| Academic Year |
2026Year |
School/Graduate School |
School of Science |
| Lecture Code |
HX206000 |
Subject Classification |
Specialized Education |
| Subject Name |
数学特別講義（Gauss測度が誘う解析学の世界） |
Subject Name
（Katakana） |
スウガクトクベツコウギ（ガウスソクドガイザナウカイセキガクノセカイ） |
Subject Name in
English |
Special Lectures in Mathematics(An exploration of analysis through Gaussian measures) |
| Instructor |
To be announced.,OKAMOTO MAMORU |
Instructor
(Katakana) |
タントウキョウインミテイ,オカモト マモル |
| Campus |
Higashi-Hiroshima |
Semester/Term |
4th-Year, Second Semester, Second Semester |
| Days, Periods, and Classrooms |
(2nd) Inte |
| Lesson Style |
Lecture |
Lesson Style
(More Details) |
Face-to-face |
| blackboard |
| Credits |
1.0 |
Class Hours/Week |
|
Language of Instruction |
J
:
Japanese |
| Course Level |
4
:
Undergraduate Advanced
|
| Course Area（Area） |
25
:
Science and Technology |
| Course Area（Discipline） |
01
:
Mathematics/Statistics |
| Eligible Students |
|
| Keywords |
Gaussian measures, Ornstein-Uhlenbeck semigroup, Functional inequalities, High-dimensional probability) |
| Special Subject for Teacher Education |
|
Special Subject |
|
Class Status
within Educational
Program
(Applicable only to targeted subjects for undergraduate students) | |
|---|
Criterion referenced
Evaluation
(Applicable only to targeted subjects for undergraduate students) | |
|---|
Class Objectives
/Class Outline |
In recent years, high-dimensional probability has garnered significant attention as a burgeoning interdisciplinary field at the intersection of statistics and data science. A central challenge in this area is the tail probability estimation for the norms of high-dimensional random vectors--a problem deeply intertwined with the phenomenon of measure concentration. Functional inequalities, which are indispensable in the study of metric measure spaces and optimal transport, play a pivotal role here as well. This course focuses specifically on Gaussian measures, exploring key functional inequalities--including the logarithmic Sobolev inequality, Poincaré inequality, Borell-TIS inequality, isoperimetric inequality, and Brunn-Minkowski inequality--along with their diverse applications in high-dimensional probability. |
| Class Schedule |
(i) Gaussian measures and the Slepian inequality
(ii) Heat semigroup and the Ornstein-Uhlenbeck semigroup
(iii) Brownian motion and the Ornstein-Uhlenbeck process
(iv) Gradient estimates for the Ornstein-Uhlenbeck semigroup
(v) Functional inequalities for Gaussian measures
(vi) Application to high-dimensional probability: Operator norm estimates for Gaussian random matrices
The above topics are tentative and do not represent a chronological schedule. The syllabus may be adjusted based on the students' comprehension and feedback.
|
Text/Reference
Books,etc. |
Further instructions will be given in due course during the lectures. |
PC or AV used in
Class,etc. |
|
| (More Details) |
|
| Learning techniques to be incorporated |
|
Suggestions on
Preparation and
Review |
Further instructions will be given in due course during the lectures. |
| Requirements |
|
| Grading Method |
The primary basis for evaluation will be reports. As the significance of this intensive course lies in attending and actively listening to the lectures in person, students who attend less than two-thirds of the total sessions will not be eligible for grading. |
| Practical Experience |
|
| Summary of Practical Experience and Class Contents based on it |
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| Message |
|
| Other |
|
Please fill in the class improvement questionnaire which is carried out on all classes.
Instructors will reflect on your feedback and utilize the information for improving their teaching. |