Spring 2016

Founded in March 2014, the Student Colloquium for Undergraduates in Mathematics (SCUM) is a series of weekly math talks by (mostly) undergraduates, for undergraduates, at MIT. Anyone 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 (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 2-131**.

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.)

Organizers: Peter Haine, 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 2-131

**The following Wednesdays are open** for talks, panels, or other activities: Wednesdays all taken, but we may be able to accommodate graduating seniors (or anyone else who won't be here in the foreseeable future), just let us know if you're interested and we should be able to find a non-Wednesday that works!

**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. Some ideas for reference:

- Panels: grad school, jobs/internships, UROPs/REUs/research, outreach, women in math, sharing stories (favorite parts about MIT/math?), anything really...
- Puzzles, problems, true/false test, etc.
- Wiki/blog/forum of some sort: class pages, advice, etc., cf. Course 2
- Periodic newsletter of some sort: fun facts, what everyone is up to, advice, etc.

(On the topic of math community: check out Math Major Cookie Breaks (Spring 2016), Undergraduate Mathematics Association (UMA), Undergraduate Society of Women in Math (USWIM), Harvard-MIT Mathematics Tournament.)

See the archive here.

Date: 11 May

Title: **Small Sumsets**

Speaker: **Mitchell Lee**

Abstract: Given sets A and B, define A + B to be {a + b : a in A, b in B}. Given the sizes of the sets A and B, how small can we make A + B? We will answer this question in the setting of an arbitrary additive group and introduce a few variations. No familiarity with groups will be assumed.

Date: 4 May

Title: **The Parity Problem in Sieve Theory**

Speaker: **Xiaoyu He** (Harvard)

Abstract: A sieve is a modified inclusion-exclusion process designed to count prime numbers. In this talk we will describe one of the most powerful and beautiful sieves, the Selberg Sieve. The Selberg Sieve is nevertheless off by exactly a factor of two from the Prime Number Theorem - this is the infamous "parity problem" in sieve theory, which says sieves by themselves cannot distinguish between primes and almost primes (i.e. products pq of two primes). We will show an easy elementary proof of the Almost-Prime Number Theorem (Selberg's Symmetry Formula), and then give some intuition for why no such proof can extend directly to counting prime numbers themselves.

Only basic combinatorics required, and no knowledge of analytic number theory necessary. We will touch on the the Riemann zeta function at times when necessary but most of the talk will avoid complex analysis.

Further references:

- (On all of sieve theory.) Friedlander, John B., and Henryk Iwaniec. Opera de cribro. Vol. 57. American Mathematical Soc., 2010.
- (On the Selberg Sieve.) http://www.math.harvard.edu/~elkies/M259.02/sieve.pdf
- (On the parity problem.) https://terrytao.wordpress.com/2007/06/05/open-question-the-parity-problem-in-sieve-theory/

Date: 27 April

Title: **Examples from birational geometry**

Speaker: **Ravi Jagadeesan** (Harvard)

Abstract: An important problem in algebraic geometry is to classify smooth
(projective) varieties. For curves, this classification is simple: a
smooth, projective curve is uniquely determined by its function field.
The Enriques-Kodaira classification explains that every smooth
projective surface can be reduced to an (essentially unique) "minimal
model" by contracting certain kinds of curves. In dimensions 3 and
higher, the situation is very different: minimal models are often
singular and non-unique, and the process of simplifying a smooth
variety is more complicated (the process is the minimal model program,
proposed by Mori). In this talk, I will present examples that
illustrate some of the general phenomena.

Some familiarity with algebra and smooth manifolds would be helpful
for motivation, but I will recall the algebraic geometry needed to
understand the talk.

Further references:

- J. Kollár and S. Mori. Birational geometry of algebraic varieties. Vol. 134. Cambridge University Press, 2008.
- K. Matsuki. Introduction to the Mori program. Springer Science & Business Media, 2013.

Date: 20 April

Title: **A generalization of the Schwarz lemma**

Speaker: **Sahana Vasudevan** (Harvard)

Abstract: A first course in complex analysis usually introduces the Schwarz lemma, one of the simplest results capturing the rigidity of holomorphic functions. In this talk, I will prove a powerful generalization of the Schwarz lemma due to Ahlfors. Then, I will describe several applications of it, including a proof of Big Picard, and a generalization of the Riemann mapping theorem to annuli.

