Founded in March 2014, the SCUM is a series of weekly math talks/panels by (mostly) undergraduates, for undergraduates, at MIT. Anyone is invited to speak about a math topic he or she finds interesting, sometimes but usually not original research, and talks are open to the MIT community (as well as interested visitors, e.g. from Harvard). With the support of the MIT Mathematics Department (and in past semesters, the Undergraduate Mathematics Association (UMA), Undergraduate Society of Women in Math (USWIM), and the Harvard-MIT Mathematics Tournament), SCUM also features free dinner each week! We'll meet weekly on **Wednesdays, 5:30--6:30 PM, in room 4-145**.

If you are interested in picking up a talk, please send a title and abstract, as well as any appropriate references (optional but recommended), to scum-exec@mit.edu. (If you have any food preferences, let us know!) In addition, please include a preference for dates, and any other accommodations you will need (e.g. projector). Talks should be fifty minutes long and accessible to first-year undergraduate math majors. We strongly recommend attending several SCUM talks before signing up to give one yourself.

(For logistical tips/advice on organizing such activities yourself, see below here.)

To receive email announcements, make sure you are either a declared MIT math major (which by definition excludes all freshmen first semester), or on the scum-interest mailing list (possibly via UMA or USWIM). If you do not have MIT certificates, email scum-exec@mit.edu and we'll manually add you to the mailing list.

Alternatively, you can like or follow us on Facebook at https://www.facebook.com/colloquium18/

Organizers: Ahaan Rungta, Sandeep Silwal, Victor Wang, and possibly others to be determined (definitely let us know if you're interested!)

**Meeting Time/Place**: Wednesdays, 5:30 PM, in 4-145

**The following Wednesdays are open** for talks, panels, or other activities: next semester (Spring 2017) is completely open!

**Let us know if you (or someone you know) might like to give a talk**, or if you have suggestions for topics/ideas you would like to see in future talks.

See the archive here.

Date: Wednesday, 14 December

Title: **Morse Theory and Symplectic Floer Homology**

Speaker: **Charles (Yuchen) Fu**

Helpful background: The talk will be self-contained so there is no prerequisite, though knowing the basics of differentiable manifolds and/or homology would be helpful.

Abstract: Morse theory, named after Marston Morse (1892-1977), is the study of the topology of a finite dimensional differentiable real manifold using the data of critical points of a "typical" real-valued function on the manifold. The associated homology turns out to be topologically invariant even though the computation relies on the differential structure. In this talk, we'll go through the basic constructions of Morse homology and see how it works on some concrete examples (e.g. the torus). We'll also briefly introduce the symplectic Floer homology, which is an infinite-dimensional version of Morse homology and serves as a fundamental tool in modern geometry.

Further reading for those interested:

- "Morse Theory and Floer Homology" by Michèle Audin and Mihai Damian

Date: Wednesday, 7 December

Title: **Elliptic curve factorization**

Speaker: **Yongyi Chen** (G)

Helpful background: Minimal group theory assumed but nothing else. (For example, skimming over the definition and some examples at https://en.wikipedia.org/wiki/Abelian_group should be enough if you haven't seen it before.)

Abstract: What are the fastest algorithms for factoring a positive integer? We all know about trial division, which takes up to O(sqrt(N)) time to find a nontrivial factor. There is a faster factoring algorithm called Lenstra’s elliptic curve factorization, using the theory of elliptic curves. To ease into the discussion, we first discuss Pollard’s p-1 algorithm, a purely number-theoretic algorithm that has many of the same ideas as Lenstra’s algorithm. Finally, we end by doing some hands-on (small or large) examples.

Further reading for those interested:

- Andrew Sutherland, 18.783 (see e.g. http://math.mit.edu/classes/index.php?course=783 or take the class next term!)

Date: **Tuesday** 22 November at 5:30p, in **MIT 5-217** (special date and room!)

Title: **Local-global principles in number theory**

Speaker: **Victor Wang**

Abstract: Often to solve Diophantine equations (usually equations, over integers, that fit in margins of notebooks) it helps to reason with primes and/or size. We will work up to (parts of) the Hasse-Minkowski theorem addressing the situation of quadratic polynomial equations.

Further reading for those interested:

- Serre, A Course in Arithmetic, available to (those with MIT certificates) for free here

Date: 16 November

Title: **TQFTs and higher category theory**

Speaker: **Sanath Devalapurkar**

Helpful background: Knowledge of tensor products of vector spaces, and the idea of what a manifold is, may be helpful, but you are welcome to come either way!

