfilling if we look at its applications rather than just studying linear algebra. Therefore, we have chosen quantum mechanics as the first topic that can be discussed as an extension of linear algebra, which is not difficult (the second is Lie algebra the next day). In terms of content, little more than a second year of undergraduate knowledge is required. We hope that the audience will realise that even 'rudimentary' content can explain a significant part of quantum mechanics.

Saturday 14 September, 2nd, 3rd, 4th, 5th and 6th periods

Speaker: Shinichiro Kakuta（Rikkyo University, College of Science, Department of Mathematics, 3rd year）

Title (translated): 'Lie algebras' -- let us get started with linear algebra of matrices.

Keywords (translated): Lie algebras, Jacobi identity, adjoint representations, Killing forms, semisimple Lie algebras, Cartan subalgebras, roots, coroots, Cartan integers, Cartan matrices, Dinkin diagram, representations of Lie algebras.

Detail (translated): 

Starting from the first year of linear algebra (in number vector spaces), we define the Lie algebra as the set of all of square matrices satisfying appropriate conditions, based on the fact that the abstract Lie algebra is isomorphic to the subspace of square matrices. Then, avoiding more abstract discussions than necessary, the paper explains the Lie algebra and some topics related to it, using many concrete examples.

Message from speaker (translated): 

We have chosen Lie algebra as the second topic that can be discussed as an extension of linear algebra, which is not difficult (the first was quantum mechanics the day before). Lie algebra is closely related to physics, specifically, it appears in quantum mechanics, quantum field theory, conformal field theory and so on. For example, it can be seen that the exchange relation of angular momentum operators in quantum mechanics is the same as the exchange relation of generators in the Lie algebra $\mathfrak{su}(2)$.

In terms of content, (again) little knowledge beyond the second year of undergraduate studies is required. However, if there is room, we hope to prepare some difficult and interesting topics for 'stories'.

Tuesday 17 September, 2nd, 3rd, 4th and 5th periods

Speaker: Yuta Watanabe（Rikkyo University, College of Science, Department of Mathematics, 3rd year）

Title (translated): Rediscussion of General Topology A

Keywords (translated): Metric spaces, metric functions, convergence and continuous map of point sequences in distance spaces, topological spaces.

Detail (translated): 

This course deals with the rudiments of topological spaces. Specifically, the aim is to start from metric spaces, touch on the basics of metric spaces such as convergence of point sequences in metric spaces and continuous map, and then reach the definition of topological spaces. A simple knowledge of sets and maps is assumed.

Message from speaker (translated):  

Resumes are distributed to the audience, and the same slides as the resumes are copied and presented. However, blackboards may be used to some extent. There will be time for the audience to solve exercises.

Thursday 19 September, 2nd, 3rd, 4th, 5th (, 6th)

Speaker: Shinichiro Kakuta（Rikkyo University, College of Science, Department of Mathematics, 3rd year）

Title (translated): Geometry in Higher Education Mathematics

Detail (translated): 

The course will cover a wide range of topics in geometry and its extensions, as studied in university mathematics and physics departments. Specifically, the following is planned.

1. Differential Geometry Part 1 -- Curves and Surfaces, and Manifolds

2. Differential Geometry Part 2 -- The gateway to abstract and advanced differential geometry

3. Topology Part 1 -- Through Morse theory (differential topology)

4. Topology Part 2 -- Understanding Poincaré duality ( algebraic topology)

5. Topics of interest to the speaker

In particular, given that geometry and physics are inseparable in mathematics, the talk will also touch on the connection with (theoretical) physics. In particular, we plan to explain the connection with general relativity, string theory and superstring theory.

Message from speaker (translated): 

I am not sure how much of the content I will actually be able to cover as I am still learning a lot myself. The level of difficulty of the content will be very high except for a few parts, but rest assured it will be the same for the speaker. However, I will try to avoid unnecessary abstraction and give concrete examples whenever possible. Also, the aim is to 'know' a lot of topics without (in most cases) going into the proofs of theorems.

Content

Speaker: Shinichiro Kakuta（Rikkyo University, College of Science, Department of Mathematics, 3rd year）

Title (translated): Riemann Integral and Reconsideration of it -- Rigorous discussion of integrals and areas and some applications

Abstract (translated): 

The Riemann integral is a rigorous formulation of the integral of the first function constructed by Bernhard Riemann. The first goal of this colloquium is to review the (ε, δ)-definition and then use it to discuss the basics of the Riemann integral (division of intervals, Riemann sums, the meaning of Riemann integrability and some equivalence paraphrases, relation to Darboux integrals, etc.).

The second aim is to prove the Riemann-Lebesgue theorem and to show how the Dirichlet integral (see appendix) can be computed without using complex function theory. We want to show that, although it is much more laborious and tedious than thinking in terms of complex line integrals, one can obtain values even by calculating muddy real functions by preparing a supplementary problem.

