Fall 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. Any MIT 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 (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. 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, 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

**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---there's always room for improvement and creativity---or ideas for other possible synergistic math activities).

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

See the archive here.

Evan Chen kindly took notes for many of the talks this semester (Fall 2015).

Date: 2 December 2015, in room 4-145

Title: **"Nonclumpiness" in the Parallel Chip-Firing Game**

Speaker: **Ziv Scully**

Abstract: The parallel chip-firing game is a deterministic automaton on graphs in which vertices "fire" chips to their neighbors when they have enough chips to do so. On finite graphs, the game has finitely many states and is therefore eventually periodic, but the possible periodic behavior on general graphs is not well understood, though it is known for some simple cases like trees [1], cycles [2], complete graphs [3], and complete bipartite graphs [4]. However, though little is known about global behavior, it turns out that certain "clumpy" local behavior is impossible [5], namely that no vertex can both fire and fail to fire twice in a row in a period. In this talk we'll introduce the parallel chip-firing game, prove some basic results about it, and conclude with a proof of the games "nonclumpiness".

References:

- Javier Bitar and Eric Goles, Parallel chip firing games on graphs, Theoret. Comput. Sci. 92 (1992), no. 2, 291--300.
- Luca Dall'Asta, Exact solution of the one-dimensional deterministic fixed-energy sandpile, Phys. Rev. Lett. 96 (2006), 058003.
- Lionel Levine, Parallel chip-firing on the complete graph: Devil's staircase and Poincare rotation number, Ergodic Theory Dynam. Systems 31 (2010), no. 3, 891.
- Tian-Yi Jiang, On the period lengths of the parallel chip-firing game, ArXiv e-prints (2010).
- Tian-Yi Jiang, Ziv Scully, and Yan X. Zhang. Motors and Impossible Firing Patterns in the Parallel Chip-Firing Game, SIAM Journal on Discrete Mathematics 29 (2015), no. 1, 615--630.

Date: 18 November 2015, in room 4-145

Title: **The Topology of Categories**

Speaker: **Colin Aitken**

Abstract: You've probably seen category theory used to study topology, but have you seen topology used to study category theory? Given a category C, we'll follow Quillen in introducing a topological space BC, called the classifying space of C, and trying to relate its topology to the categorical structure of C.
Highlights will (hopefully) include computing the fundamental group of BC in terms of purely categorical information, showing that adjoint functors induce homotopy equivalences, and showing that a poset viewed as a category is weak equivalent to the same poset viewed directly as a space.

References:

- D. Quillen, Higher Algebraic K-Theory: I

Date: 4 November 2015, in room 4-145

Title: **Graph Magnitude Homology**

Speaker: **Yuzhou Gu**

Abstract: Graph magnitude is an algebraic invariant of graphs defined by Leinster [1]. It has a lot of good properties such as inclusion-exclusion under certain conditions and is preserved under certain Whitney twists. Hepworth and Willerton [2] then defined graph magnitude homology, a categorification of graph magnitude. They translated properties of graph magnitude into graph magnitude homology language, and proved most of them. We will give the definitions, talk about the properties, and give some open problems.

References:

- T. Leinster, The magnitude of a graph. arXiv: 1401.4623.
- R. Hepworth and S. Willerton. Categorifying the magnitude of a graph. arXiv: 1505.04125.

Date: 28 October 2015, in room 4-145

Title: **A categorical proof of the orbit-stabilizer theorem.**

Speaker: **Peter Haine**

Abstract: In this talk we'll start with a review the basics of groups and group actions, and we'll explain the orbit-stabilizer theorem, noting a few applications. We'll then shift gears and talk about categories and functors, how to look at groups as special cases of categories, and group actions as functors. We'll use this perspective to provide a non-standard proof of the orbit-stabilizer theorem. If time permits, we'll discuss how to look at the representation theory of (finite) groups from a categorical perspective, and how this all relates to algebraic topology and categorical homotopy theory. Though familiarity with groups and group actions will be helpful, this talk is meant to be accessible to all and we won't assume any prior knowledge.

