Boston College G.I.S.T Seminar
(Getting Into Some Theorems!)
Fall 2015
Mondays at 4pm in Maloney 560
This is the homepage of the BC math department's graduate student seminar. Interested in speaking? Email Siddhi!

Sept 142015
Adam Saltz
A transverse invariant from annular Khovanov homology
Annular Khovanov homology is a refinement of Khovanov homology for links embedded in an annulus. Braid closures are natural examples of such links, and there is a wellknown correspondence between braids and transverse links. Expanding on work of Plamenevskaya, I will present a computable conjugacy class invariant whose minimum we hope to be an effective transverse invariant. The invariant has applications to the word problem, the lengths of certain spectral sequences, and some classical questions about braids. This is joint work with Diana Hubbard.

Sept 212015
Peter Feller
Three roads to knot theory
Not assuming any knowledge of knot theory, we will shortly discuss three different roads into knot theory each ending at a theorem (after all this is about "Getting Into Some Theorems").
Firstly, a diagrammatical and combinatorial approach (initiated by a pipe dream of Lord Kelvin) leading up to Tait's Conjectures.
Secondly, 3dimensional surgery on knots and the LickorishWallace theorem that states that every 3manifold arises as surgeries on knots.
Thirdly, 4dimensional Kirbycalculus and the notion of sliceness yielding a "simple" construction for 'large' exotic smooth structure on R^4.
The latter is (arguably) one of the weirdest results in math. It crucially uses Freedman's results on topological 4manifiolds and topological sliceness.
We take it as a motivation to study generalizations of sliceness and discuss an upper bounds for the topological slice genus in terms of the Alexander polynomial.

Sept 282015
Beth Romano
The Local Langlands Correspondence: New Examples from the Epipelagic Zone
The Local Langlands Correspondence (LLC) conjectures that for every representation of a padic group there should be a unique corresponding field extension of Q_p. I'll talk a bit about filtrations related to padic fields and how these show up on each side of the LLC. I'll then talk about the "epipelagic zone" of the correspondence, and explicitly construct an example of the LLC. I won't assume much prior knowledge, aside from some familiarity with Q_p.

Oct 52015
Shucheng Yu
An Introduction to Modular Forms
A modular form is a holomorphic function on the upper half plane satisfying certain conditions. I'll give some basic examples of modular forms, and then introduce the L functions attached to modular forms. I'll talk about certain properties of such Lfunctions, e.g. analytic continuation and functional equation.

Oct 192015
Spencer Leslie
Modular forms and Elliptic curves: A glimpse into the Langlands Correspondence
Most of us have heard of the famous connection between elliptic curves and modular forms, conjectured by ShimuraTanayama and proved by Wiles, TaylorWiles and others. This is the Modularity theorem, and implies Fermat's Last Theorem. Well, we're not going to talk about all of that, but thanks to Shucheng's introduction to modular forms last week we can discuss the connection between these two types of object by discussing their Lfunctions. We will introduce (in a hopefully friendly way) the Lfunctions of a modular form as well as the Lfunction of an elliptic curve and state the Lfunction version of the Modularity theorem.

Oct 262015
Kyle Hayden
Lagrangian fillings of (positive) Legendrian knots: One theorem to rule them all
Many questions in knot theory are centered around determining what types of surfaces a given knot can bound. For example, every knot in the 3sphere bounds a compact, orientable surface. We'll introduce the notion of a Lagrangian surface in a symplectic manifold and ask: When does a Legendrian knot in the contact 3sphere bound a Lagrangian surface in the symplectic 4ball? To investigate this question, we'll discuss a combinatorial tool called a ruling of a knot diagram. I'll promote a conjectured answer to the question and sketch of a proof that all positive knots possess such surfaces. I will also prove that there are no upper bounds on the number of chalk colors you can use in one drawing. (This is old joint work with Joshua Sabloff.)

Nov 22015
Melissa Zhang
Does P=NP? An Introduction to the Theory of Computation
Along with "WHAT IS THE MEANING OF LIFE?" and "HOW DO I FIND EVERLASTING LOVE?", "DOES P = NP?" is one of the most popular questions of our day, for the answer potentially bears enormous consequences on our futures and our fantasies. In this talk we'll take quick look at the fundamentals in the theory of computation, and quickly move our way toward understanding the problem statement. We'll see some examples of common examples in the theory, such as HALT (an undecidable problem), "is this number prime?" (a polynomialtime problem), and CLIQUE and 3SAT (two popular NPcomplete problems). If we have time, we'll talk about what kinds of proofs might work, which can't, and what the current frontiers for research in this area look like.

Nov 92015
Tom Crawford
Gordon's Conjecture: Counting Manifolds' Exceptional Dehn Surgeries
Given a compact irreducible hyperbolic 3manifold with torus boundary, all but a finite number of Dehn fillings yield a hyperbolic 3manifold. Cameron Gordon conjectured that no such manifold could have more than 10 of these exceptional (nonhyperbolic) fillings. Marc Lackenby and our own Robert Meyerhoff proved this in 2000. We will discuss an attempt to improve this bound to Gordon's stronger conjecture: that other than the figure8 knot complement (which realizes the bound of 10), no manifold can have more than 8 exceptional fillings.

Nov 162015
Scott Mullane
Polygons, Donuts and Snakes: Degenerating abelian differentials
Through the use of lots of pictures of polygons, donuts and snakes, I'll discuss the different connected components of the moduli space of abelian differentials and how abelian differentials degenerate. Maybe I'll get to talking about my research, but who knows.
P.S. There will be lots of pictures.

Nov 232015
Siddhi Krishna
Spectral Sequences in the Wild
I will give a brief introduction to spectral sequences. We won't get too much into the gory homological algebra details  rather, we'll talk about the pieces that go into defining a spectral sequence, and the vocabulary needed to talk about them. In addition to getting our hands dirty and proving the 5ive Lemma (!), we will discuss how spectral sequences naturally show up in topology and number theory.

Nov 302015
Ari Shnidman
Counting cusps and isogeny volcanoes
I'll present a solution to a counting problem that shows up in many contexts: in number theory (how many orbits are there for the action of a congruence subgroup on the projective line?), topology (how many cusps are there on a hyperbolic 3manifold?), and algebraic geometry (how many elliptic curves lie on a fixed abelian surface?). The key tool is an explicit description of the Ext groups of a pair of CM elliptic curves. The general formula involves taking walks along certain graphs called isogeny volcanoes. This is joint work with Julian Rosen.

Dec 72015
Anand Patel
Rational Distance Problems
I will talk about the "Rational Distance Problem." No familiarity will be assumed, though it would be nice if the audience has had some exposure to rational numbers, distances, and problems.