### CONTACT INFO

OFFICE 711 (7TH FLOOR), THACKERAY HALL
E-MAIL: lupucezar@gmail.com, cel47@pitt.edu

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## The Putnam Seminar (MATH-1010), Fall 2017, University of Pittsburgh

Information about the competition and seminar (course description)

The William Lowell Putnam Mathematical Competition is the premiere competition for undergraduate students in North America. More than 500 universities compete in this contest organized by the Mathematical Association of America (MAA).

### Putnam Examination (2017)

The Seventy Eight Putnam Examination will be held on Saturday, December 2nd, 2017.

It will consist of two sessions of three hours each:

• Morning Session: 10:00am-1:00pm, in location TBD.
• Afternoon Session: 3:00pm-6:00pm, in location TBD.

The test is supervised by faculty members of each participating school. Every problem is graded on a scale of 0-10. The problems are usually listed in increasing order of difficulty, with A1 and B1 the easiest, and A6 and B6 the hardest. Top 5 scoring students on the Putnam exam are named Putnam Fellows. A student can take this exam maximum four times and the Putnam official team of the university consists of 3 members.
The purpose of this class is to provide a comprehensive introduction into the world of problem solving in different branches of mathematics such as: real & complex analysis, linear algebra, abstract algebra, combinatorics, probability, geometry and trigonometry and number theory.

The 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. By the end of this course, students should develop fundamental problem solving skills, and become accustomed to concentrating on a problem for an extended period of time. Indeed, this seminar concentrates on the raw creative problem-solving skills which can serve as an essential ingredient in almost every field of activity.  On the other hand, starting this Fall, the Putnam seminar has honors designation.

Last but not least, since the department wants to revamp the Putnam tradition at Pitt, Thomas Hales will work with the undergraduate committee to establish a special prize for performers.

In 2015, Pitt official Team ranked 24th in the nation and this marks the best performance since 2002. More details about this can be found here and here.

Course number, lecturers and webpage
Schedule and locations


This is a lecture given by the instructor on a certain topic. The students will learn different concepts and techniques. Moreover, the lecturer will also present solutions of some problems and will assign other problems as homework for the students.

This is more like a recitation rather than a lecture. The students will meet and discuss with the coordinators the problems assigned by the lecturer as homework.

Syllabus, grading criteria and references


The grade will be determined by the following three factors:

• Seminar attendance: 20%
• Homework and seminar activity: 60%
• Participation in the Putnam exam: 20%

Any student must attend at least 10/14 seminars to get full credit. Homework will be assigned weekly and posted on the teaching section of my webpage (https://lupucezar.wordpress.com/teaching/) at the end of each lecture on Tuesdays. It will consist of 3-4 problems A1-B1 from previous Putnam exams. The homework will be discussed in the recitation on Thursday and will be placed in Cezar’s mailbox by Friday. It will be returned graded the following week. The participation in the Putnam exam is mandatory for any student who wants to get full credit.

The main references include the following:

1. R. Gelca, T. Andreescu, Putnam and Beyond, Springer Verlag, 2007.
2. K. Kedlaya, B. Poonen, R. Vakil- The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions and Commentary, The Mathematical Association of America, Washington, D.C., 2002.
3. L. Larson, Problem-Solving Through Problems, Springer Verlag, 1983.
Detailed schedule (lectures & recitations)

Week 1. (Elementary) Algebra I

Abstract: This seminar will cover problems on topics such as algebraic identities and inequalities.

Lecturer: Cezar Lupu
Date: August 29

Recitation instructors: Cezar Lupu & George Sparling
Date: August 31

Week 2. (Elementary) Algebra II

Abstract: This seminar will focus more on mathematical induction, functional equations and polynomials (integer polynomials, roots of polynomials).

Lecturer: Cezar Lupu
Date: September 5

Recitation instructors: Cezar Lupu & George Sparling
Date: September 7

Week 3. Geometry and Trigonometry

Abstract: This will cover problems on topics such as vectors, conics, quadratics, and other curves in the plane as well as trigonometric formulae.

Lecturer: Derek Orr
Date: September 12

Recitation instructors: Derek Orr and George Sparling
Date: September 14

Week 4. Abstract Algebra

Abstract: This will cover problems on topics such as groups, rings, and finite fields.

