A good practice in math, especially when studying a type of mathematical objects, is by considering what doesn’t fit the definition.

# Student

# LSRI LaTeX Workshop: Activity Page

Activity Resources:

Here’s the link to the actual latex code used to build the above file.

# Introduction to LaTeX

LaTeX is a powerful document preparation system. This system is *free* and is available across many operating systems, such as Windows, Mac OS and most distributions of Linux. A successor of TeX, LaTeX was created in the 1983 by Leslie Lamport. This computer typesetting software was originally created for academic papers in STEM fields, and since has grown in popularity in other fields because of its power to create many types of paper and digital documents common in academic fields. For those who are aiming to write a long text such as a dissertation, thesis or a book, a major attraction to learning LaTeX is its ability to handle large papers and provide uniform, beautiful, and professional formatting. LaTeX handles all the mundane tasks of formatting, so you can focus on writing the content!

## Why use LaTeX

For some samples of what you can create with LaTeX, visit the Notes and Work section of this blog. Below are some reasons why you might be interested in using LaTeX:

- Manage large documents by using include directives, which allows for reuse or quick modification of content as well as manage large documents.
- The technique of using multiple files to represent one documents also allows for the flexibility to quickly create of many versions of an assessment.
- Automate citations, bibliographies and formatting of bibliographies. (Quickly switching from numeric style to APA when you transition from math to social sciences, for example)
- Natural to integrate into workflow if you use popular reference management software such as Zotero, Mandeley, Citavi and most others. (Check here to see if your reference manager is compatible with Bibtex.)
- Automate the creation of list of figures/tables/graphs, and the associated numbering
- Automate the creation, sorting and creation of index, acronym and glossary lists
- Typeset complex mathematical formulae, chemical equations
- Create more complicated documents like posters, slides, and double-sided flash cards
- Precise typesetting for many languages, this is especially useful for languages outside of the romantic languages family such as Arabic

## Comparing LaTeX with word-processors

There are some contrasts with WYSIWYG (What You See is What You Get) word processors like Microsoft Word or Open Office.

There are some advantages of using WYSIWYG processors:

- you wish to take advantage of the templates provided by MS Word, such as Resumes or business cards (You can create this in LaTeX also, especially if you choose Overleaf)
- if you’re creating documents that have specific syntax, such as markup language for github readmes,
- and if you’re creating very short files where formatting doesn’t matter (such as notes not meant for publication).

When common word processors attempt to handle large files, any major modifications to the file can break the table of contents, and automatic numbering of the tables/figures/lists are not reliable. Furthermore, for students working in multiple operating systems, transitioning between different operating systems using the same latex code will generate the exact same document – this precision is simply not guaranteed with most WYSIWYG processors.

With the above in mind, if you’re interested in learning about LaTeX as a beginner, I’ll be giving an introductory workshop at LSRI on November 13th, 2019.

*Coming soon: *

- Getting Started with LaTeX: Choosing a LaTeX environment
*Basic structure of a document**Links to sample latex files*

Disclaimer: Featured image source https://en.wikibooks.org/wiki/LaTeX

# Masters Thesis

I defended my thesis, “Computational Verification of the Cone Conjecture”, in December 2018, and submitted all final edits in May 2019.

My advising committee:

- Dr. Joseph Gubeladze
- Dr. Matthias Beck
- Dr. Serkan Hosten

Abstract:

The set of polyhedral pointed rational cones form a partially ordered set with respect to elementary additive extensions of certain type. This poset captures a global picture of the interaction of all possible rational cones with the integer lattice and, also, provides an alternative approach to another important poset, the poset of normal polytopes. One of the central conjectures in the field is the so called Cone Conjecture: the order on cones is the inclusion order. The conjecture has been proved only in dimensions up to 3. In this work we develop an algorithmic approach to the conjecture in higher dimensions. Namely, we study how often two specific types of cone extensions generate the chains of cones in dimensions 4 and 5, whose existence follows from the Cone Conjecture.

A naive expectation, explicitly expressed in a recently published paper by Dr. Gubeldaze, is that these special extensions suffice to generate the desired chains. This would prove the conjecture in general and was the basis of the proof of the 3-dimesional case. Our extensive computational experiments show that in many cases the desired chains are in fact generated, but there are cases when the chain generation process does not terminate in reasonable time. Moreover, the fast generation of the desired chains fails in an interesting way—the complexity of the involved cones, measured by the size of their Hilbert bases, grows roughly linearly in time, making it less and less likely that we have a terminating process. This phenomenon is not observed in dimension 3. Our computations can be done in arbitrary high dimensions. We make a heavy use of SAGE, an open-source mathematics software system, and Normaliz, a C++ package designed to compute the Hilbert bases of cones.

Full Text is below:

The actual latex code is hosted on github.

# Notes: Real Analysis I, II and Graduate Analysis

I took Real Analysis I & II with Professor Alex Schuster in 2016-2017. These comprehensive notes were compiled using lecture notes and the textbook, An Introduction to Analysis (Fourth Edition), by William Wade.

Please feel free to download and print these notes for your convenience.

Disclaimer: my notes are meant to be a toolbox while doing proofs and studying/practicing the course in general. There may contain typos or mistakes. Please feel free to let me know if you find any errors!

