For other semesters, see the archive here.
Organizers: Carl Lian, Soohyun Park, Sam Trabucco
Meeting Time/Place: Wednesdays, 5:30 PM, E17-139
10 September 2014
Rationality of (Cubic) Hypersurfaces
How can you compute the integral of any function made out of trigonometric functions (e.g., sin(x)^2cos(x)^3/(1-cos(x)^5))? We will relate this to the following problem: Take a random polynomial P of degree d in variables x_1,x_2,...,x_n. When can we parametrize the points (x_1,...,x_n) such that P(x_1,...,x_n)=0 by n polynomials in n-1 variables? After drawing some pictures, the solution for d=2 will become clear. We will then consider the d=3 problem for various values of n. For n=1,2,3 we can draw some more pictures. After that, we will describe the n=4 and n=5 cases, and a recent proof that the n=5 case would follow from the so-called "Cancellation Conjecture."
17 September 2014 (Special Time: 6:00 PM)
Topology and PIE
How can we compute the volume of a collection of n intersecting shapes? The classic Principle of Inclusion-Exclusion (PIE) provides a formula that expresses this volume exactly. However, application of PIE is in practice extremely inefficient because the number of terms is exponential in n. This computational difficulty has spurred two lines of research: the first, to find general methods of approximation, and the second, to see if more efficient formulas exist for specific families of shapes. In this talk, we will explore the second line of research, proving a beautiful theorem by Naiman and Wynn (1992): for any collection of d-dimensional spheres, there exists a formula for the volume of the union that is polynomial in the number of spheres! This proof makes ingenious use of ideas from elementary topology, combinatorics, and geometry. No prior knowledge is assumed, but be prepared to confront the presence of plenty of intersection and union symbols.
24 September 2014
Growth Diagrams and RSK-like Correspondences
The RSK correspondence gives a bijection between matrices with nonnegative integer entries and pairs of semistandard Young tableaux. The correspondence is important in the theory of symmetric functions and the representation theory of the symmetric group. We give a less well-known description of the RSK correspondence via Fomin growth diagrams. This viewpoint is more natural, as it makes a number of fundamental properties of RSK immediate. The RSK correspondence is also one of four related correspondences, including the dual RSK, whose inputs are matrices with entries either 0 or 1. The growth diagram setting lets us generalize previous combinatorial results like the RSK and dual RSK into a single "super RSK correspondence." We will also see an application to symmetric function identities that encapsulates results from both the RSK and dual RSK correspondences.
1 October 2014
Parameter Spaces in Algebraic Geometry
This talk will be an introduction to the world of algebraic geometry through the lens of parameter spaces. Roughly speaking, algebraic geometry is the study of the geometry of polynomial equations, like x+2y+3z+4w=0 or x^7+y^7=1; the solution set to a system of polynomial equations is called an algebraic variety. Classifying varieties completely via their geometric properties is too ambitious of a goal, but one way to get a handle on an interesting family of varieties (say, planes in 4-dimensional space) is to understand the geometry of a space that parametrizes this family. That is, instead of trying to write down a list of every plane in 4-dimensional space, we instead look at a single variety whose points naturally correspond to planes in 4-dimensional space (this particular parameter space is known as a Grassmannian). We will give examples of parameter spaces, how to construct and study them, and see what they're good for. No prior exposure to algebraic geometry will be assumed -- the focus will be on examples, rather than general theory.
8 October 2014
Sets of Finite Perimeter and "Applications"
Suppose you are given some oddly-shaped piece of land, and would like to divide it in half using a fence of the least possible length. Is there a best way of doing this? Will the fence even be possible to construct (that is, how smooth is the path it will take)? In this talk we'll go over the theory of sets of finite perimeter, which is one tool that can be used to make these kinds of questions rigorous. We'll also go over some basic properties of sets of finite perimeter, and describe the arguments that are used to tackle these questions. Prerequisites: Some familiarity with multi-variable calculus is assumed.
