Cezar Lupu

Primes constitute the holy grail of analytic number theory, and many of the famous theorems and problems in number theory are statements about primes. Analytic number theory is a branch of number theory that uses techniques from analysis to solve problems about the integers. Its origins go back at least to Euler’s proof of the infiniteness of the set of prime numbers using the zeta function — to be called Riemann’s zeta function in the 19th. century — and was pursued notably by, Dirichlet (1837) (through his work on primes in arithmetic progression in which he introduced systematically L-functions) and, in a different direction, by Gauss when he counted the number of integral points within a circle of large radius.

In analytic number theory texts there is a frequent occurrence of asymptotic formulas, in which a complicated, not well understood function is approximated by a simple, well understood function, and an estimate for the order of magnitude for the error is given. In other words, we will study different number theoretical objects such as arithmetical functions dwelling more on the analytical techniques such as asymptotic estimates and summation methods.

Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique.

Multiplicative number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number theorem and Dirichlet’s theorem on primes in arithmetic progressions.

Additive number theory is concerned with the additive structure of the integers, such as Goldbach’s conjecture that every even number greater than 2 is the sum of two primes. One of the main results in additive number theory is the solution to Waring’s problem.

Video Lecture 1–The big picture of analytic number theory

This Spring Semester, I am teaching again Creative Problem Solving at the Qiuzhen College, Tsinghua University.

This course is an elective course for pre-college students who want to enter Qiuzhen College. The purpose of this course is to teach important skills in problem solving that are not taught in a systematic way in any other course.  These skills are extremely valuable in preparing students for jobs and for graduate-level research.

The teaching style will be a mixture of a lecture and a problem-solving session. Training will involve the study of problems from previous Putnam competitions, for which this course can be regarded as a useful preparation. An attempt will be made to look for unifying mathematical ideas. General strategies for solving problems will also be discussed. The ultimate goal of this course is to expose students to the Shadow Putnam Competition which will take place on Sunday, December 8th. All Qiuzhen College students are invited to solve the challenging problems from the 3rd Shadow Putnam Competition.

More details about the course can be found on my teaching page.

Course Syllabus

This Fall Semester, I am teaching a YMSC course (Creative Problem Solving) at the Qiuzhen College, Tsinghua University.

This course teaches important skills in problem solving that are not taught in a systematic way in any other course.  These skills are extremely valuable in preparing students for jobs and for graduate-level research.

The teaching style will be a mixture of a lecture and a problem-solving session. Training will involve the study of problems from previous Putnam competitions, for which this course can be regarded as a useful preparation. An attempt will be made to look for unifying mathematical ideas. General strategies for solving problems will also be discussed. The ultimate goal of this course is to expose students to the Shadow Putnam Competition which will take place on Sunday, December 8th. All Qiuzhen College students are invited to solve the challenging problems from the 3rd Shadow Putnam Competition.

More details about the course can be found on my teaching page.

Course Syllabus

COmbinatorial Number Theory and Connected Topics (CONTACT-III) is a two-day ONLINE conference (March 16 – March 17, 2024). The main aim of this conference is to create an opportunity for young researchers to get an overview of the current research trends in both combinatorics and number theory. In previous years we organized CONTACT-I, CONTACT-II and it was a great success. You can find out the details below.

Plenary Speakers:

Arvind Ayyer, Indian Institute of Science, Bangalore
Atul Dixit, Indian Institute of Technology, Gandhinagar
Fan Ge, College William & Mary
Krishnan Sivasubramanian, Indian Institute of Technology, Bombay

Invited Speakers:

Cezar Lupu, Beijing Institute of Mathematical Sciences and Applications & Tsinghua University
Bibekanada Maji, Indian Institute of Technology, Indore
Sudhir Pujahari, National Institute of Science Education and Research, Bhubaneswar
Manjil Suika, Ahmedabad University
Rajat Gupta, University of Texas at Tyler

Organizing Committee:

Arindam Roy, University of North Carolina at Charlotte
Eshita Mazumdar, Ahmedabad University
Debika Banerjee, Indraprashta Institute of Information Technology Delhi

Pittsburgh Links among Analysis and Number Theory (PLANT) is a joint conference between University of Pittsburgh and Carnegie Mellon University. The purpose of the conference is to bring together representatives of two disciplines with multiple points of interface: number theory and analysis. This is funded by the National Science Foundation (DMS-2334874).

