GIST will be online this semester. Contact Braeden or Eric if you want to speak or attend!
Title: Why would a 3-manifold topologist care about the E_8 lattice? (and how!)
Abstract: In this talk, I will give a very informal sketch of why and how the E_8 root lattice is involved in obstructing L-space surgeries on knots in the Poincaré homology sphere. In particular, I will describe a curious desired property of vectors in an integer lattice suggested by Heegaard Floer homology.
Title: Representations of sl_2(C)
Abstract: In past GIST talks you’ve heard me make bold claims such as “semisimple Lie algebras are made up of copies of sl_2.” In this talk we will look at one of the many reasons why this statement is so important. In particular, we will look at wonderfully simple structure of irreducible representations of sl_2(C).
Title: Fractional Dehn twist coefficients and capping off open books
Abstract: I'll introduce fractional Dehn twist coefficients of open book decompositions and talk about how they change under the operation of "capping off." I'll keep everything introductory, focusing on motivation for the definitions and what's known currently. Expect lots of pictures!
Title: The axiom of the fold
Abstract: In US mathematics curricula straightedge and compass constructions is a standard topic. These constructions, while natural in some setting, have well-known limitations such as trisecting the angle and doubling the cube. In this talk we will confront these limitations and show how to overcome them with a new axiom: paper-folding. Pictures will abound: I encourage the audience to bring a straightedge, compass, and plenty of paper to write on and fold!
Title: Enumerative questions on differentials on curves.
Abstract: I'll go through two enumerative problems on differentials forms on the Riemann Sphere by giving some conditions to their zeros and poles and trying to determine how many those differentials forms exists.
Title: Why modular forms?
Abstract: Modular forms and their generalization to automorphic forms are part of a deep and interesting story about number theory, but how did it get started? Why did people care in the first place? I won’t be attempting a thorough historical overview, but I will be attempting to show some concrete things that people noticed starting with Riemann, Dirichlet, and Ramanujan, and where that puts us today.
Title: Left-orderability of 3-manifold groups
Abstract: In this talk, I will first discuss some basics of left-orderable groups and quote (and maybe sketchily prove) some classical (but not necessarily basic) results about them. After that, I will illustrate the proof of a theorem due to Boyer, Rolfsen and Wiest, which asserts that the fundamental group of a 3-manifold is left-orderable iff it admits a non-trivial homomorphism to a left-orderable group. This is not true for arbitrary group and hence distinguishes 3-manifold groups, which is nice.
Title: Shortest paths in the Sierpiński carpet
Abstract: Everybody's first favorite fractal, the Cantor set, has a number of path-connected higher-dimensional generalizations. We will look at the construction of paths in the Sierpiński carpet and Menger sponge, with one application being finding the diameter of the carpet. I promise y'all, this one will be accessible because we'll mostly be doing geometry in the plane, albeit a ridiculous subset thereof, and there will be lots of pictures.
Title: The HOMFLY polynomial of an algebraic link and the topology of punctual Hilbert schemes
Abstract: If f(x,y) is a complex polynomial in two variables which vanishes at the origin, one can take the intersection of the vanishing locus of f with a small ball about the origin to obtain a link. Such links are called algebraic links. If f(x,y) cuts out a curve that is singular at the origin, the associated algebraic link is non-trivial and encodes information about this singularity. In this expository talk, we discuss a conjecture of Oblomkov and Shende, which states that the HOMFLY polynomial of an algebraic link is determined entirely by the Euler characteristics of certain closely related Hilbert schemes. This conjecture has been verified for torus knots and the (2, 13) cable of the right-handed trefoil knot. We briefly discuss connections to topics in number theory and arithmetic geometry. Expect lots of examples, and hands-on computation!
Title: Links, Skein modules, and SL_2(C)
Abstract: In this talk we will discuss a surprising connection between representations of the fundamental group into SL_2(C) and invariants of 3-manifolds inspired by the Jones polynomial. At varying times, the math in this talk will be adjacent to many different topics people at BC think about. To celebrate how many different areas of math might come up I'll be making BINGO cards so you can play along during the talk!
Title: On plane tilings
Abstract: The unifying theme of this expository talk is plane tiling. In the first half, we'll discuss classical results of periodic tiling---when patterns admit translational symmetries. In the second half, we'll explore the total opposite: a set of tiles that never admits a periodic tiling. Below is an example taken from the cover of the January 1977 issue of Scientific American.