Boston College G.I.S.T Seminar
(Getting Into Some Theorems!)
Fall 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!

Sept 192016
Matt Grimes
Moduli spaces and the fundamental theorem of algebra
It is one of life's funny ironies that the fundamental theorem of algebra is usually not proved with algebra. In this talk, we will develop some of the language of moduli spaces and use tools from geometry and topology to prove the fundamental theorem of algebra. We will place heavy emphasis on accessibility and motivation.

Sept 262016
Spencer Leslie
Intersection Homology, Poincare Duality and Perverse Sheaves
In the context of closed, orientable manifolds, Poincare duality is a beautiful and important relationship between the singular homology and cohomology of the space which also has connections to intersection pairing of cycles. In the context of singular spaces, however, this relationship breaks down.
In this talk, we will discuss the theory of intersection homology, a theory developed to extend Poincare duality and intersection pairings to singular spaces. It is from this very topological construction that the theory of perverse sheaves emerges.

Oct 32016
Ian Banfield
Constructions of fibered knots
A knot K is fibered if it bounds a surface S, such that the knot exterior fibers over the circle with S a fiber. This is a very strong geometric condition; it implies for example that S is minimal genus amongst all surfaces bounding the knot. We’ll review constructions by Stallings, Milnor and Rudolph to construct fibered knots. It turns out that these constructions are all related to braid theory (and complex geometry), and we’ll explain why. If time permits, we may talk about homological invariants.

Oct 172016
Scott Mullane
Moduli spaces, geometric interpretations of residues and orbit closures of null ngon surfaces
I plan on providing many different ways of thinking about Riemann surfaces, holomorphic and meromorphic differentials on Riemann surfaces and geometric ways to consider the residue at a pole of a meromorphic differential. Providing an upper bound for the number of irreducible components of the strata of zero residue differentials in general genus can be reduced to the question of the SL(2,R) orbit closures of null ngon surfaces in genus one. I'll explain this question and give examples.

Oct 242016
Shucheng Yu
A proof of Oppenheim conjecture
Given a non degenerate indefinite and irrational real quadratic form Q in n (n\geq 3) variables, the Oppenheim conjecture states that the set of values of Q on integer vectors is a dense subset of the real numbers. This conjecture was proved by Margulis in 1987 using the method arising from ergodic theory and the study of discrete subgroups of semisimle Lie groups. In this talk I'll sketch a proof of Oppenheim conjecture assuming Ratner's orbit closure theorem.

Oct 312016
Mustafa Cengiz
Nonzero degree maps between Seifert fibered spaces of infinite fundamental group
A. Edmonds (1979) proved that a nonzero degree map between closed connected orientable surfaces is homotopic to a composition of branched coverings and pinches, i.e. crushing a subsurface to a point. An analogous theorem holds for Seifert fibered 3manifolds which are circle bundles over 2orbifolds (surfaces with singularities). A Theorem of Y. Rong (1993) says that a nonzero degree map between closed connected orientable Seifert fibered 3manifolds with infinite fundamental groups is homotopic to a
composition of fiber preserving branched coverings and vertical pinches, i.e. taking a subsurface in the base orbifold and crushing all corresponding circle fibers to a circle. I’ll talk about the interesting parts of Rong’s proof.

Nov 72016
Cristina Mullican
Simple closed curves on the punctured torus and the primitives of F(s,t)
The free group on two generators is the fundamental group of the once punctured torus. A word in F(s,t) is primitive if and only if it corresponds to a simple closed curve on the punctured torus. A key to this correspondence is Nielsen's theorem(1923) that Nielsen moves generate Aut(Fn). Moreover, Osborne and Zieshang (1981) give a nice geometric algorithm for finding the primitive elements of F(s,t) corresponding to the abelianized element s^mt^n and for finding an associative primitive. We will sketch the proof of the correspondence between primitives and scc and illustrate the geometric algorithm to find primitives of F(s,t).

Nov 142016
Tom Crawford
Gordon's Conjecture: Counting Exceptional Dehn Fillings
Dehn surgery on a cusped hyperbolic 3 manifold involves removing a torus neighborhood of the cusp and filling the region back in with a solid torus. The different ways this can be done can be parameterized by the class of simple closed curves on the tours boundary of the cusp which bound a solid disk in the resulting manifold. Thurston proved that for all but a finite set of exceptional slopes (classes of simple closed curves on a torus), the resulting manifold yields a hyperbolic metric. Marc Lackenby and Rob Meyerhoff showed that this finite number of exceptional slopes is at most 10. We will sketch the proof of the AgolLackenby 6Theorem which was crucial in this finding.

Nov 282016
Maria Fox
Points and Curves and Honda and Tate
We will discuss rational points on genus zero curves, points on genus one curves over finite fields, a theorem of Honda and Tate, and how these topics all fit together! The only prerequisite is some familiarity with circles, lines, and points. There will be examples!

Dec 52016
Tom Cremaschi
Results on Haken Manifolds
In this talk, we will introduce Haken manifolds and their hierarchies. Then we will state some results on the rigidity of their Homotopy Type due to Waldhausen, Scott, and Johansson.