## Schedule

9:30 -10:25 am - Topology (Ortiz-Navarro – EC 246)

10:35 – 11:30 am - Real Analysis (Stan - EC 246)

12:45 – 1:40 - Abstract Algebra (Roman/Lopez – EC 246)

1:50 – 2:45 - Computational Science (Overman – EA 285)

2:55 – 3:50 - Combinatorics & Probability (Johnson – MW 154)

## Topology

### Instructor: Juan Ariel Ortiz-Navarro

• Office: MW 706

• Telephone: 614- 292-4974

• E-mail: juan.ortiz35@upr.edu

• Office Hours: Tuesday, Thursday, 2:30 – 3:30 p.m.

• Classes: 9:30 – 10:25 am (EC 246)

### TA: Gabriel Jaime Montoya

• Email: Gabriel.montoya@upr.edu

• Office: MA 234

### Lessons:

1. Open Sets

2. Closed Sets

3-4. Quotient Topology

5. Continuity

6-7. Homeomorphisms

8. Manifolds

9. Orientability

10. Examples of Manifolds

11. Surfaces with boundary

12-13. Classification of 2-Manifolds

14. Euler Characteristic

15. Introduction to Knots

16. Reidemeister Moves

17. Tricolorability

18. Invariants

19. Types of Knots

20. Polynomials

## Real Analysis

### Instructor: Aurel I. Stan

• Office: MW 602

• E-mail: stan.7@osu.edu

• Phone: 614-292-5754

• Office Hours: TBD

• Classes: 10:35 – 11:30 a.m. (EC 246)

### TA: Mario Antonio Zepeda

• E-mail: Mario.zepeda@upr.edu

• Office: MA 234

### Textbooks:

• W. Rudin, Principles of Mathematical Analysis, Third Edition, McGraw- Hill, 1976

2

• G. Polya and G. Szego, Problems and Theorems in Analysis I, Springer- Verlag, 1978

### Topics:

• Inequalities: Arithmetic and Geometric Means, Bernoulli, Cauchy-Schwarz, Jensen.

• Sequences: ϵ–δ definition of convergence, Cauchy, monotone and bounded, recursively defined.

• Series: conditionally and absolutely convergent, criteria of convergence.

• Continuous Functions: ϵ–δ definition, uniform continuity, some functional equations.

### Syllabus:

• Monday, July 14, 2014: The Arithmetic–Geometric–Harmonic Means in- equality. Cauchy proof.

• Tuesday, July 15, 2014: Bernoulli and Cauchy–Schwarz inequalities.

• Wednesday, July 16, 2014: Applications.

• Thursday, July 17, 2014: Jensen inequality.

• Friday, July 18, 2014: Applications of Jensen inequality and a new look at the Arithmetic–Geometric and Bernoulli inequalities.

• Monday, July 21, 2014: Convergent sequences, ϵ–δ definition.

• Tuesday, July 22, 2014: Monotone and bounded sequences.

• Wednesday, July 23, 2014: Composition of monotone functions. Recursively defined sequences.

• Thursday, July 24, 2014: Applications.

• Friday, July 25, 2014: Cauchy sequences.

• Monday, July 28, 2014: Series. Necessary conditions for convergence.

• Tuesday, July 29, 2014: Absolutely convergent series. Cauchy condensation criterium.

• Wednesday, July 30, 2014: Applications.

• Thursday, July 31, 2014: Conditionally convergent series. Abel–Dirichlet Theorems.

• Friday, August 1, 2014: Applications.

• Monday, August 4, 2014: Continuous functions. ϵ–δ definition.

• Tuesday, August 5, 2014: Studying the continuity of continuous functions using sequences. Examples: Dirichlet and Riemann functions.

• Wednesday, August 6, 2014: Uniformly continuous functions.

• Thursday, August 7, 2014: Functional Equations involving continuous functions.