This talk will *most* benefit people who have taken some complex analysis before, and for example have already seen the Riemann mapping theorem, and the proof of Big Picard using analysis about families of functions (or at least the statement if Big Picard). However, everyone is welcome; the relevant definitions and theorem statements will be reviewed in the course of the talk.

Further references:

- S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings
- L. Ahlfors, Complex Analysis
- E. Stein and R. Shakarchi, Complex Analysis

Date: 13 April

Title: **Pattern formation in soft and biological matter**

Speaker: **Prof. Jörn Dunkel**

Abstract: Identifying the generic ordering principles that govern multicellular and intracellular dynamics is essential for
separating universal from system-specific aspects in the physics of living organisms. In this talk, we will survey
and compare three recently proposed nonlinear continuum theories, which aim to describe pattern formation and
topological defect structures in soft elastic bilayer materials, dense bacterial suspensions and ATP-driven active
liquid crystals. We will discuss the phase diagrams of the three models, relate their predictions to experiments,
and emphasize the underlying universality ideas. The good agreement with experimental data supports the idea that
non-equilibrium pattern formation in a broad range of soft and active matter systems can be described effectively
within the same class of higher-order partial differential equations.

Further references:

- Phys Rev Lett 116: 104301, 2016
- Nature Physics 12: 341-345, 2016
- Nature Materials 14: 337-342, 2015

Date: 6 April

Title: **Using insights from fractal geometry to accelerate similarity search in
biological data**

Speaker: **Yun William Yu** (G)

Abstract: Many datasets exhibit well-defined structure that can be exploited to
design faster search tools, but it is not always clear when such
acceleration is possible. Here, we introduce a framework for similarity
search based on characterizing a dataset’s entropy and fractal
dimension. We prove that searching scales in time with metric entropy
(number of covering hyperspheres), if the fractal dimension of the
dataset is low, and scales in space with the sum of metric entropy and
information-theoretic entropy (randomness of the data). In this talk,
I'll work through some of the underlying theory and present biological
applications where these techniques can speed things up (e.g. BLAST
sequence search, protein similarity, and small molecule overlap).

Further references:

- http://www.cell.com/cell-systems/abstract/S2405-4712(15)00058-7

There was no talk on 30 March.

Date: 16 March

Title: **Quantum Groups**

Speaker: **Yuchen (Charles) Fu**

Abstract: Quantum groups are certain deformations of (the enveloping algebras of) the classical semisimple Lie algebras and have been intensively studied since mid 1980s. We will introduce their basic properties, some basic structures of the category of their representations, and their relationship with other topics (e.g. theoretical physics, knot theory, differential geometry). As this is an informal overview, results will be stated without proofs. We will assume familiarity with basic algebra. Necessary Lie-theoretic facts will be reviewed throughout the talk.

Date: 9 March

Title: **Panel**

Speakers: **Matthew Brennan '16, Peter Haine '16, Soohyun Park (G), Lynnelle Ye (G, Harvard)**

Abstract: It's up to you! (A panel of seniors and grad students will answer questions about anything you want to talk about that they are willing to talk about, especially but not necessarily topics related to math and grad school. Examples of potential topics: their experiences with classes, internships, research, job and graduate school applications, life at MIT/Harvard, broader philosophy and outreach.)

There was no talk on 2 March. Instead there was a UMA-SPS mixer; on Thursday there was also a UMA talk.

Date: 24 February

Title: **"Li+e/near" Algebra**

Speaker: **Janson Ng** (Hong Kong University of Science and Technology)

Abstract: Starting from definitions, we introduce a few basic concepts for the beautiful representation theory of Lie algebras. We then utilize these concepts to classify the finite-dimensional representations of sl(2,C). Only knowledge of linear algebra is assumed.

Further references:

- James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, vol. 9, Springer New York, 1972.
- Alexander Kirillov, Jr., An introduction to Lie groups and Lie algebras, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 2008.

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.)