Writing a short survey
The final and sole assignment of the course is to write a short survey accounting for (all) the articles listed below. You are to survey the results written in the featured articles and present a cohesive and concise account of the results reported in these articles.
Your account need not offer proofs of the main results appearing in the featured articles; as that would entail writing the whole article. In your account you should make a sincere attempt to address the following:
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Chronological order of the results.
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Provide the mathematical context for all results.
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How are the results related to one another if at all?
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Who improved who and in what sense?
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What is the importance of the results encountered?
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What is the intuition underlying the results?
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Provide examples or simple constructions illustrating the results.
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What methods are being used?
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What are the advantages of using these methods?
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What problems the authors had to overcome along the way?
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What is considered novel in each result and the methods used to prove it?
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How are the results in the featured articles related?
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How are the results in the featured articles inspire one another and in what ways?
Your account cannot be simply answering the above questions one after another or per article; these questions are mere guidelines and you are expected to answer this in a streamline fashion through a cohesive essay and not through a "shopping" list. Your account cannot be broken down by-article, so to speak.
Overall, your account should be a stable, well-structured, and precise report regarding the results featured in the articles. The articles are not listed in any particular order hinting to the connections between them; this you are to find out on your own. The break-down of the entire body of results featured in these articles that you should present to your reader is to be conceptual and driven by math as opposed to any other ordering. Your overall aim should be that a (reasonable) person reading your account would be able to grasp all of the above and subsequently delve into the technical details knowing the "big picture". The introductory part of each of the articles on the list is a good mini-example of what is expected of your survey; though your survey is to encompass many more results.
Pay close attention to the "added value" that you bring to the reader in your analysis; simply copying the results one after another is meaningless. This assignment simulates the writing of a proper introduction to a scientific research article.
Do not delve into topics in your account without the proper definitions and background set up prior to doing that. Do not leave your reader guessing what you mean. Think carefully and intensely on the notation that you plan to employ to convey your points.
Guidelines
You are to adhere to the following guidelines:
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Up to 5 students can submit a single report.
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There is no space limitation.
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It is preferred that you use LaTeX to format your report. This, however, is not obligatory. If using a different text formatter (like, say, Word) harms your presentation it is on you.
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PDF files only will be accepted.
DEADLINE: 8th of March, 2024 by 14:00
One member of your team should send me the assignment via
email to my GMAIL account only. Be sure to state your ID
numbers on the assignment.
Instructions continued:
The articles provided to you cite many other articles and the latter are not supplied in the list below. For your account to be complete you must take these into consideration and interleave these into your survey. Use the articles citing them to learn of their results; if things are not clear, then you are to open the cited article to make sure that what you are writing is correct.
The very top grades are reserved to those students who will delve more ambitiously into the math portrayed by the featured articles. We will distinguish between students using math to make their points and those that are content with rough "qualitative" accounts. Whilst you are not being asked to prove the results you can still use math to motivate these, justify their tightness, examplify that certain methods fail, and that other methods ought to work. Here lies that "added benefit" that you bring into the you account.
A proper Cambridge level use of the English is expected.
A masterful LaTeX style presentation is expected.
Precise bibliography written flawlessly is expected; be precise with the names of researchers. Precise mathemathical formulaitons are expected.
Anti-concentration
invertibility of random matrices &
related topics
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Singularity of random symmetric matrices revisited, by M. Campos, M. Jenssen, M. Michelen, and J. Sahasrabudhe.
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The Littlewood-Offord problem and invertibility of random matrices, by M. Rudelson and R. Vershynin.
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On the smoothed analysis of of the smallest singular value with discrete noise, by V. Jain, A. Sah, and M. Sawhney.
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Singularity of discrete random matrices, by V. Jain, A. Sah, and M. Sawhney.
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On the smallest singular value of symmetric random matrices, by V. Jain, A. Sah, and M. Sawhney.
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Random symmetric matrices are almost surely non-singular, by K. Costello, T. Tao, and V. Vu.
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Bilinear and quadratic variants on the Littlewood-Offord problem, by K. Costello.
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An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs, by M. Kwan and L. Sauremann.
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Resolution of the quadratic Littlewood--Offord problem, by M.Kwan and K. Sauremann.
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Anticoncentration in Ramsey graphs and a proof of the Erdős-McKay conjecture, by M. Kwan, A. Sah, L. Sauermann, M. Sawhney.