• Friday, August 8, 2014: Closing ceremony (no classes)

## Abstract Algebra

### Instructors: Cosmin Roman/Sergio López-Permouth

• Office: MW 748

• E-mail: cosmin@math.osu.edu and lopez@ohio.edu

• Phone: 614-292-4015

• Office Hours: M, T, W, TH, F: TBA and by appointment

• Classes: 12:45 – 1:40 (EC 246)

### TA: Alexander Diaz

• E-Mail: adiaz4@nd.edu

• Office: MA 236

### Syllabus.

Week 1:What is a ring?

Axiomatics, definitions, examples (and counterexamples!). We will (re-)visit concepts such as monoids, groups, sub-objects and factor objects, morphisms, kernels and images. The goal would be to arrive at modules over a ring, and some basic results and properties of modules.

Week 2: Properties of rings.

We will go firstly over arithmetic properties of elements in rings, but then also over properties concerning the lattice of ideals (e.g. the so-called chain conditions: Noetherian and Artinian).

Week 3: Division rings, fields, simple and semisimple rings.

We will go, as much as possible, over the details of Wederburn-Artin Theorem, the structure of semisimple Artinian rings.

Week 4: Applications.

Depending on the interests and readiness of the students who participate in the algebra we will follow one of the following two strategies:

- Plan A: We will attempt a Research Experience based on a very generally phrased question. The main idea stems from the fact that every time one has a binary operation on a set S, that operation induces an operation on the power set P(S) of S according to which, for any two subsets A and B of S, the set AB consists of all "products" ab with a in A and b in B. This operation is not very good on P(S) itself; for example, the product of any A with the empty set will yield the empty set as a result, ruling out the possibility of a cancellation property. On the other hand, good properties are attainable by choosing suitable subsets of P(S) which are closed under the induced operation. Many research question are easy to phrase and have various levels of difficulty as one explores properties of these "power structures".
- Plan B: If tackling a research problem is not a feasible option, we will use the fourth week to learn about the Algebraic Theory of Error-Correcting Codes, a very useful real-life application that will fit very nice with the original intention of the program to sample advanced topics in mathematics.

## Computational Science

### Instructor: Ed Overman

• Office: MW 440

• E-mail: overman@math.osu.edu

• Phone: 614-292-1046

• Office Hours: TBD

• Classes: 1:50 – 2:45 (EA 285)

### TA: Carlos Theran

• E-mail: carlos.theran@upr.edu

• Office: MA 236

### List of Topics:

Week 1 - Introduction to MATLAB/Octave:

- using it as a scalar computer.
- everything you need to know about vectors and matrices and how to use
- them.
- how to generate two-dimensional and three-dimensional plots,
- including histograms (which are very important in probability and
- statistics).
- how to program using if tests and for and while loops.
- how to write functions.

Week 2 - It is not required, but probably a good idea, to be also taking the

Combinatorics & Probability course because we will be taking some of

the problems from that course and "exploring" them computationally.

This is a good way to increase your knowledge of MATLAB using

interesting problems.

Weeks 3,4 - Work on problems that interest you.

I can do some lecturing if you wish or work with individual groups.

### References:

I am writing the book "Learning MATLAB, Problem Solving, and

Numerical Analysis Through Examples", which has lots of examples

from different topics. You can have a hardcopy of the first few

chapters when you arrive and you will get a pdf of the entire book.

## Combinatorics & Probability

### Instructor: John Johnson

• Office: MW 756

• E-mail: Johnson.5316@osu.edu

• Phone: 614-292-6353

• Office Hours: Tuesday and Thursday 11:00 – 12:30

• Classes: 2:55 – 3:50 (MW 154 except for 7/28 – 7/30)

### TA: Liz del Rosario Teran

• E-mail: liz.teran@upr.edu

• Office: MA 236

### Syllabus.

Week 1

For the first week, we will study some consequences of the pigeonhole principle, its probabilistic version, a few “recurrent type” problems, and binomial coefficients.

Week 2

For the second week, we will study generating functions and the probability of dice games.

Week 3

For the third week, we will study Beatty's theorem (and its generalization: the Lambek--Moser theorem).

Week 4

For the last week, we study some classical Ramsey theory results, such as, Ramsey's theorem, Van der Waerden's theorem, Schur's theorem, Folkman's theorem, and Hindman's theorem.