## Daily Schedule

### 9:30 – 10:25 am – Topology (Ortiz-Navarro – UH 0043)

10:35 – 11:30 am – Real Analysis (Stan – UH 0043)

11:30 am – 12:45 pm – Lunch – on own

12:45 – 1:40 pm – Abstract Algebra (López-Permouth – UH 0043)

1:50 – 2:45 pm – Computational Science (Hiary – EA 265)

2:55 – 3:50 pm – Combinatorics & Probability (Poole – UH 0043)

4:00 – 5:00 pm – Recitations (MW 724)

### Graduate School Prep Class: Thu 7/12, 9:30 – 11:30 am; Thu 7/19, 12:45 – 2:45 pm; Thu 7/26, 9:30 – 11:30 am; Thu 8/2, 12:45 – 2:45 pm (Zelada Cifuentes/Kerler – UH 0043)

## Reserved rooms

### • Study space: Math Building (MA) 052, the Undergraduate Student Lounge 8:00 - 5:00 M - F.

• TA Offices: MA 052

• Recitations: MW724, scheduled for 4:00 – 5:00 pm each day.

## Topology

### Instructor: Juan Ariel Ortiz-Navarro

• Office: MA 114

• Telephone: 614-292-0819

• E-mail: Juan.Ortiz35@upr.edu

• Office Hours: 10:30 - 11:30 am

• Classes: 9:30 – 10:25 am (UH 0043)

### TA: Victor Adolfo Cardenas

• Email: Victor.Cardenas@upr.edu

• Office: MA 052

### Topics

• Students will sample different topics that range from basic point set topology, classification of surfaces. The final topics are used to encourage students further than a typical point set topology course into graduate courses ideas and concepts. It will involve challenging questions presented in a research – type emphasis.

### Textbooks

• Introduction to Topology: Pure and Applied, C. Adams and R. Franzosa

• Topology of Surfaces, L.C. Kinsey

• The Knot Book, C. Adams

### Weekly schedule

• Week 1: Open Sets, Closed Sets, Quotient Topology, Continuity

• Week 2: Homeomorphisms, Manifolds, Orientability, Examples of Manifolds

• Week 3: Surfaces with boundary, Classification of 2-Manifolds, Euler Characteristic, Introduction to Knots

• Week 4: Reidemeister Moves, Tricolorability, Invariants, Types of Knots, Polynomials

## Real Analysis

### Instructor: Aurel I. Stan

• Office: MA 114

• E-mail: Stan.7@osu.edu

• Phone: 614-292-5131

• Office Hours: TBD

• Classes: 10:35 – 11:30 am (UH 0043)

### TA: Carlos Carvajal Ariza

• E-mail: Carlos.Carvajal@upr.edu

• Office: MA 052

### 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.

### Textbooks

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

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

### Weekly Schedule

• Week 1: The Arithmetic–Geometric–Harmonic Means inequality. Cauchy proof; Bernoulli and Cauchy–Schwarz inequalities; applications. Jensen inequality; applications of Jensen inequality and a new look at the Arithmetic–Geometric and Bernoulli inequalities.

•Week 2: Convergent sequences, ϵ–δ definition; monotone and bounded sequences; composition of monotone functions. Recursively defined sequences; applications; Cauchy sequences.

•Week 3: Series. Necessary conditions for convergence; absolutely convergent series. Cauchy condensation criterium; applications; conditionally convergent series. Abel–Dirichlet Theorems; applications.

•Week 4: Continuous functions. ϵ–δ definition; studying the continuity of continuous functions using sequences. Examples: Dirichlet and Riemann functions; uniformly continuous functions; functional equations involving continuous functions.

## Abstract Algebra

### Instructors: Sergio López-Permouth

• Office: MA 414

• E-mail: Lopez@ohio.edu

• Phone: 614-292-5431

• Office Hours: TBD

• Classes: 12:45 – 1:40 (UH 0043)

### TA: Jose Emilio Calderón Gómez

• E-Mail: JoseEmilio.Calderon@upr.edu

• Office: MA 052

### Topics

• For starters, some concepts of the theory of Error-Correcting Codes, a very useful real life application that fits nicely with the program intention of sampling advanced topics in mathematics.