## Real Analysis I

## Real Analysis II

## Graduate Analysis

# Notes: Graduate Algebra

I took Graduate Algebra with Professor Matthias Beck in Spring 2017. These comprehensive notes were compiled using lecture notes and the textbook, David S. Dummit & Richard M. Foote, Abstract Algebra (3rd edition), Wiley 2004. [errata]

Please feel free to download and print these notes for your convenience.

Disclaimer: my notes are meant to be a toolbox while doing proofs and studying/practicing the course in general. There may contain typos or mistakes. Please feel free to let me know if you find any errors!

Topics Covered:

- Polynomial rings,
- irreducibility criteria,
- Gröbner bases & Buchberger’s algorithm,
- field extensions,
- splitting fields,
- Galois groups,
- fundamental theorem of Galois theory,
- applications of Galois extensions,
- introduction to the polynomial method with applications in graph theory and incidence geometry.

# Reflection: Planning/Time Management

### As a Teacher

- Pre-Calculus
- Lesson Planning is taking a lot less time this time around, since I spent a lot of time documenting the lessons from last year. Over-planning was actually an issue last semeseter; I would never actually complete everything I planned, and that was mildly frustrating. Also, there’s an element of improvisation that happens depending on the mood of the classroom, which means that scripting every sentence is simply not possible anyway.

- Calculus II
- The first two weeks I only prepared by rereading the text on the sections that the lead professor went over, but on the second week students asked me questions that definitely stomped me because I had not seen the problem before hand. Now, I ask students to email me their questions before the TA session, so that I can prepare with care. Planning short lectures on Calculus II material has been easier after that change in preparation, especially with access to great tools.

### As a Student

- Algebra
- I’m really glad to be practicing with SAGE, but I’ve honed in on specific problems and end up spending significant time cleaning up code instead of doing proofs. I need to find a balance in what I’m focusing on when learning.

- Real Analysis II
- I should really be spending more time on this course. The lectures in this cource has been a blessing for preparing for Calculus II TA sessions, since it helps get me into the “mood” to do Calculus. 🙂

- Combinatorics
- Obsessing over the details in this class takes too long. I’m in a similar situation with Algebra, but there’s this one is more like I spend 4-5 hours on one proof and run out of time for the other problems.

There’s a lot of juggling, and I have to get better at this soon before the midterm season.

# Notes: Modern Algebra II

I took Modern Algebra II with Professor Matthias Beck in Spring 2016. These comprehensive notes were compiled using lecture notes and the textbook, David S. Dummit & Richard M. Foote, Abstract Algebra (3rd edition), Wiley 2004. [errata]

Please feel free to download and print these notes for your convenience.

Featured Image: **Icosahedron and dodecahedron Duality****
**Credit: Images from Algebra: Abstract and Concrete by Frederick M. Goodman

Disclaimer: my notes are meant to be a toolbox while doing proofs and studying/practicing the course in general. There may contain typos or mistakes. Please feel free to let me know if you find any errors!

Topics Covered:

- Review of basic properties of groups and rings and their quotient structures and homomorphisms,
- group actions,
- Sylow’s theorems,
- principal ideal domains,
- unique factorization,
- Euclidean domains,
- polynomial rings,
- modules,
- field extensions,
- primitive roots,
- finite fields.

# San Francisco State University Math Resources: Mathematica & Wolfram Alpha Pro

SF State has a site license for Mathematica. Under this site license, Mathematica can be installed without further cost on any computer owned by San Francisco State University, by SF State faculty, or by SF State students. This software works on Windows, Macintosh, and Unix.

Under the terms of the site license negotiated by CSU, free WolframAlpha Pro accounts are also now available to our faculty, students, and staff.

If you are an SFSU student, click here for a free WolframAlpha Pro account.

### Mathematica

If you are a SFSU student and you want to download the *Mathematica* software to your computers, then follow the instructions below:

- Create an account
*(New users only)*:- Go to user.wolfram.com and click “Create Account”
- Fill out form using a @sfsu.edu email, and click “Create Wolfram ID”
- Check your email and click the link to validate your Wolfram ID

- Request the download and key:
- Fill out this form to request an Activation Key
- Click the “Product Summary page” link to access your license
- Click “Get Downloads” and select “Download” next to your platform
- Run the installer on your machine, and enter Activation Key at prompt

For both *Mathematica* and WolframAlpha Pro requests on the links above, **it is important that only sfsu.edu email addresses be used.** Requests from other email addresses will be rejected.

# Precalculus Resources: Intro & Review of Functions

In the first couple of chapters, Dr. Axler covers the properties of real numbers, and gives a throughout exploration of functions in his textbook for Math 199 (Precalculus at SFSU).

### Flashcards

You can take the file to a print shop to have directly printed and cut for you, or print it double sided on regular paper and cut + paste onto index cards. Get creative!

### Helpful Khan Academy Videos:

- Graphing Absolute Value Functions
- Finding Domain of Functions
- Manipulating Functions (Shift, stretch and reflect)
- Composing Functions
- Parallel and Perpendicular Lines
- Finding Vertex of a Quadratic Function from Standard Form
- Intro to Inverse Functions

More resources to come!