18.204 Term Paper Workshop

1. 18.204 Term Paper Workshop Spring 2019 Laszlo Lovasz and MalcahEffron

2. What differences do you notice between reading and listening to mathematics? • Amount of detail needed to understand on own • Amount of formality capable of taking in

3. What should introductions to math expository papers do? • In groups of 2-3, discuss the answers to this question based on the reading and thinking you did in preparation for class.

4. Math exposition introductions should… • Set up the overarching goal of the paper • Provide general and/or historical context to the problem • Motivate the problem by explaining its mathematical interest • Provide a structure for the paper

5. Strategies to set up the overarching goal of the paper… • Select problems that relate to each other (or else that differ in important ways you intend to describe) • Be able to articulate this relationship to yourself • Explain this relationship explicitly in the introduction • Examples from readings: • Our intention is to give a complete proof of this theorem [CFZ]. • The goal of this paper is to discuss several such results and to use them to illustrate various measures that can be used to study the robustness of graph properties. [S].

6. Strategies to provide context to problem • Offer an explanation of the problem that can be understood without diving into the technical details: • Example: Let G be a graph and P a graph property. Many results in graph theory are of the form “under certain conditions, G has property P”. Once such a result is established, it is natural to ask how strongly does G possess P? In other words, we want to determine the robustness of G with respect to P. [S] • Explain the steps/approaches that have been taken to solve the problem: • Example: Although there have been numerous other expositions [21, 22, 28, 29, 43, 45, 47], we were prompted to write this note because of our recent work [7, 51] simplifying one of the key technical ingredients in the proof. Together with work of Gowers [21], Reingold, Trevisan, Tulisiani, and Vadhan [33], and Tao [42], there have now been substantial simplifications to almost every aspect of the proof. [CFZ] • Sometimes details of the history can be interesting, too (e.g. what aspects were discovered first)

7. Strategies to motivate interest • Identify the challenges in solving the problem: • Example: The problem of deciding the Hamiltonicity of a graph is one of the NP-complete problems that Karp listed in his seminal paper [53], and accordingly, one cannot hope for a simple classification of such graphs. Nonetheless, there are many results deriving conditions that are sufficient to establish Hamiltonicity. [S] • Identify where the insightful discoveries appear: • Example: If not by density considerations, how do Green and Tao prove their theorem? The answer is that they treat Szemer´edi’s theorem as a black box and show, through a transference principle, that a Szemer´edi-type statement holds relative to sparse pseudorandom subsets of the integers, where a set is said to be pseudorandom if it resembles a random set of similar density in terms of certain statistics or properties. [CFZ]

8. Strategies to motivate interest (con’t) • Call out the value added to the field • Example: Dirac’s theorem is one of the most influential results in the study of Hamiltonicity, and by now many related results are known (see, e.g., [17]). It is therefore very natural to try to strengthen this theorem, by asking whether Dirac graphs are robustly Hamiltonian. It turns out that there are several ways to answer this question using different measures of robustness. [CFZ]

9. Strategies to provide an organization for the paper • Follow the order set up in the context provided in the introduction • Conclude with paragraphs that explain the organization of the paper and the logic of that organization

10. How should the body of math expository papers be organized? • In groups of 2-3, discuss the answers to this question based on the reading and thinking you did in preparation for class.

11. Math exposition paper body structures should… • Organize material into sections that have cohesive goals • Organize sections into a global structure with a logical ordering • Support decisions to provide proofs or just proof strategies

12. Strategies to organize material into sections that have cohesive goals • Identify the points you want to make • Create sections of papers organized by those points • Sort contents into sections based on to which points they most closely relate

13. Structure the body logically • Building block structure: begin with information needed to understand later information • Most important first: begin with the biggest idea, then move to tangential points • This can help make decisions about problems, examples, etc. • This can also help with when to defer proofs of claims • Simple to complicated: begin with easiest ideas and then move to more involved and/or less obvious ones • Chronological: move through the history of the problem * These structures can overlap.

14. Use transitions to articulate the structure (Examples from [S]) • Section 2: The above mentioned theorem of Dirac states that h(G) > 0 for every n-vertex graph with minimum degree at least n/2. One obvious way to strengthen this theorem is to show that every Dirac graph contains several Hamilton cycles. • Section 3: In the previous section we showed that any Dirac graph contains not only one but exponentially many Hamilton cycles. Another interesting way to extend Dirac’s theorem is to show that a minimum degree of at least n/2 implies the existence of many edge-disjoint Hamilton cycles. Note that there is no obvious way to deduce such a statement from the results in Section 2.

15. Proof v. Proof Strategy • In general, you should include proofs, unless • The claims are obvious • Well-known • On a different topic (e.g. a result from another (sub)field) • For some topics, it might make sense to omit some proofs, or only do sketches, to be able to cover a broader amount of material.

16. Let’s build a rubric for the term paper • Based on the principles you’ve distilled and the strategies we’ve suggested, you can come up with standards for a good term paper. • In groups of 3, find space at the board and put together a rubric for the term paper. This rubric should include: • Required components of the paper • Ways to evaluate the quality of the paper and its components