The third objective is to discuss what it means for a set to have a volume. We introduce the notion of a zero set and consider what it means for a set to be volume definite and when a bounded function on a volume definite bounded set is Riemann integrable. This discussion is similar to the theory of measures and may help those interested in learning Lebesgue integrals to learn the concept of 'measure'.

Content

Tuesday 19 February

10:45-12:15

Self-introductions, small talk, etc.

Tuesday 19 February

13:15-19:50

Shinichiro Kakuta（Rikkyo University, College of Science, Department of Mathematics, 2nd year）

Title (translated): Field Theory and Geometry I

Keywords (translated): vector fields, work (mechanics), line integrals, scalar potentials, curves, parameters, continuity equation, surfaces, coordinate transformations, divergence theorem, Stokes’ theorem.

Tuesday 5 March

10:45-12:15, 13:15-16:30

Tsubasa Kitamura（Rikkyo University, College of Science, Department of Mathematics, 2nd year）

Title (translated): Introduction to Algebra

Keywords (translated): Algebraic systems, rings, homomorphisms, ideals, unique factorization domain, Noetherian rings, etc.

Tuesday 5 March

16:40-19:50

Kisui Tazawa（Rikkyo University, College of Science, Department of Mathematics, 2nd year）

Title (translated): Continuation of ‘Probability and Statistics’ - Study for the Mathematics of Actuarial Exams.

Keywords (translated): Skewness, kurtosis, geometric distribution, first success distribution, negative binomial distribution.

Wednesday 6 March

10:45-12:15, 13:15-16:30

Keigo Horikoshi（Rikkyo University, College of Science, Department of Physics, 1st year）

Title (translated): Special relativity

Keywords (translated): Speed of light and the aether, Lorentz hypothesis, Einstein's principle of special relativity , Lorentz transformations, relativistic motion, intrinsic time, etc.

Wednesday 6 March

16:40-18:10

Kisui Tazawa（Rikkyo University, College of Science, Department of Mathematics, 2nd year）

Title (translated): Continuation of ‘Probability and Statistics’ - Study for the Mathematics of Actuarial Exams.

Keywords (translated): Skewness, kurtosis, geometric distribution, first success distribution, negative binomial distribution.

Wednesday 6 March

18:10-19:50

Free time.

Friday 22 March

10:45-12:15, 13:15-19:50

Shinichiro Kakuta（Rikkyo University, College of Science, Department of Mathematics, 2nd year）

Title (translated): Field Theory and Geometry II

Keywords (translated): Coulomb's law, Gauss's law, boundary conditions, potential, Poisson's equation, harmonic function, dielectric constant, Dirichlet problem, Biot-Savart law, linking numbers, Ampere's law, vector potential, gauge transformation, Lorentz force, electromagnetic induction, Maxwell's equations, electromagnetic waves.

Tuesday 26 March

10:45-12:15, 13:15-18:10

Yuta Watanabe（Rikkyo University, College of Science, Department of Mathematics, 2nd year）

Title (translated): Construction of numbers II

Keywords (translated): Ordered set and order relation．

Tuesday 26 March

18:10-19:50

Free time.

Content

Thursday 5 September

10:45-16:30

Tsubasa Kitamura（Rikkyo University, College of Science, Department of Mathematics, 2nd year）

Title (translated): Elementary number theory

Abstract (translated): Starting with a review of elementary number theory, the session covers indefinite equations, continued fraction, and Pell's equation.

Thursday 5 September

16:40-19:50

Yuta Watanabe（Rikkyo University, College of Science, Department of Mathematics, 2nd year）

Title (translated): Construction of numbers (I)

Abstract (translated): The main theme of the session is to define the natural numbers with Peano axioms as an introduction. The focus will be on constructing the theory by logical procedures.

9月11日（火）

10：45～16：30

Kisui Tazawa（Rikkyo University, College of Science, Department of Mathematics, 2nd year）

Title (translated): Introduction to actuarial mathematics

Abstract (translated): This course covers the field of probability in mathematics for the actuarial examinations. The aim is to deepen understanding by deriving formulae that arise in the actual solution of similar problems.

Tuesday, 11 September

16:40-19:50

Shinichiro Kakuta（Rikkyo University, College of Science, Department of Mathematics, 2nd year）

Title (translated): Introduction to complex analysis 1

Abstract (translated): The main topics will be concepts such as limits and series considered in complex numbers, and basic issues in complex function theory. In particular, the discussion of real functions can be approached prospectively using complex function theory, and topics unique to complex functions will be introduced.

Wednesday 12 September

10:45-19:50

Shinichiro Kakuta（Rikkyo University, College of Science, Department of Mathematics, 2nd year）

Title (translated): Introduction to complex analysis 2

Abstract (translated): Continuation of Introduction to Complex Analysis 1, covering material not covered in 1.