## Daily Schedule

### 9:30 – 10:25 am – Topology (Ortiz-Navarro – UH 051)

10:35 – 11:30 am – Real Analysis (Stan – UH 051)

11:30 am – 12:45 pm – Lunch – on own

12:45 – 1:40 pm – Abstract Algebra (López-Permouth – UH 051)

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

2:55 – 3:50 pm – Combinatorics & Probability (Johnson – UH 051)

4:00 – 4:55 pm – Recitations (MW 724, except 7/27 – 7/29 in MW 154)

### Graduate School Prep Class: Thu 7/16, 9:30 – 11:30 am; Thu 7/23, 12:45 – 2:45 pm; Thu 7/30, 9:30 – 11:30 am; Thu 8/6, 12:45 – 2:45 pm (Poole/Kerler – MW 154)

## Reserved rooms

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

• TA Offices: MW426

• Recitations: MW 724, scheduled for 4:00 – 5:00 pm each day (MW 154 7/27-7/29).

## Topology

### Instructor: Juan Ariel Ortiz-Navarro

• Office: MW 750

• Telephone: 614-292-5585

• E-mail: juan.ortiz35@upr.edu

• Office Hours: 3:00 - 4:00

• Classes: 9:30 – 10:25 am (UH 051)

### TA: Gabriel Jaime Montoya

• Email: Gabriel.montoya@upr.edu

• Office: MW 426

### 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: MW 700

• E-mail: stan.7@osu.edu

• Phone: 614-292-6243

• Office Hours: TBD

• Classes: 10:35 – 11:30 am (UH 051)

### TA: Mario Zepeda

• E-mail: Mario.zepeda@upr.edu

• Office: MW 426

### 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: MW 748

• E-mail: lopez@ohio.edu

• Phone: 614-292-4015

• Office Hours: TBD

• Classes: 12:45 – 1:40 (UH 051)

### TA: David Mendez

• E-Mail: davidfernando.mendez@upr.edu

• Office: Mw 426

### Topics

• 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 questions are easy to phrase and have various levels of difficulty as one explores properties of these "power structures".

• Alternatively: if tackling a research problem is not a feasible option, we will use our time to learn about the Algebraic Theory of Error-Correcting Codes, a very useful real-life application that will fit very nicely with the original intention of the program to sample advanced topics in mathematics.

### Textbooks

(mostly as a quick reference)

• anything from Gallian’s to Hungerford’s Algebra

• Coding Theory, Hankerson et al

### Weekly schedule

• Week 1: Algebraic structures and “smaller structures” (substructures and homomorphic images): groups, subgroups and normal subgroups, morphisms, kernels and images, rings, subrings, ideals, factor rings.

• Week 2: Powergroups (reading through research paper by SAMMS 2014 participants.)

• Week 3: Research.

• Week 4: Continuing research or learning about error correcting codes.

## 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 265)

### TA: Hiva Samadian

• E-mail: hiva.samadian@upr.edu

• Office: MW 426

### Topics

• Applied mathematics problems are generally solved using techniques from calculus, linear algebra, and ordinary partial differential equations. However, "real" problems can only rarely be solved analytically, so numerical algorithms and codes are normally required; and MATLAB is (in my opinion) the easiest computer language to use, both because it is quite easy to learn and because it has a huge number of functions which allow you to carry out complicated computational tasks in one or a few statements.

• This is an introduction to MATLAB by "playing around" with various aspects of the mathematical topics of chaos, fractals, and randomness, all of which are heavily computational. Those of you who don't already know MATLAB will "play around" by writing simple computer programs, while those who do can write somewhat more involved programs.

• This is NOT a class on numerical analysis; it is a class on learning MATLAB --- or learning it better --- by writing programs to "play around" with interesting mathematical topics.

### 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 graphical images, and how to program in MATLAB

• Week 2: Chaos Theory – we will explore continuous and discrete dynamical systems that are highly sensitive to initial conditions, what this means, what the requirements are for a chaotic system, how to calculate the dimension of a chaotic system, and how to generate interesting graphical images.

• Week 3: Fractals – many “fractal-like” physical systems occur in the world around us, but we are more concerned with mathematical fractal systems, their properties, their dimensions, and how to generate them recursively.

• Week 4: Randomness – to the general public “chaotic” and “random” are synonymous, but they are very different; the question we will address is “How random is the world around us?” by focusing on a few specific areas, particularly sports. An interesting discussion can be found in the Wikipedia article “Hot-hand fallacy”.

## Combinatorics & Probability

### Instructor: John Johnson

• Office: MW 756

• E-mail: Johnson.5316@osu.edu

• Phone: 614-292-6353

• Office Hours: TBD

• Classes: 2:55 – 3:50 (UH 051)

### TA: Dan Poole

• E-mail: poole.89@osu.edu

• Office: MW 529

### 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 Overman'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: Introduction to basic counting principles, binomial coefficients, Fibonacci numbers, and Stirling numbers

• Week 2: Probabilistic pigeonhole principle, generating functions, and basic definitions and principles of probability

• Week 3: Important Ramsey's theoretic theorems (Ramsey's theorem, van der Waerden's theorem, Schur's theorem, Folkman's theorem, and Hindman's theorem)

• Week 4: Student presentations on some combinatorial or probabilistic theorem

## Graduate School Prep Class

### Instructors: Dan Poole/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