Abstract: In 1988, Michael Atiyah proposed axioms to formalize Edward Witten's theory of "topological quantum field theories" (TQFTs), that interestingly involved topological notions like that of cobordisms. TQFTs are important because studying them yields interesting topological invariants. In order to simplify computations, Atiyah's formalism was extended to higher dimensions. This extension motivated the cobordism hypothesis, which says that you can generate a bunch of TQFTs simply by looking at its value on the point. A sketch of a proof of this conjecture using higher category theory was given by Jacob Lurie in 2010, but there are still many aspects of his proof that still need to be made formal. I'll talk about his main tool in the proof, namely the theory of "infty-categories". Time permitting, I'll say a little about a project of mine that is a test for how one might formalize higher dimensional TQFTs.

Further reading for those interested:

- Atiyah, "Topological quantum field theory". See http://www.math.ru.nl/~mueger/TQFT/At.pdf. Best for a first read.
- Lurie, "On the Classification of Topological Field Theories". Available at http://www.math.harvard.edu/~lurie/papers/cobordism.pdf. A good reference in general if you know homotopy theory.
- Freed, "The Cobordism Hypothesis". See http://www.ams.org/journals/bull/2013-50-01/S0273-0979-2012-01393-9/S0273-0979-2012-01393-9.pdf. Good for math folks.
- Witten, "Topological quantum field theory". Comm. Math. Phys. Volume 117, Number 3 (1988), 353-386. It's probably good if you understand physics, but I don't.

Date: 9 November

Title: **Ordering, tournaments, and triples**

Speaker: **Sidharth Sidharth**

Abstract: A classical result of Erdos and Szekeres (1935) states that every sequence of N numbers has an increasing or decreasing subsequence of length N^(1/2), and the exponent 1/2 is the best possible. We will discuss natural variants of this question, shamelessly emphasizing questionably related themes and connections (or lack thereof) such as saturation, amplification, local-global phenomena, probability, incidence geometry, or representation theory.

Further reading for those interested:

- W. T. Gowers and J. Long, https://arxiv.org/abs/1609.08688 (most recent paper on the problem)
- L. Guth, http://math.mit.edu/~lguth/PolyMethod/lect4.pdf (lecture notes on the joints problem via polynomial method)
- M. Lee, The Young-Frobenius Identity, available at http://web.mit.edu/uma/www/mmm/mmm0401.pdf
- D. Romik, The Surprising Mathematics of Longest Increasing Subsequences, available at https://www.math.ucdavis.edu/~romik/book/ (thanks to Mark Sellke for this reference)
- T. Tao, https://terrytao.wordpress.com/2007/09/05/amplification-arbitrage-and-the-tensor-power-trick/ (exposition on amplification via the tensor power trick)

Date: 12 October (notes by Sanath Devalapurkar are available here)

Title: **Modularity of Elliptic Curves**

Speaker: **Asra Ali**

Abstract: In 1995, Andrew Wiles, with assistance from Richard Taylor, showed
modularity for semistable elliptic curves, proving Fermat's Last
Theorem. The goal for this talk is to understand what it means for an
elliptic curve over $\Q$ to be modular. We will introduce elliptic
curves and modular forms, and define an $L$-series for each of the two
objects. This will lead us to the question, given an elliptic curve
$E/\Q$, is there a modular form whose $L$-series coincides with the
$L$-series for $E$? We'll discuss what the correspondence is
explicitly, and see some concrete examples.

Further reading for those interested:

- J-P. Serre: A Course in Arithmetic
- J.S. Milne: Elliptic Curves

If you would like to organize such math activities yourself, you can request room reservations in the Math Department HQ (next to Academic Services). (In particular, Barbara Peskin covers recurring, e.g. weekly, reservations. Make sure to specify room specifications, e.g. "flat, seats 30, tables and chairs, video projector". It is difficult to request specific rooms, so instead it is probably wise to request to *not* have certain bad rooms. You can also probably easily find out the right people to ask for logistical things like funding or term-to-term seminar-listing adjustments; email us if you have questions.

Food-related: MIT students do not need to pay tax (see here for tax exemption forms). One convenient option is to set up a tax-exempt account on Foodler, as described in their FAQ. (You can often request plates, utensils, cups, and napkins. Drinks are often cheaper separately, e.g. at convenience stores.)