Lecturer: George Sparling
Date: September 19

Special lecture! Generating Functions and Applications

Lecturer: Cody Johnson
Date: September 21

Week 5. Linear Algebra I

Abstract: This will cover topics on $2\times 2$ and $3\times 3$ matrices and determinants.

Lecturer: Bogdan Ion
Date: September 26

Recitation instructors: Bogdan Ion & George Sparling
Date: September 28

Week 6. Linear Algebra II

Abstract: This will cover problems on topics such as vectors spaces, linear transformations, characteristic and minimal polynomials, eigenvalues, eigenvectors.

Lecturer: Cezar Lupu
Date: October 3

Recitation instructors: Cezar Lupu & George Sparling
Date: October 5

Week 7 Number Theory I

Abstract: This will cover problems on topics such as integer-valued sequences and functions, congruences, divisibility and arithmetic functions.

Lecturer: Thomas Hales
Date: October 10

Lecturer: Thomas Hales
Date: October 12

Week 8. Number Theory II

Abstract: This will cover problems on topics such as quadratic residues and diophantine eqations.

Lecturer: Roman Fedorov
Date: October 17

Recitation instructors: Roman Fedorov & George Sparling
Date: October 19

Week 9. Real Analysis I

Abstract: This will cover problems on topics such as sequences and series of real numbers.

Lecturer: Cezar Lupu
Date: October 24

Recitation instructors: Cezar Lupu & George Sparling
Date: October 26

Week 10. Real Anaysis II

Abstract: This will cover problems on topics such as intermediate value property, continuity and differentiability of functions of a single variable.

Lecturer: Cezar Lupu
Date: October 31

Recitation instructors: Cezar Lupu & George Sparling
Date: November 2

Week 11. Combinatorics

Abstract: This will cover problems on topics combinatorial arguments in set theory and geometry, graph theory, binomial identities and counting strategies.

Lecturer: Bogdan Ion
Date: November 7

Recitation instructors: Bogdan Ion & George Sparling
Date: November 9

Week 12. Real Analysis III

Abstract: This will cover problems on topics such as Riemann integral and continuity of integrals.

Lecturer: Cezar Lupu
Date: November 14

Recitation instructors: Cezar Lupu & George Sparling
Date: Novermber 16

Week 13. Problems and Theorems in Linear Algebra

Abstract: This will cover some special topics in linear algebra and beyond.

Lecturer: Cezar Lupu
Date: November 21

Thanksgiving break: No recitation this week!

Week 14. Real Analysis IV

Abstract: This will cover problems on topics such as applications of multivariable calculus.

Lecturer: Cezar Lupu
Date: November 28

Special lecture! Problems and Theorems in Real Analysis

Lecturer: Piotr Hajlasz
Date: November 30

Week 15. The 2017 Putnam Competition-Problems discussion

Abstract: This week we discuss the the 2017 Putnam exam.

Lecturer: Cezar Lupu
Date: December 5

Recitation instructors: Cezar Lupu & George Sparling
Date: December 7

THE 2017 PUTNAM SEMINAR POSTER

## The 2017 Mathematical Olympiad Summer Program, Carnegie Mellon University, Pittsburgh, PA, June 7-July 1

The 2017 Mathematical Olympiad Summer Program will take place at Carnegie Mellon University between June 7-July 1. The camp is organized by the Mathematical Association of America  and it is run by the CMU faculty Po-Shen Loh (director) with the help of its associate director Razvan Gelca of Texas Tech University. They will be accompanied by other instructors (faculty, postdocs and Ph.D. students) from schools such as Massachusetts Institute of Technology (MIT), Stanford University, Harvard University, Columbia University, Carnegie Mellon University (CMU), and University of Pittsburgh.  Moreover, this year is for the second time when MOP welcomes 16 international students from countries such as China, Hong Kong, India, Romania, Russia, and Singapore. The two amazing Romanian students are Ciprian Bonciocat and Mihnea Ocian from Tudor Vianu High School of Bucharest.  Another news is the fact that four Romanians will serve as academic instructors. Besides the associate director, Razvan Gelca and myself, Bogdan Ion and Irina-Roxana Popescu of University of Pittsburgh are in the staff as well. This is for the third time in the history of MOP when this happens. The last two times when MOP had four Romanian instructors was back in 2002 and 2016.

