Representations of S_n
In this talk, I will be discussing the irreducible representations of the symmetric group S_n, and prove their connection with Young tableaux. I will also discuss the Frobenius character formula and some of its application.
Braids in algebra and topology
Braids are not just for hair and challah; they are mathematically interesting objects in their own right. The study of braids has contributed to many areas of mathematics and has applications ranging from cryptography and fluid mechanics to literary analysis. Our story begins with a description of the abstract braid group discovered by Emil Artin in the early 1900's. Then we'll see how braids show up in topology, especially when studying knotted curves in three-dimensional space and surfaces in four-dimensional space. Finally, we weave these two threads together to see how a special subset of the braid group corresponds to a set of knots that arise when studying singularities of complex functions.
Examples of Moduli Spaces
A moduli space is a topological space/manifold/scheme whose points parametrize interesting objects. In this talk, we'll explore several examples of moduli spaces from the perspectives of topology, differential geometry, and number theory/algebraic geometry. We'll end with my favorite example of a moduli space, which also happens to be a Shimura variety. This talk has been prepared with all audiences in mind!
A counting problem and Eisenstein series
In this talk, I'll discuss a classical counting problem studied by Schmidt. I'll show how Eisenstein series shows up naturally in this counting problem and how the spectral theory can help us in solving it. I'll try to avoid most of the computations. Meanwhile, as probably the unique analytic number theorist (graduate student) in BC, I'll try to present some of the computations. If time allows, I'll discuss some of my own results. Most of this talk should be accessible to everyone.
Schur-Weyl duality and maybe some other examples...
An important idea in representation theory is that it is useful to study to action of two commuting sets of linear operators. This principle motivates the theta correspondence, geometric approaches to the local Langland correspondence, and many other topics. Instead, I want to talk about the most classical case: Schur-Weyl duality, which gives an explicit way of relating the representation theory of general linear groups to that of (finite) symmetric groups.
Double slice genus and related invariants
I will be speaking about some preliminary invariants in knot theory leading up to the definition of double slice genus, and then follow up with some related invariants/obstructions via embeddings of 3 manifolds in 4 manifolds.
Geometric invariant theory
In this talk, I will present the fundamental definitions of geometric invariant theory (GIT). GIT asks the question: What is the correct way to take a quotient of a variety by a group action, so that this quotient also admits a variety structure? It turns out that given a fixed variety under a fixed group action, there are many ways to take such a quotient, and some quotients may be more desirable than others. I will focus on a single example: C* acting on C^2 by scaling coordinates in the natural way. If time permits, I will briefly discuss my own work, which aims to characterize semi-stable replacements.
Fibrations, Foliations, and Branched Surfaces
One of my favorite classes of knots are those that are fibered; these are knots in S3 whose complements are surface bundles over S1. In addition to having some cool algebraic properties, they also lend themselves to some beautiful geometric constructions. In this talk, I'll provide some intuition for how to visualize fibered knots, and try to convince you that one generalization is via the theory of taut foliations. Time permitting, I'll explain how to use branched surfaces to construct taut foliations in the Trefoil exterior.
Little topology background will be assumed. Please bring colored pens!
Octonions and G2
Maybe most of you have heard the story of how Hamilton discovered the quaternions. One evening, while walking along the Royal Canal in Dublin with his wife, Hamilton felt a “galvanic circuit of thought close; and the sparks which fell from it were the fundamental equations between i, j, k; exactly such as I have used them ever since.” Paper being scarce at hand, Hamilton carved his equations into the stone of Brougham Bridge: "i^2 = j^2 = k^2 = ijk = -1”.
I really like the quaternions, however, this talk is not, strictly speaking, about the quaternions, but rather their “double”, the octonions. They’re more eccentric than the quaternions in that they are non-associative. I’ll talk about what the octonions are and how one can use them to construct the compact Lie group of type G2.
Mutation, positive Hopf plumbings and knot Floer homology
In knot theory, mutation is a procedure that turns a knot into another (possibly different) knot. In terms of knot diagrams, this means that one cuts out a part of a given diagram, rotates it and glues it back into the original diagram.
Distinguishing mutant knot by means of knot invariants is a notoriously difficult task, because many invariants are insensitive to mutation.
In this talk I will first talk about a family of knots (resp. links) that allows for a particular type of mutation which I call tree-like mutation. Then I will give a little background on knot Floer homology, which is a quite young knot invariant and known to be (in general) sensitive to mutation. After that, I will state a result that can be regarded as a partial answer to the question whether knot Floer homology is invariant under tree-like mutation.
Hilbert functions, Hilbert polynomials, and Hilbert schemes: an example driven introduction
In this talk, I will introduce two invariants of projective schemes: Hilbert functions and Hilbert polynomials. After some examples, I will define the Hilbert scheme H_p(P^n), which is a moduli space whose closed points parametrize the set of all subschemes of the projective space P^n with Hilbert polynomial p(m). I will then do the example of H_2(P^2), the Hilbert scheme parametrizing length 2 subschemes in P^2 and state some general facts. No knowledge of schemes will be necessary to understand this talk.
Free Bicuspid Groups in Small-Cusped Hyperbolic 3-Manifolds
One approach to the study of cusped hyperbolic 3-manifolds is to look at the shape of the cusp itself. Cusps lift to collections of horospheres in H^3. By specifying parameters which govern aspects such as the the size, shape, and twist of the isometries of a manifold, we can determine the relative size and positioning (known as the packing) of these horospheres. Typically hyperbolic 3-manifolds with low volume exhibit cycles of tangent horospheres known as bracelets (or necklaces). We consider manifolds with cusps that have no such bracelets. Put another way, we consider manifolds M with the property that pi_1(M) has a subgroup B congruent to Z^2 * Z, known as a free bicuspid group. Specifically we attempt to determine how small a cusp can be if it does not have any bracelets.