Student Colloquium for Undergraduates in Mathematics (SCUM)
Spring 2015

Information

Founded in March 2014, the Student Colloquium for Undergraduates in Mathematics (SCUM) is a series of weekly math talks by undergraduates, for undergraduates, at MIT (or occasionally Harvard). Any MIT (or willing Harvard) undergraduate is invited to speak about a math topic he or she finds interesting, not necessarily 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, 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 in E17-129.

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 colloquium-exec@mit.edu. In addition, please include a brief talk outline, 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.)

Upcoming Talks

Organizers: Peter Haine, Soohyun Park, Victor Wang
Meeting Time/Place: Wednesdays, 5:30 PM, E17-129

Let us know if you (or someone you know) might like to give a talk, or want to discuss SCUM- or math community- related things in general (e.g. suggestions for SCUM or ideas for other possible synergistic math activities).

Past Talks

See the archive here.

Date: 13 May 2015
Title: Heat Flow and the Poincaré Inequality
Speaker: Cole Graham
Abstract: The Poincaré inequality quantifies the idea that a function with a small derivative can't deviate much from its average. In this talk I'll use a variational argument to prove the inequality (in L^2) on bounded domains in R^n. The proof illuminates a neat connection between tight cases of the inequality, heat flow on the domain, and the spectrum of the Laplacian operator. I'll examine this connection in greater detail, present an important result due to Payne and Weinberger, and discuss some related open problems.
References:

Date and Location: 7 May 2015 (Thursday, not the usual Wednesday date) in room 4-153 (not the usual E17 room), with food provided afterwards in the Math Majors' Lounge (26-110)
Event: Special SCUM
Speakers: Allen Yuan, Carl Lian, Cole Graham, Nathan Pinsker

Date: 29 April 2015
Title: Solutions of Polynomial Equations
Speaker: Gary (Ka Yu) Tam
Abstract: In this talk I will discuss some basic notions in complex analysis and sketch a proof of the fact that every polynomial equation has a "solution" modulo a branched covering.
References:

Date: 22 April 2015
Title: The Standard Young Tableaux, RSK, and Planar Partitions
Speaker: Adit Radha
Abstract: I will discuss some elementary results concerning the Young Tableau. First I will discuss enumerating Standard Young Tableaux using the Hook length formula. I will then describe the RSK algorithm and use it to provide a simple bijection between permutations and pairs of Standard Young Tableaux. I will then (hopefully) spend most of my time discussing how an extension of the RSK algorithm gives a nice generating function for planar partitions.
References:

Date: 15 April 2015
Title: Categories: much more than abstract nonsense
Speaker: Peter Haine
Abstract: Many criticize category theory just saying that it’s abstract nonsense. The goal of this talk is to show that there’s much more to category theory than diagram chases and abstract nonsense proofs — it’s an interesting subject in it’s own right and a handy organizational principle when working in any mathematical field. We’ll draw on examples from algebra, topology, analysis, and other branches of math to contextualize category theory and show how it naturally arises in these areas. Some interesting examples include classical product and quotient constructions, groups, monoids, the orbit–stabilizer theorem, (co)homology theories, and the Riesz Representation Theorem. If time allows, we’ll talk about (co)limits and how category theory is the natural setting for abstract homotopy theories.
References:

Date: 8 April 2015
Title: Polygon Problem
Speaker: Zipei Nie
Abstract: Given a simple polygon M with n+3 edges such that any three of its vertices are not collinear. Let M_d (resp. M_e) be the set of diagonals (resp. epigonals), i.e., chords which lie entirely in the interior (resp. exterior) of P. For 0 \le i \le n, let d_i (resp. e_i) be the number of i-subset of M_d (resp. M_e) whose elements are pairwise disjoint chords. We'll prove that if M is convex, then \sum (-1)^i d_i = (-1)^n; otherwise \sum (-1)^i d_i = \sum (-1)^i e_i =0.
References:

There was no SCUM talk on Wednesday, April 1. Instead we encouraged people to go to HUMA's "Gender Gap in Mathematics Discussion" at Harvard in Emerson 105 (see here for directions), from 4pm--5:30pm.

Date: 18 March 2015
Title: 2
Speaker: Carl Lian
Abstract: Given four general lines L1, L2, L3, L4 in three dimensional space, how many lines L intersect all four?
References:

Date: 11 March 2015
Title: Measure Theory Crash Course
Speaker: Mark Sellke
Abstract: In 18.01 and 18.02, we learn about the Riemann integral and how it gives us a way to assign a "volume" to some nice sets in R^n by exhausting it with rectangles. But the Riemann integral has some annoying problems: for example, integrable functions may converge to non-integrable functions. We'll describe the Lebesgue measure, which allows us to extend Riemann integration to a more powerful theory. We'll also discuss general measures, and hopefully apply this to finding all (Lebesgue) measurable solutions to the Cauchy functional equation.
References:

Related Logistical Advice

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. For instance, E17-129 is known to be unbearably hot at times...)

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.)