• Introductory Group and Ring theory notions.

Textbooks

(mostly as a quick reference)

• anything from Gallian’s to Hungerford’s Algebra

• Coding Theory, Hankerson et al

### Weekly schedule

• Week 1: Error-correcting codes concepts

• Weeks 2-3: Algebraic structures and their properties (substructures and homomorphic images): groups, subgroups and normal subgroups, morphisms, kernels and images, rings, subrings, ideals, factor rings.

• Week 4: Possible enrichment using materials from past years OR continuing group and ring theory study.

## Computational Science

### Instructor: Ghaith Hiary

• Office: MW 646

• E-mail: hiary.1@osu.edu

• Phone: 614-292-4013

• Office Hours: TBD

• Classes: 1:50 – 2:45 (EA 265)

### TA: Arlin Alvarado

• E-mail: arlin.alvarado@upr.edu

• Office: MA 052

### Topics

• Applied mathematics problems are generally solved using techniques from calculus, linear algebra, probability, and ordinary and partial differential equations. In practice, such problems are rarely solved in exact form, so approximation and simulation algorithms are normally required. MATLAB is a popular computer language to implement such algorithms, both because it is relatively easy to learn and because it has a large number of built-in functions allowing you to carry out complicated computational tasks in a few statements of computer code.

• This is an introduction to MATLAB by "playing around" or "experimenting" with various problems from the topics of probability theory, combinatorics, and other topics. The goal is to write computer programs to investigate interesting math problems, and to learn how to fruitfully approach math problems with a computational mindset.

• The course will be run in conjunction with the Combinatorics & Probability class which follows (so that course should also be taken). We will investigate various computational aspects of the birthday problem and the Monty Hall problem, which are well-known problems with apparently counter-intuitive solutions. We will also write code to find the longest increasing subsequence of a random permutation of the integers {1, ..., n}, a problem that arises frequently in combinatorics. And we will explore various mathematical games via computation.

• This is a class on learning MATLAB --- or learning it better --- and on how to effectively employ computation to your advantage when investigating math problems.

### Textbook

• Learning MATLAB, Problem Solving, and Numerical Analysis Through Examples (pdf)

### Weekly schedule

• Week 1: MATLAB as a scalar calculator, how to handle arrays, how to generate 2D and 3D graphical images, and how to program in MATLAB.

• Week 2: Finishing off programming in MATLAB, particularly using Monte Carlo simulations. MATLAB functions which are particularly appropriate for probability theory will be stressed.

• Week 3: "Playing around" with particular topics that arise in the Combinatorics & Probability class.

• Week 4: "Playing around" with whatever strikes our fancy.

## Combinatorics & Probability

### Instructor: Dan Poole

• Office: MW 529

• E-mail: Poole.89@osu.edu

• Phone: 614-292-1923

• Office Hours: TBD

• Classes: 2:55 – 3:50 (UH 0043)

### TA: Lesbia López Limas

• E-mail: Lesbia.Lopez@upr.edu

• Office: MA 052

### Topics

• In this course we will study some important combinatorial and probabilistic principles and see how they are applied to derive important and nontrivial results. This course will be loosely coordinated with Professor Hiary's Computational Science course, and, if possible, I encourage you to take Computational Science concurrently with this course.

### Textbook

• Combinatorics Through Guided Discovery by Kenneth P. Bogart. (Freely available on the web.)

### Weekly schedule

• Week 1: Colorings in graph. In particular, vertex colorings, edge colorings and total colorings. Strategies for finding the various chromatic numbers as well as the big theorems and conjectures in the area (4 color theorem, Vizing's Theorem, Hadwiger-Nelson problem, etc.)

• Weeks 2-3: Probability. Birthday problem and Monty Hall problem. Coin Flipping Game. Introduction to random graphs.

• Weeks 3-4: Ramsey Theory. In particular, Schur's Theorem, Van der Waerden's Theorem (Green-Tao Theorem), Ramsey's Theorem.

## Graduate School Prep Class

### Instructors: Rigo Zelada Cifuentes/Thomas Kerler

### Topics

• How to apply to graduate school

• The GRE Subject test – practice & Strategy

• Writing CV’s and Letters of Intent

• Finding out about schools

• Document preparation