Spring 2015

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.)

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).

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:

- The short, readable, beautiful paper by Payne and Weinberger: "An optimal Poincaré inequality for convex domains."

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**

- Allen Yuan:
**Homotopy Groups of Spheres**. Homotopy groups are the fundamental invariants in the study of homotopy theory. If we want to study CW-complexes, then it is important to understand the homotopy theory of their building blocks: the spheres. However, this is a difficult question to answer in general and an active area of study. I will define the homotopy groups, show some basic properties and constructions and time permitting, give an idea of what we currently know about homotopy groups of spheres. - Carl Lian:
**Algebraic curves and their moduli**. A curve is called algebraic if it can be described as the set of solutions to some system of polynomial equations in finitely many variables. A common strategy for studying algebraic curves, an idea with origins in the 19th century but later formalized by Grothendieck and others, has been to study their moduli space(s), a "space" whose points correspond "naturally" to curves. For instance, if you knew the "dimension" of this space, you would know how many "parameters" you would need to describe an algebraic curve. I will mention two questions of current research about the moduli space of curves and applications. - Cole Graham:
**Near-Equality and Robustness in Analysis**. The Cauchy-Schwarz inequality is one of the most valuable tools in analysis, and begs close study. When is equality attained? When is equality *almost* attained? What causes equality to break? Questions of this form are ubiquitous in classical and modern analysis. We view several problems through the lens of "robustness" around equality cases. We'll examine how small deviations from algebraic perfection can lead to fascinating open analytic questions. We hope to consider Cauchy-Schwarz, the Schrodinger equation, and black holes. - Nathan Pinsker:
**The Halting Problem**. What are the limits of computers? Are there problems that they can't solve, even given infinite time and space? Are some computers "more powerful" than others? We'll be looking at one of the most important results in computability theory, which paved the way to many other active fields in theoretical CS.

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:

- Otto Forster,
*Lectures on Riemann Surfaces*.

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:

- Richard Stanley’s text
*Topics in Algebraic Combinatorics*, freely available here.

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:

- Emily Riehl's
*Category Theory in Context*notes (currently being written): link

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:

- Lee, Carl W. "The associahedron and triangulations of the n-gon." European Journal of Combinatorics 10.6 (1989): 551-560.
- Shephard, G. C. "A Polygon Problem." American Mathematical Monthly (1996): 505-507.
- Devadoss, Satyan L., et al. "Visibility graphs and deformations of associahedra." arXiv preprint arXiv:0903.2848 (2009)

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:

- David Eisenbud & Joe Harris,
*3264 & All That: Intersection Theory in Algebraic Geometry*. (This is probably searchable online; there is currently a PDF here. Among other things, Chow rings are covered here.) - Carl Lian's notes from 18.315 (Schubert Calculus), taught by Alex Postnikov at MIT, Fall 2014. (Among other things, Grassmanian (spaces) and Schur polynomials are covered here.)

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:

- (Mostly Chapter 1 of) "Big Rudin", i.e.
*Real and complex analysis*, by Rudin (see the Chicago undergraduate mathematics bibliography for comments and other analysis books). - (For an alternative treatment, see Chapters 1 & 2 of
*Real Analysis: Measure theory, Integration, and Hilbert Spaces*, by Stein & Shakarchi.) - For questions about the existence of non-measurable sets vs. axiom of choice, see MathOverflow.

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.)