## Daily Schedule

### 9:30 – 10:25 am – Topology (Ortiz-Navarro – JR 0353)

10:35 – 11:30 am – Real Analysis (Stan – JR 0353)

11:30 am – 12:45 pm – Lunch – on own

12:45 – 1:40 pm – Abstract Algebra (López-Permouth – JR 0353)

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

2:55 – 3:50 pm – Combinatorics & Probability (Poole – JR 0353)

4:00 – 4:55 pm – Recitations (MW 724; except for 7/14, 7/21, 7/24, and 7/28, when we will use MA 010)

### Graduate School Prep Class: Thu 7/13, 9:30 – 11:30 am; Thu 7/20, 12:45 – 2:45 pm; Thu 7/27, 9:30 – 11:30 am; Thu 8/3, 12:45 – 2:45 pm (Xiong/Kerler – NEW ROOM JR 0353)

## 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 except as noted above.

## Topology

### Instructor: Juan Ariel Ortiz-Navarro

• Office: MA 236

• Telephone: 614-292-7898

• E-mail: Juan.Ortiz35@upr.edu

• Office Hours: 10:30 - 11:30 am

• Classes: 9:30 – 10:25 am (JR 0353)

### 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: MW 610

• E-mail: Stan.7@osu.edu

• Phone: 614-292-5310

• Office Hours: TBD

• Classes: 10:35 – 11:30 am (JR 0353)

### TA: Hassam Hayek Barrius

• E-mail: Hassam.Hayek@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: MW 610

• E-mail: Lopez@ohio.edu

• Phone: 614-292-5310

• Office Hours: TBD

• Classes: 12:45 – 1:40 (JR 0353)

### TA: David Fernando Méndez Aguela

• E-Mail: DavidFernando.Mendez@upr.edu

• Office: MA 052

### Topics

• We will attempt a Research Experience based on a very generally phrased question.

Mostly, Modern Algebra deals with binary operations; in this experience we will try to do something different and focus on n-ary operations with n greater than or equal to 3.

For starters, note that binary operations can be used to create ternary (or higher dimensional) operations. For example, given a binary operation *, one can define a ternary operation m given by m(a,b,c) = a*(b*c). Alternatively, one can define n(a,b,c)=(a*b)*c. The two ternary operations m and n coincide when * is associative. The kinds of problems we anticipate begin with questions such as: given a ternary operation on a set S, how can one recognize whether it was induced by a binary operation? What kind of axioms on a ternary operation would allow you to recognize properties of an underlying binary operation that may have induced it? Can the same ternary operation be induced by more than one binary operation? Can this kind of research be extended to n-ary operations for n larger than 3?

A similar kind of inquiry can be carried looking at ternary operations that are induced by not one but two binary operations. If one has two binary operations # and * on the same set S, one can define a ternary operation in various ways: a(x,y,z) = x*(y#z), b(x,y,z)= (x#y)*z, c(x,y,z) = (x*y)#(x*z), etc. Some of these definitions would coincide, for example, if * distributes over #. The challenge is to start with ternary operations and try to identify when they were induced by binary operations in one of the above ways (or similar ones.)

As with the first kind of problem, one may also try to identify properties of induced ternary operations which reflect properties of their underlying binary operations.

• Alternatively: if tackling a research problem is not a feasible option (sometimes innocent-looking problems can prove to be too difficult) , 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 their properties (substructures and homomorphic images): groups, subgroups and normal subgroups, morphisms, kernels and images, rings, subrings, ideals, factor rings.

• Week 2: Research.

• Week 3: Research and possible enrichment reading a paper written by SAMMS participants on a previous year.

• 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: Arlin Alvarado

• E-mail: arlin.alvarado@upr.edu

• Office: MA 052

### 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 combinatorics and probability theory. 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 ones.

• The course will be run in conjunction with the Combinatorics & Probability class which follows (so that course should also be taken). In the first week we will code up various aspects of the birthday problem and also code the Monty Hall problem (which confused even professional mathematicians). In the second week we will write a code to find the longest arithmetic sequence in a sequence of positive, monotonically increasing integers, which arises frequently in combinatorics. And in the third week we will numerically explore various games discussed in the following course.

• 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 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 (JR 0353)

### TA: Luis Miguel Mestre

• E-mail: Luis.Mestre2@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 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: Jue Xiong/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