The talks are open to the public. See you at PLANT!

Invited Speakers:

Olivia Beckwith, Tulane University
Alex Iosevich, University of Rochester
Robert Lemke Oliver, Tufts University
Zane Li, North Carolina State University
Benjamin Linowitz, Oberlin College
Cezar Lupu, Beijing Institute of Mathematical Sciences and Applications 
Amita Malik, Pennsylvania State University
Alexandria Rose, Institute for Advanced Studies
Caroline Turnage-Butterbaugh, Carleton College

Organizing Committee:

Theresa Anderson, Carnegie Mellon University
Armin Schikorra, University of Pittsburgh
Carl Wang-Erickson, University of Pittsburgh

GADEPs = Geometry, Arithmetic and Differential Equations of Periods.

Title: Multiple zeta values: from arithmetic to geometry and physics
Abstract: In this talk, we give a survey on multiple zeta values with an emphasis on their applications ranging from arithmetic properties of odd zeta values to Dirichlet eigenvalues in a regular polygon and periods of zig-zag graphs in quantum field theory. We explore various identities involving different families of multiple zeta values and we emphasize their importance in proving some very important conjectures.

Video: https://www.youtube.com/watch?v=r0xxieV0Kg8

The Second IWMZVRF (2nd International Workshop on Multiple Zeta Values and Related Fields) will take place in Zhejiang Sci-Tech University, Hangzhou, China, August 4-6, 2023.

The goal of 2nd IWMZVRF is to promote, encourage, and provide a forum for the academic exchange of ideas and recent research works on Multiple Zeta Values and Related Fields. The workshop will present new results and future challenges, in a series of keynote lectures.

Keynote Speakers

Kam Cheong Au, University of Bonn (区锦昌,波恩大学)
Masanobu Kaneko, Kyushu University (金子昌信,九州大学)
Li Lai, Peking University (赖 力,北京大学)
Zhonghua Li, Tongji University (李忠华,同济大学)
Cezar Lupu, Yanqi Lake Beijing Institute of Mathematical Sciences and Applications (Cezar Lupu,北京雁栖湖应用数学研究院)

Zhi-Wei Sun, Nanjing University (孙智伟,南京大学)
Xiaosheng Wu, Hefei University of Technology (吴小胜,合肥工业大学)
Hongyu Xiang, Sichuan University (向洪玉,四川大学)
Houyi Yu, Southwest University (喻厚义,西南大学)
Bin Zhang, Sichuan University (张 斌,四川大学)
Jianqiang Zhao, The Bishop’s School (赵健强,毕夏普学校)
Yajun Zhou, Peking University (周亚俊,北京大学)


Organizing Committee

Jiangtao Li, Central South University (李江涛,中南大学)
Zhonghua Li, Tongji University (李忠华,同济大学)
Weiping Wang, Zhejiang Sci-Tech University (王伟平,浙江理工大学)
Ce Xu, Anhui Normal University (徐  策,安徽师范大学)

This Summer, between July 11th and August 18th, I will be teaching a course, Analysis and Number Theory at The 2023 Yau Tsinghua Mathcamp. I will be assisted in this endevour by my collegues Dongsheng Wu 吴冬生 and Tianyu Zhang 张天羽. The syllabus is available and the course outline is listed below.

• Chapter 0. What is the connection between analysis and number theory?
  The big picture.
  
• Chapter 1. Infinite sequences and series of real numbers. Convergence and
  examples. Convergence criteria.
  
• Chapter 2. Derivatives and integrals of functions on the real line and their
  properties.
  
• Chapter 3. Sequences and series of real numbers. Pointwise and uniform
  convergence.
  
• Chapter 4. Asymptotics and summation formulas.
  
• Chapter 5. Arithmetic functions. Elementary properties and asymptotic
  estimates.
  
• Chapter 6. Prime numbers and their properties and distribution. Elementary properties.
  
• Chapter 7. The Riemann zeta function for real argument and its special
  values.

Moreover, I will assign 11 research projects during my course. These projects involve applications of famous combinatorial results, but will also include applications in analysis, topology, linear algebra or number theory.

This math camp is organized by the the Department of Mathematical Sciences of Tsinghua University, Yau Mathematical Sciences Center and Yanqi Lake Beijing Institute of Mathematical Sciences and Applications.

