Boston College G.I.S.T Seminar
(Getting Into Some Theorems!)
Spring 2016
Mondays at 4pm in Maloney 560
This is the homepage of the BC math department's graduate student seminar. Interested in speaking? Email Siddhi!

Feb 12016
Cihan Soylu
Siegel Modular Varieties
I will explain the moduli interpretation of Siegel Modular Varieties as moduli spaces of abelian varieties with extra structure. We start with a brief introduction to complex abelian varieties, so no familiarity will be assumed.

Feb 82016
Jonah Gaster
Combinatorics and geometry in the curve complex
The study of Harvey's "complex of curves" (or the curve complex) of a topological surface has had profound impact on the study of Teichmuller spaces, mapping class groups, and 3manifolds. I will define this simple and complicated object, introduce some basic properties, indicate some deep theorems that have resulted from its study, and describe some other aspects of the combinatorics of the curve complex that remain opaque. This last bit will include ongoing joint work with Nick Vlamis and Josh Greene. Most of the talk should be accessible to any grad student, and the rest of it I'll do my best to make comprehensible.

Feb 152016
Dongtai (Tyson) He
Khovanov homology and the slice genus.
Based on Lee's work on Khovanov (co)homology, Jacob Rasmussen defined a knot concordance invariant s. I will first review Lee's modification of the differential in Khovanov (co)homology. Then I will go over the definition and basic properties of s. I will show that s provides a lower bound for the slice genus of a knot. This gives a purely combinatorial proof of Milnor's conjecture which saids the slice genus of a torus knot equals its genus.

Feb 222016
Lev TovstopyatNelip
HFK^{} concordance maps
After reviewing an interpretation of HFK^{} due to Etnyre, VelaVick, and Zarev, I will quickly define the concordance/cobordism maps on HFK(HAT) due to Juhasz. Combining these two ideas gives us concordance, and possibly cobordism, maps for HFK^{}. This is a work in progress.

Feb 292016
Adam Saltz
Khovanov dreams, Khovanov memes
Meier and Zupan introduced triplane diagrams for knotted surfaces in their study of bridge trisections. A triplane diagram is just three simple tangles, but they may allow us to use knottheoretic tools on knotted surfaces. I use Szabo's perturbation of Khovanov homology and some novel algebraic constructions due to Manolescu and Ozsvath to associate an A_\inftyalgebra to a triplane diagram.

Mar 142016
Yuanqing Cai
Theta functions
The quadratic theta function is a modular form of half integer weight. In this talk, I will discuss the properties of the quadratic theta function and its connection with metaplectic automorphic forms. Then I will define the generalized theta functions on higher rank groups and discuss their applications in number theory.

Mar 212016
Ian Banfield
3 Braid Links
Links that can be represented by 3braids are a very nice class of links to study. While they admit a complete classification, they are not "too trivial" and often serve as a very nice class of links to test out conjectures. I'll explain Xu's classification of 3braid links and will discuss properties of 3braids, including an application to counting the number of genus one fibred knots in any 3manifold. I might also talk about the sutured Khovanov homology of positive 3braid links.

April 42016
Mustafa Cengiz
Heegaard splittings of irreducible 3manifolds are not unique.
In 1975 Birman, GonzalesAcuna and Montesinos gave the first counterexample to the conjecture that same genus Heegaard splittings of an irreducible 3manifold are isotopic. The example is constructed as homeomorphic branched covers of S^3 along two nonisotopic knots suggesting two distinct Heegaard splittings. I will explain the details of the construction in this talk. We start with an introduction to Heegaard splittings, and I will define basic tools throughout the talk, so no familiarity is needed.

April 112016
Sangsan (Tee) Warakkagun
Distribution of values: Nevanlinna Theory
One can rephrase the Fundamental Theorem of Algebra (FTA) as follows: A nonconstant polynomial with complex coefficients takes every finite complex value finitely many times. A direct generalization of FTA to entire functions or meromorphic functions is not true. Instead, we ask how the zeros of a function are distributed on the complex plane. In this talk, I will provide some results of Nevanlinna Theory that will help us understand the value distribution of meromorphic functions.

April 252016
Siddhi Krishna
Lefschetz Fibrations Going Wild
I will introduce Lefschetz Fibrations, which are relevant to lowdimensional topologists and algebraic geometers alike. Lefschetz Fibrations give a method of studying 4 (real) dimensional manifolds. Throughout the talk, we'll discuss (real and complex) Morse Theory, the braid group, and the topology of (real) surfaces. My goal will be to state some theorems of Thurston, Gompf, and Donaldson, which prove that understanding Lefschetz Fibrations gives a method for classifying symplectic 4manifolds. Along the way, I'll derive the PicardLefschetz formula. No prior knowledge about any of the above words will be assumed (except maybe "manifold").