Classrooms are Wean 5403, Wean 5421, Margaret Morrison A14, Wean 8220, Gates 5222.

• Students will be separated into four groups.
• Black (24): USAMO winners and IMO team and many IMO-level students from other countries. The group will split dynamically into two parts for each class.
• Blue (14): next top few from USAMO
• Green (13) / Red (24): students in grades 9 and 10, plus girls, split into two groups. Green includes all returning students, and will be faster.
• Red students will come with no prior MOP experience. Black level is quite impressive.
• The timetable will be:
• 8:30am – 10:00am (Lecture 1)
• 10:15am – 11:45am (Lecture 2)
• 1:15pm – 2:45pm (Lecture 3), or 1:15pm – 5:45pm
• 7:30pm: optional-attendance evening research seminar

My schedule consists of 19 lectures and one seminar. More details are given below:

Lecture 1. Real analysis techniques in solving elementary problems I.

Group: Black 2

Time and date: 8.30-10 AM,June 7

Location: Gates Hall 5222

Lecture 2. Real analysis techniques in solving elementary problems II.

Group: Black 2
Time and date: 10.15-11.45 AM, June 9

Location: Gates Hall 5222

Lecture 3. Real Analysis techniques in solving elementary problems.

Group: Blue

Time and date: 1.15-2.45 PM, June 9

Location: Margaret Morrison A 14 Hall

Lecture 4. Algebraic integers and applications.

Group: Green

Time and date: 8.30-10 AM, June 12

Location: Wean Hall 5421

Lecture 5. Algebraic integers and applications.

Group: Red

Time and date: 10.15-11.45 AM, June 12

Location: Wean Hall 5403

Lecture 6. Real Analysis techniques in solving elementary problems.

Group: Black 1

Time and date: 8.30-10 AM, June 13

Location: Wean Hall 8220

Lecture 7. Sequences, series of real numbers and inequalities.

Group: Black 1

Time and date: 8.30-11, June 15

Location: Wean Hall 8220

Lecture 8. Sequences, series of real numbers and inequalities.

Group: Black 2

Time and date: 1.15-2.45 PM, June 15

Location: Gates Hall 5222

Lecture 9Sequences, series of real numbers and inequalities.

Group: Blue

Time and date: 8.30-10 AM, June 16

Location: Margaret Morrison A 14 Hall

Seminar. Euler’s formula(s) for Apery’s constant $\zeta(3)$.

Group: Black (1 & 2), Blue, Red & Green

Time and Date: 8-9 PM, June 17

Location: Stever dorm

Group: Blue
Time and date: 8.30-10 AM, June 20

Location: Margaret Morrison A 14 Hall

Group: Black 2
Time and date: 1.15-2.45 PM, June 21

Location: Gates Hall 5222

Lecture 12. Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities.

Group: Blue

Time and date: 8.30-10 AM, June 26

Location: Margaret Morrison A 14 Hall

Lecture 13. Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities.

Group: Black 2
Time and date: 8.30-10 AM,June 27

Location: Gates Hall 5222

Lecture 14. Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities.

Group: Green
Time and date: 1.15-2.45 PM, June 27

Location: Wean Hall 5421

Lecture 15. Advanced analytic methods in number theory I.
Group: Black 2
Time and date: 8.30-11 AM, June 28

Location: Gates Hall 5222

Lecture 16. Advanced analytic methods in number theory II.

Group: Black 2
Time and date: 8.30-11 AM, June 29

Location: Gates Hall 5222

Lecture 17. Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities.