15 October 2014
Linear Transformations of Stable Polynomials
Stable polynomials are complex polynomials with roots in some prescribed region of the complex plane. They play key roles in PDE theory, statistical mechanics, combinatorics, control theory, and the Riemann Hypothesis.* In this talk we'll look at linear transformations which preserve stability. We'll cover results ranging from the classical Gauss-Lucas theorem to a recently discovered characterization of such transformations. With time, we'll extend some results to classes of entire functions, and mention some hotly pursued open problems. *Millennium Prize not guaranteed.
22 October 2014
Math Majors' Panel (Special SCUM)
A panel of junior and senior math majors at MIT will answer questions about their experiences with classes, internships, UROPs, job and graduate school applications, life at MIT, and anything else you want to talk about.
29 October 2014
Physical Analogies in Wave Equations
Wave equations are obviously at the heart of physics, but the "physics" associated to a general, abstract wave equation arising in a non-physical context is also very important. Physical analogies typically provide the most intuitive, geometric route towards understanding the complex behavior of non-linear waves. In this lecture, we will discuss two of the most important physical ideas that come into play in a particular variety of wave equations: the optics, and scattering theory, of a curved spacetime. We will also provide intuition about how these physical ideas are connected to the more traditional tools available to PDE analysts. Prerequisites: A good understanding of calculus and linear algebra.
5 November 2014
Spontaneous Symmetry Breaking and the Physics of Phase Transitions
Symmetries play a special role in physical systems, a role which was formally codified in Noether's theorem back in the 19th century. For many years it was believed by physicists that Noether's theorem completely explained the extent to which symmetries played a role in any physical theory. However, in recent decades, the development of field theory has inspired research into symmetries that goes beyond simply classifying them and identifying the consequences of their existence. Now physicists also consider the breaking of symmetries to play an important role in modern physics. This talk will start with a brief examination of symmetries and Noether's theorem in classical mechanics. We will then give a quick introduction to the main ideas of statistical mechanics and the physics of phase transitions, where there are many examples of spontaneous symmetry breaking. In particular, we will consider superfluids as an example, and show that spontaneous symmetry breaking leads to immediate consequences for the spectrum of excitations in the superfluid. This result, known as Goldstone's theorem, has an analog in quantum field theory which is closely tied to the famous Higgs mechanism. If time allows, we will discuss in more detail the relationship between the content of this talk and the more general formalism of quantum field theory.
12 November 2014
In number theory (resp. calculus) it often helps to look at (rational) numbers modulo arbitrarily large prime powers p^k (resp. Taylor series expansions of functions *locally* at x=a). In a somewhat related vein, we might call a number "close to 0" if it has a large exponent of p in its prime factorization; this leads to a "p-adic absolute value" weakly analogous to the usual "real absolute value". With these analogies in mind, we will give a morally incorrect view of "p-adic analysis" by going through some of the following concrete problems: (1) most naturally, verifying some very unnatural but funny p-adic identities like 2^1/1 + 2^2/2 + ... = 0; (2) reading Rudin_p (replace || with ||_p, R with Q_p, etc.); - studying the zeros of linear recurrences (such as the Fibonacci sequence or more relevantly, 1+(-1)^n) and possibly some related (exponential) Diophantine equations; (3) studying divisibility of polynomials and (formal) power series, e.g. with Newton polygons (from "tropical geometry") and related methods.
19 November 2014
A beginning course in algebra presents the student with a multitude of mathematical structures (groups, rings, fields, vector spaces) and slightly different notions of maps between a pair of each of these structures. One might wonder whether a more abstract mathematical structure can capture essential facts about all of these simultaneously. One is led naturally to the notion of a category. I will explain the basics of category theory in an example-oriented fashion. Then, time-permitting, I will explain some of the basic intuition behind the theory of higher categories. The talk will aim to be accessible to any student taking an introductory course in algebra and interesting to a significantly wider audience.
10 December 2014
The Combinatorial Euler Characteristic
If you ask a combinatorialist what a natural number is, they will probably say that it is a cardinality (of a finite set). Similarly, if you ask a combinatorialist what an integer is, they might say that it is an Euler characteristic. The Euler characteristic is a generalization of cardinality which can take negative integer values. We will define the Euler characteristic, and then discuss some of the combinatorics of "negative sets."