The Mathcamp is designed for students who love mathematics, are passionate about this subject, and have some basic competency in this respect. This event aims to enable students to learn about mathematics of modern times and have access to modern mathematical research. The Mathcamp mainly invites outstanding teenagers. We will rigorously select 70-80 students from those who have submitted applications for the Mathcamp held at Tsinghua University. These students with potential are expected to devote themselves to mathematical research in the future through the enlightenment by this summer camp so that our goal of training a new generation of young mathematical research talents can be achieved.

There will be offered three mathematical courses for four weeks, including linear algebra, algebraic combinatorics, and analysis and topology. These courses cover essential tools and applications for modern scientific research. The Mathcamp is not limited to standard classroom learning. Some courses are also combined with special research projects and students are encouraged to deliver special lectures and publish their research results in the last few days of the camp. During the Mathcamp, a tutorial class will be arranged each afternoon. Coaches will lead students in discussing about and doing exercises or talking about the special research.

On behalf of the organizing committee, and with the support of the International Consortium of Chinese Mathematicians, Beijing Association of Science and Technology, Children and Youth Science Center of China Association of Science and Technology and Yau Mathematical Sciences Center of Tsinghua University, Yanqi Lake Beijing Institute of Mathematical Sciences (BIMSA) inaugurated the 1st International Mathematics Summer Camp-#IMSC23.

The 1st International Mathematics Summer Camp is a two weeks program that is designed to strengthen high school students’ problems solving skills in elementary mathematical fields such as algebra, combinatorics, geometry and number theory. The purpose of this camp is to bring together and train the most passionate students from all around the world for international competitions such as the International Mathematical Olympiad (IMO). Academic activities will include lectures, problem sessions, mock-tests, and evening seminars, among others.

IMSC23 is the largest mathematical camp of this kind ever organized, and it reunites students, leaders, instructors, guest speakers, and staff from 28 countries from all over the world. The camo is run by Cezar Lupu (Academic director) and Supanat Kamtue (Associate director) and the host is the Yanqi Lake Beijing Institute of Mathematical Sciences and Applications (BIMSA), and it will take place between June 18th-July 6th, 2023, in Beijing, China.

The participating teams/countries include the following:

Armenia Azerbaijan Botswana Brazil Bulgaria Canada China Estonia Ghana Hungary Italy Iran Kyrgyzstan Mexico Mongolia Poland Romania (unofficially) Rwanda Saudi Arabia South Africa Serbia Thailand Uzbekistan

We have assembled a team of instructors who are very well-known for their dedication in preparing high-school students for the highest level competition such as the International Mathematical Olympiad (IMO). These instructors are the following:

Cezar Lupu-Academic Director, Yanqi Lake Beijing Institute of Mathematical Sciences and Applications (BIMSA), China

Supanat (Phil) Kamtue-Associate Director, Yau Mathematical Sciences Center (YMSC), Tsinghua University, China

Andrei Bud, Goethe University Frankfurt am Main, Germany

Stijn Cambie, Institute for Basic Science-Daejeon, South Korea

Joao Cesar Campos-Vargas, Stanford University, United States

Octav Dragoi, Citadel LLC-London, United Kingdom

Zilin Jiang, Arizona State University, United States

Wei Luo, Zhejiang University, China

Tudor Popescu, Brandeis University, United States

Zhuo Qun (Alex) Song, University of Illinois at Urbana Champaign, United States

Ben Yang, Beijing, China

Junyao Peng, Princeton University, United States

Also, there will be seminars delivered by outstanding guest speakers such as

Shing-Tung Yau, Tsinghua University & Yanqi Lake BIMSA, China (president of IMSC23)

Zhengwei Liu, Tsinghua University & Yanqi Lake BIMSA, China (vice-president of IMSC23)

Caucher Birkar, Tsinghua University & Yanqi Lake BIMSA, China

Nicolai Reshetikhin, Tsinghua University & Yanqi Lake BIMSA, China

Highlights of the camp include the following:

27 (IMO and non-IMO) participating teams from Europe, Asia, Africa and America
180 lectures delivered by instructors and team leaders
4 mock tests for the high-school students participating at the camp
4 seminars delivered by outstanding scholars
IMSC23 competition (IMO-style) which will take places in the last weekend of the camp
group study (students and their leaders)
excursions to the Mutianyu Great Wall of China and other scenic areas around Beijing