Group: Red
Time and date: 1.15-2.45 PM, June 29

Location: Wean Hall 5403

Group: Green
Time and date: 8.30-10 AM, June 30

Location: Wean Hall 5421

Group: Red
Time and date: 10.15-11.45 AM, June 30

Location: Wean Hall 5403

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## The 3rd Algebra, Geometry and Topology Graduate Student Conference, June 2-4, Philadelphia, PA

The 3rd Annual Graduate Student Conference in Algebra, Geometry and Topology will take place between June 3-5 at Temple University in Philadelphia, PA. This meeting reunites Ph.D. students and postdoctoral scholars from the most prestigious universities from all around US and Canada. The organizers are graduate students and faculty from the department of mathematics from Temple University. The talks were delivered by Ph.D. students from universities such as Princeton, University of Pennsylvania, Tufts, University of Virginia, Cornell, Caltech, University of Maryland College Park, University of Iowa, University of Chicago, University of New Hampshire, University of Pittsburgh, Dartmouth College, Purdue University, University of Illinois Chicago, CUNY, University of California Davis, North Carolina State University, University of Toronto and University of Michigan.

## Overview of the Calculus III (MATH 0240) course, University of Pittsburgh, 1st 6weeks, Summer 2017

This Summer I shall be teaching Calculus III (MATH 0240) course at the University of Pittsburgh, between May 15-June 25. Mainly, we shall cover vectors, vector functions and space functions, arc length and curvature, functions of several variables, double and triple integrals, line integrals, surface integrals and some other related topics.

This is the third sequence of three calculus courses for science and engineering students. The goal is to prepare you to make use of calculus as a practical problem-solving tool.

Prerequisite

Math 0230 or equivalent, with grade of C or better.

Lectures and Recitations

• Lectures are from Monday to Thursday, 6-7.45 PM  in 226 Benedum Hall.
• Recitations are from Monday to Thursday, 8-9 PM  in Benedum Hall. The recitation instructor for this class is Mohan Wu. His office hours are Wednesday, 3-5 PM in the MAC.
• Moreover, you can also check the regular teaching page for this class: https://lupucezar.wordpress.com/teaching/ (Calculus 3-Summer 2017).
• Quizzes and exams will appear on the teaching section above.

Office Hours

My office hours are from Monday to Thursday, 5-6 PM in the MAC. If you want to schedule an additional meeting with me in my office (Thackeray hall, room 711) do not hesitate to send me an e-mail to lupucezar@gmail.com.

References (Lecture notes, textbook, and practice exams)

1. P. Hajlasz, Lecture Notes in Calculus III, Part 1, Part 2, Part 3.
2. J. Stewart, Essential Calculus, Early Transcendentals, 2nd edition.
3. A. Athanas, Calculus 3-practice exams.
4. I. Sysoeva, Calculus 3-practice exams.
5. E. Trofimov, Calculus 3-webpage.
6. A. Yarosh, Calculus 3-webpage.

• Homework: 20% (2 homeworks of 10% each)
• Quizzes: 20% (4 quizzes of 5% each)
• Two exams: 60% (2 exams of 30% each)
• LON CAPA homework: 5% (extra credit)

Homework

I shall assign two homeworks which will appear on the teaching section. The first homework is hereEach one will be due before the exams. Additionally, one can do problems from the LON CAPA system. If you find it really annoying, you can also do this.

A: 90-100%, A-: 85-90%, B+: 80-85%, B: 75-80%, B-: 70-75%, C+: 65-70%, C: 60-65%.

Exam Dates

The first exam will be on Thursday, June 1st, 6-8 PM in Benedum 226.

The second exam will be on Thursday, June 22nd, 6-9 PM in Benedum 226.

Disability Resource Services

If you have any disability for which you are or may be requesting an accomodation, you are encouraged to contact both your instructor and the Office of Disability and Services, 216 William Pitt Union (412) 624-7890 as early as possible in the term.

## The 7th Upstate New York Number Theory Conference, Binghamton, NY, 5-7 May, 2017

The 7th Upstate New York Number Theory Conference took place at the SUNY-University of Binghamton, Binghamton, NY between 5-7 May. More than 30 number theorists and related areas affiliated with universities from USA, Canada, and Germany participated.
The conference was organized by Alexander Borisov, Jaiung Jun, Marcin Mazur,  and Adrian Vasiu of SUNY-University of Binghamton and it included plenary speakers such as Michael Filaseta, Thomas Hales, Jeffrey Lagarias, Melvyn Nathanson, Alexandra Shlapentokh, and Joseph Silverman. The 3 contributed sections involved all sorts of number theoretical aspects. More details about the conference including the abstract of the talks are given below:

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## An introduction to the Riemann zeta function, multiple zeta function, multiple zeta values (MZV or Euler-Zagier sums) and some of their special evaluations

1. The Riemann zeta function and values is defined, for $s\in\mathbb{C}$ as

$\displaystyle\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}, Res>1$.

