Cezar Lupu

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

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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 and 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


Lecture 10. Romanian Olympiad gems.

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

Location: Margaret Morrison A 14 Hall


Lecture 11. Romanian Olympiad gems.

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


Lecture 18. Romanian Olympiad gems.

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

Location: Wean Hall 5421


Lecture 19. Romanian Olympiad gems.

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.

Conference poster                            Conference Schedule                       Title and Abstract List

 

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.

About the course

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.

Grades

Your final grade will be determined as follows:

  • 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.

Final Grade Policy

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:

  Talk abstracts                                                                          Detailed Schedule

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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 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:

\displaystyle \zeta(2k)=\sum_{n=1}^{\infty}\frac{1}{n^{2k}}=(-1)^{k+1}\frac{2^{2k-1}B_{2k}}{(2k)!}\cdot\pi^{2k},

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.  

(more…)

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:

     Plenary Speakers                 Participants                Abstract of Talks                Schedule

header-logo UTK


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Some important integral inequalities in mathematical analysis and beyond

Integral inequalities are often a very important tool in mathematical analysis, number theory, partial differential equations, differential geometry, probability, statistics, etc.

The most basic integral inequality is given by the following: Given a continuous function (Riemann integrable is sufficient) h:[a, b]\to\mathbb{R}, we have

\displaystyle\int_a^bh^2(x)dx\geq 0.

For any real number \lambda, consider h(x)=f(x)-\lambda g(x), h\in C([a, b]). After computations, this us give us the celebrated Cauchy-Schwarz’s inequality,

 \displaystyle\left|\int_a^bf(x)g(x)dx\right|\leq\left(\int_a^bf^2(x)dx\right)^{1/2}\left(\int_a^bg^2(x)dx\right)^{1/2}-\textbf{(Cauchy-Schwarz}).

Moreover, the Cauchy-Schwarz’s inequality can be obtained by integrating over [a, b]\times [a, b] the symmetrizing functions u\mapsto f(x)g(y) and v\mapsto f(y)g(x) together with the elementary inequality 2uv\leq u^2+v^2 and Fubini’s theorem.

A generalization of Cauchy-Schwarz’s inequality was given by Rogers (1888) and Holder (1889),

\displaystyle\int_a^b|f(x)g(x)|dx\leq\left(\int_a^b|f(x)|^pdx\right)^{1/p}\left(\int_a^b|g(x)|^qdx\right)^{1/q}-\textbf{(Holder}),

where p, q>1 such that \frac{1}{p}+\frac{1}{q}=1. This follows from Young’s inequality, \displaystyle \frac{a^p}{p}+\frac{b^q}{q}\geq ab for all a, b\geq 0 applied for a=\frac{|f(x)|}{||f||_{p}} and b=\frac{|g(x)|}{||g||_{q}} and an integration afterwards. Here ||f||_{p} is the p-norm and it is defined as \displaystyle ||f||_{p}:=\left(\int_a^b|f(x)|^pdx\right)^{1/p} .  An self-extension of Holder’s inequality reads as follows:

\displaystyle\left(\int_a^b|f(x)g(x)|dx\right)^{1/r}\leq\left(\int_a^b|f(x)|^pdx\right)^{1/p}\left(\int_a^b|g(x)|^qdx\right)^{1/q}-\textbf{(extended Holder}),

where p, q, r>0 such that \displaystyle \frac{1}{p}+\frac{1}{q}=\frac{1}{r}. This last inequality follows easily from the first Holder inequality applied for x\mapsto |f(x)|^r and x\mapsto |g(x)|^r with exponents \displaystyle p_{1}=\frac{p}{r}, q_{1}=\frac{q}{r}. By an easy induction, the above inequality can be generalized as follows:

\displaystyle\left(\int_a^b |f_{1}(x)f_{2}(x)\ldots f_{k}(x)|^rdx\right)^{1/r}\leq \left(\int_a^b|f_{1}(x)|^{p_{1}}\right)^{1/p_{1}}\left(\int_a^b|f_{2}(x)|^{p_{2}}\right)^{1/p_{2}}\ldots \left(\int_a^b|f_{k}(x)|^{p_{k}}\right)^{1/p_{k}}-\textbf{(generalized Holder}),

where \displaystyle p_{i}, r\geq 1 and \displaystyle\sum_{i=1}^k\frac{1}{p_{i}}=\frac{1}{r}. Applications of Holder’s inequality are the following inequalities due to Minkovki:

\displaystyle\left(\int_a^b|f(x)+g(x)|^pdx\right)^{1/p}\leq \left(\int_a^b|f(x)|^pdx\right)^{1/p}+\left(\int_a^b|g(x)|^pdx\right)^{1/p}-\textbf{(Minkowski}),

and

\displaystyle\left(\int_a^b\left|\int_c^d f(x, y)dy\right|^pdx\right)^{1/p}\leq \int_c^d\left(\int_a^b|f(x, y)|^pdx\right)^{1/p}dy-\textbf{(generalized Minkowski})

All these inequalities presented above are discussed in details here. Some of these inequalities such as Cauchy-Schwarz, Holder and Minkowski (with their proofs) are also presented in the video below.