## The 2018 Mathematical Olympiad Summer Program (June 3-27), Carnegie Mellon University, Pittsburgh, PA

The 2018 Mathematical Olympiad Summer Program will take place at Carnegie Mellon University between June 3-27. 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 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), Princeton University, Ohio State University and University of Pittsburgh. Moreover, this year is for the second time when MOP welcomes 20 international students from countries such as Bulgaria, China, Czech Republic, Hong Kong, India, Romania, Ukraine and Singapore. The two amazing Romanian students are Denis Chirita and George Picu from International High School of Bucharest. Another news is the fact that two Romanians will serve as academic instructors. Also, Irina-Roxana Popescu of University of Pittsburgh is in the staff as well.

- Classrooms are Gates 5222, Margaret Morrison A14, Wean 8201, Wean 8220, Gates 4101, Gates 4102.

- Students will be separated into four groups.
- Black (22 students): approx IMO gold.
- Blue (21 students): approx IMO silver
- Green (24 students): approx IMO bronze
- Red (13 students): approx IMO honorable mention.

- The timetable will be:
- 8:30am – 10:00am (Lecture 1)
- 10:15am – 11:45am (Lecture 2)
- 7:30pm: optional-attendance evening research seminar

My schedule consists of 7 lectures and one seminar. More details are given below:

**Lecture 1**.* Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities.*

Group: Black 2

Location: Gates Hall 4102

**Lecture 2**.

*Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities*.

Group: Blue 2

Location: Wean Hall 8220

**Lecture 3**.* Romanian Olympiad (Algebra) gems*.

Location: Margaret Morison A14

**Lecture 4**.* Romanian Olympiad (Algebra) gems*.

Location: Gates Hall 5222

**Lecture 5**.* Romanian Olympiad (Algebra) gems*.

Location: Wean Hall 8220

**Seminar**. *Zeta regularization and the golden nugget: ““*.

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

Location: Stever dorm

**Lecture 6**. *Algebraic integers and applications*.

Group: Red

Location: Gates Hall 5222

**Lecture 7**. *Algebraic integers and applications*.

Group: Blue 2

Location: Wean Hall 8220

## The 2018 Joint Mathematics Meeting of the AMS & MAA, San Diego, January 10-13, 2018

The 2018 Joint Mathematics Meeting of the American Mathematical Society and Mathematical Association of America is the largest mathematics meeting in the world. More than 6000 mathematicians and math enthusiasts met in the beautiful city of San Diego, CA between January 10-13. The location is San Diego Convention Center. This is the 101th annual winter meeting of MAA and the 124rd annual meeting of AMS. My talk (slides) is part of the AMS Contributed Paper Session on Number Theory I , Wednesday, January 10, 2017, 10.15 a.m.-10.30 a.m at the Mezzanine, room 13A.

** Abstract book Program Presenters**

## The 2017 William Lowell Putnam Competition Exam at the University of Pittsburgh

The 77th annual William Lowell Putnam Mathematical Competition took place on *December 2nd in 705 Thackeray Hall*. Twelve Pitt students had the mission of solving the challenging Putnam problem sets. As usual, there were two sessions of 6 problems each. The official Pitt Team members were Andrew Tindall, Tianke Li and Haihui Zhu. Other participating students were: Matthew Gerstbrein, Abraham Harris**,** Andrew Klang, Eric Peterson, Tanmoy Sarker,** **Anthony Sicillia, Ed Terrell, Carlos Vazquez Gomez, and Lu Zhao. Below one can find this year’s Putnam problems. Congratulations to all participants!

**Problem A1**. Let be the smallest set of positive integers such that

a) is in $latex S$,

b) is in whenever is in , and

c) is in whenever is in .

Which positive integers are not in

(The set is “smallest” in the sense that is contained in any other such set.)

**Problem A2**. Let , , and

for all . Show that, whenever is a positive integer, is equal to a polynomial with integer coefficients.

**Problem A3**. Let and be real numbers with and let and be continuous functions from to such that but For every positive integer define

.

Show that is an increasing sequence with

**Problem A4**. A class with students took a quiz, on which the possible scores were Each of these scores occurred at least once, and the average score was exactly Show that the class can be divided into two groups of students in such a way that the average score for each group was exactly

**Problem A5**. Each of the integers from to is written on a separate card, and then the cards are combined into a deck and shuffled. Three players, and take turns in the order choosing one card at random from the deck. (Each card in the deck is equally likely to be chosen.) After a card is chosen, that card and all higher-numbered cards are removed from the deck, and the remaining cards are reshuffled before the next turn. Play continues until one of the three players wins the game by drawing the card numbered

Show that for each of the three players, there are arbitrarily large values of for which that player has the highest probability among the three players of winning the game.

**Problem A6**. The edges of a regular icosahedron are distinguished by labeling them How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color?

**Problem B1. **Let and be distinct lines in the plane. Prove that and intersect if and only if, for every real number and every point not on or there exist points on and on such that

**Problem B2. **Suppose that a positive integer can be expressed as the sum of consecutive positive integers

for but for no other values of Considering all positive integers with this property, what is the smallest positive integer that occurs in any of these expressions?

**Problem B3. **Suppose that

is a power series for which each coefficient is or . Show that if , then must be irrational.

**Problem B4. **Evaluate the sum

.

(As usual, denotes the natural logarithm of )

**Problem B5. **A line in the plane of a triangle is called an *equalizer* if it divides into two regions having equal area and equal perimeter. Find positive integers with as small as possible, such that there exists a triangle with side lengths that has exactly two distinct equalizers.