Moreover, it has an integral representation in terms of Euler’s gamma function, $\displaystyle\zeta(s)=\frac{1}{\Gamma(s)}\int_0^{\infty}\frac{x^{s-1}}{e^x-1}dx$. It can actually be extended to a meromorphic function on the whole complex plane with a simple pole at $s=1$.

Although, it is undoubtebly the most important function in mathematics, the Riemann zeta function still keeps many misteries. The most important and impenetrable of them is the Riemann hypothesis (all non-trivial zeroes of the $\zeta(s)$ lie on the line $Res=\frac{1}{2}$) with tremendous consequences in number theory and beyond!

Let us recall what happens when we evaluate $\zeta(s)$ at integers. First, let us start with Euler’s result from 1734 which asserts that

$\displaystyle \zeta(2)=\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$.

Later, in 1740, the same Euler proved the following generalization:

where $B_{k}$ are the Bernoulli numbers and they are given by the Taylor series expansion $\displaystyle\frac{z}{e^z-1}=\sum_{k=0}^{\infty}\frac{B_{k}}{k!}\cdot z^k, |z|<2\pi$. More about Euler and the zeta values can be found in this paper.

Let us remark that the key ingredient in the classical proof of Euler’s formula is the following cotangent identity which is also due to Euler:

$\displaystyle \pi\cot(\pi x)=\frac{1}{x}+\sum_{n\geq 1}\frac{2x}{x^2-n^2}$.

Expanding the quotient inside the sum sign as a geometric series and interchanging the order of summation, we obtain the following identity:

$\displaystyle \pi\cot(\pi x)=\frac{1}{x}-2\sum_{k\geq 1}\zeta(2k)x^{2k-1}$.

Remarks.

•  Euler’s formula implies the following equality of subrings of $\mathbb{R}$:

$\displaystyle\mathbb{Q}[\zeta(2), \zeta(4), \ldots]=\mathbb{Q}[\pi^2]$.

•   Thanks to the functional equation

$\displaystyle \pi^{-s/2}\Gamma\left(\frac{s}{2}\right)\zeta(s)=\pi^{-\frac{1-s}{2}}\Gamma\left(\frac{1-s}{2}\right)\zeta(1-s)$,

one can deduce the values of $\zeta(s)$ at negative integers: $\zeta(-k)=-\frac{B_{k+1}}{k+1}$ for all $k\geq 1$. In particular, $\zeta(-2k)=0$ for all $k\geq 1$; we call these values the trivial zeros of the function $\zeta$. Also, we have $\zeta(0)=-\frac{1}{2}$ and $\zeta(-1)=-\frac{1}{12}$.

Question: What can we say about $\zeta(s)$ when $s$ is odd?

Unfortunately, not too much is known, We cannot even find a closed formula for $\zeta(2n+1)$ in terms of $\pi$. This led to the following:

CONJECTURE. (Transcedence conjecture). The numbers $\pi, \zeta(3), \zeta(5), \ldots$ are algebraically independent, that is, for each $k\geq 0$ and each nonzero polynomial $P\in\mathbb{Z}[x_{0}, x_{1}, \ldots, x_{k}]$, we have

$\displaystyle P(\pi, \zeta(3), \zeta(5), \ldots, \zeta(2k+1))\neq 0$.

## SEAM’17-The 33rd SouthEastern Analysis Meeting, March 17-19, 2017

The 33rd Southeastern Analysis Meeting (SEAM’17) took place at the University of Tennessee, Knoxville, TN between 17-19 March. More than 60 mathematicians affiliated with universities from USA, Canada and Sweden participated at the meeting.
The conference was organized by Stefan Richter and Carl Sundberg of University of Tennessee-Knoxville and it included 3 contributed sections. Although, the contributed sections involved more operator theory, it featured many other fields such as PDEs, number theory, combinatorics and probability. My talk is part of the contributed sections, and it will take place on Saturday March 18, 2017, 10:20 a.m.-10:40 a.m. in Room 524. More details about the conference including the abstract of the talks are given below:

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