References:

- Michael Artin, Algebra, 2nd ed., Pearson, 2010.
- Emily Riehl, Categorical homotopy theory, Cambridge University Press, 2013.
- ______, Category theory in context. Preprint. Available at http://www.math.jhu.edu/~eriehl/727/context.pdf.

Date: 14 October 2015, in room 4-145

Title: **Kakeya Needle Problem**

Speaker: **Sandeep Silwal**

Abstract: In 1917, Japanese mathematician Soichi Kakeya posed the following problem: "What is the planar figure of least area in which a unit line segment may be turned through 360 degrees by a continuous movement?" These figures came to be known as Kakeya needle sets. In this talk, we present the clever solution of this problem given by the Russian mathematician Abram Besicovitch in 1928. We also give a brief discussion about how "large" these sets can be through various definitions of dimension.

References:

- Markus Furtner: http://www.mathematik.uni-muenchen.de/~lerdos/Stud/furtner.pdf
- Wolfram MathWorld: http://mathworld.wolfram.com/Hypocycloid.html
- Daniel Glasscock: https://people.math.osu.edu/glasscock.4/
- Indiana University: http://www.math.indiana.edu/gallery/kakeya.phtml

Date: 7 October 2015, in room 4-145

Title: **Roots of Unity Filter**

Speaker: **Victor Wang**

Abstract: We cover some examples of the ``roots of unity filter'' (based on the rotational symmetry of a regular polygon), i.e. ``finite Fourier analysis'', possibly drawing from

- sums of binomial coefficients over arithmetic progressions (classic contest problem);
- solution of the cubic using a third roots of unity filter, with brief connection to Galois theory perspective;
- a Fourier-analytic proof of Roth's theorem (from Ramsey theory) that one can always find a length-3 arithmetic progression (3-AP) in a subset of the integers with positive density.

- T. Gowers, https://gowers.wordpress.com/2007/09/15/discovering-a-formula-for-the-cubic/
- Y.-T. Siu, http://www.math.harvard.edu/~siu/math55a/from_solving_polynomial_equations_to_groups_part1.pdf
- A. Venkatesh, http://math.nyu.edu/~venkatesh/ent.html
- G. Dospinescu and T. Andreescu, Problems from the Book, Ch. 7 and 8

Date: 30 September 2015, in room 4-145

Title: **Combinatorial Nullstellensatz and Graph Colorings**

Speaker: **Evan Chen**

Abstract: We begin by presenting the so-called combinatorial nullstellensatz, also known as the "polynomial method".
After dispensing of a couple easy applications, we'll move on to talk about the more general problem of a list coloring:
given a graph and a list of colors at each vertex, is it possible to color the graph using the colors at that vertex?

References:

- http://www.tau.ac.il/~nogaa/PDFS/null2.pdf
- http://www.tau.ac.il/~nogaa/PDFS/chrom3.pdf
- http://www.mit.edu/~evanchen/handouts/SPARC_Combo_Null_Slides/SPARC_Combo_Null_Slides.pdf

Date: 23 September 2015, **in room 3-442** (in later weeks it will be in 4-145)

Title: **An introduction to Wigner matrices**

Speaker: **Ravi Bajaj**

Abstract: How can you integrate over a variable taking place in the space of real symmetric matrices? A Wigner matrix is a symmetric, infinite matrix whose diagonal is comprised of a sequence of independent, identically distributed random variables of variance 2 and whose off-diagonal entries are a sequence of independent, identically distributed random variables of variance 1. What is the spectrum of this matrix? For each dimension N, we take the first NxN entries and (after a normalization) prove that the spectrum converges to a semi-circle distribution supported on [-2,2].

References:

- Tao, T. (2012). Topics in random matrix theory (Vol. 132). American Mathematical Society.

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...) 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.)