**Problem B6. **Find the number of ordered -tuples such that are distinct elements of and

is divisible by .

## Job Application Materials

I am currently a 6th year graduate student at the University of Pittsburgh, Department of Mathematics and I shall be graduating in June, 2018. I am currently on the job market for Fall 2018. My advisors are professors Piotr Hajlasz and William C. Troy. Most of my research is centered around special values of L-functions and multiple zeta functions which play an important role at the interface of analysis, number theory, geometry and physics with applications ranging from periods of mixed Tate motives to evaluating Feynman integrals in quantum field theory. I employ methods from real analysis and special functions. Broadly I am interested in analytic number theory and real, geometric, functional and harmonic analysis even with some PDEs flavour in it. Feel free to contact me at cel47@pitt.edu or lupucezar@gmail.com. For more details, you can find below some of my application materials:

- CV
- Resume
- List of Publications
- Research Statement
- Teaching Statement
- Teaching Portfolio
- Teaching Challenge
- Teaching Videos
- Some Evaluations
- Cover Letter
- Diversity Statement
- Recent and Upcoming Talks:

- Algebra Seminar, University of Connecticut, February 21, 2018
- Algorithms, Combinatorics and Optimization Seminar, Carnegie Mellon University, February 1, 2018
- Analysis Seminar, Cornell University, January 29, 2018
- Contributed Talk at Number Theory I: Joint Mathematics Meeting of AMS & MAA, San Diego, January 10-13, 2018
- Analysis Seminar, University of South Florida, December 1, 2017
- Colloquim, University of South Florida, December 1, 2017
- Algebra, Geometry and Topology Seminar, University of Pittsburgh, November 28, 2017
- Algebra & Number Theory Seminar, Texas Tech University, November 15, 2017
- Number Theory Seminar, University of California Irvine, November 9, 2017
- Special Session in Preparing Students for AMC: AMS Sectional Meeting, University of California Riverside, November 5, 2017
- Algebra, Geometry and Topology Seminar, University of Pittsburgh, October 24, 2017
- Undergraduate Mathematics Seminar, University of Pittsburgh, October 17, 2017
- Talk at the Parallel Session I: Northeastern Analysis Meeting (NEAM 2), SUNY-University at Albany, October 13-15, 2017
- “Nicolae Popescu” Number Theory Seminar, Simion Stoilow Institute of Mathematics of the Romanian Academy, September 27, 2017
- Undergraduate Mathematics Seminar, University of Pittsburgh, September 12, 2017
- Algebra, Geometry and Combinatorics Graduate Student Research Seminar, University of Pittsburgh, September 7, 2017

EDIT: **I accepted a postdoc offer at Texas Tech University, Department of Mathematics & Statistics, Lubbock, TX, USA.**

## 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 Thackeray 705.*Afternoon Session:*3:00pm-6:00pm, in Thackeray 705.

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

- Putnam Seminar-MATH 1010, Main Campus (Thackeray Hall), Fall 2016
- Coordinators: Cezar Lupu (Ph.D. student) and George Sparling (faculty)
- Invited lecturers: Roman Fedorov (faculty), Piotr Hajlasz (faculty), Thomas Hales (faculty), Bogdan Ion (faculty), Derek Orr (Ph.D. student), Cody Johnson (undergraduate student-Carnegie Mellon University)
- Webpage: https://lupucezar.wordpress.com/competitions/

Schedule and locations

**Tuesday, 5.15-7.00 PM in Thackeray Hall, room 427**

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.

**Thursday, 5.15-7.00 PM in Thackeray Hall, room 427**

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:

- R. Gelca, T. Andreescu, Putnam and Beyond, Springer Verlag, 2007.
- 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.
- 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 and 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 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

Location: Gates Hall 5222

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

Location: Gates Hall 5222

**Lecture 3**. *Real Analysis techniques in solving elementary problems*.

Group: Blue

Location: Margaret Morrison A 14 Hall

**Lecture 4**. *Algebraic integers and applications*.

Group: Green

Location: Wean Hall 5421

**Lecture 5**. *Algebraic integers and applications*.

Group: Red

Location: Wean Hall 5403

**Lecture 6**. *Real Analysis techniques in solving elementary problems*.

Group: Black 1

Location: Wean Hall 8220

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

Group: Black 1

Location: Wean Hall 8220

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

Group: Black 2

Location: Gates Hall 5222

**Lecture 9**. *Sequences, series of real numbers and inequalities*.

Group: Blue

Location: Margaret Morrison A 14 Hall

**Seminar**. *Euler’s formula(s) for Apery’s constant *.

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

Location: Stever dorm

**Lecture 10**.* Romanian Olympiad gems*.

Location: Margaret Morrison A 14 Hall

**Lecture 11**. *Romanian Olympiad gems*.

Location: Gates Hall 5222

**Lecture 12**.

*Maxima and minima in Euclidian geometry and beyond. Geometric and trigonometric inequalities*.

Group: Blue

Location: Margaret Morrison A 14 Hall

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

Location: Gates Hall 5222

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

Location: Wean Hall 5421

**Lecture 15**.

*Advanced analytic methods in number theory I.*

Location: Gates Hall 5222

**Lecture 16**. *Advanced analytic methods in number theory II.*

Location: Gates Hall 5222

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

Location: Wean Hall 5403

**Lecture 18**. *Romanian Olympiad gems*.

Location: Wean Hall 5421

**Lecture 19**. *Romanian Olympiad gems*.

Location: Wean Hall 5403

## 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.

Conference poster Conference Schedule Title and Abstract List