Stable reduction for curves
Consider a family of geometric objects where the general object is nice, but one particular object is not. Stable reduction gives us a way to modify such a family in a way that preserves the general object while improving the degenerate one. This process requires machinery from birational geometry. In this talk, I will first describe a fundamental tool in birational geometry: blowing up. I will use this to improve a family of nodal curves degenerating to a cuspidal curve.
The Kirby Torus Trick for Surfaces
This talk will be expository. The fact that every topological surface admits a smooth structure is a classical result generally attributed to Radó in 1925, but that is generally stated without proof in contemporary courses on surface topology. Kirby's Torus Trick, which led to Kirby & Siebenmann's classification of PL structures on high-dimensional topological manifolds in the 1960s, can be used to prove a theorem which can be used to prove this fact--without using the Schönflies theorem, which Radó's proof uses. This is the subject of a short 2014 paper by Allen Hatcher, which I will explicate this week.
The $\mathrm{GL}_{4}$ Rapoport-Zink Space
The $\mathrm{GL}_{2n}$ Rapoport-Zink space is a moduli space of supersingular $p$-divisible groups of dimension $n$ and height $2n$, with a quasi-isogeny to a fixed base point. After the $\mathrm{GL}_2$ Rapoport-Zink space, which is zero-dimensional, the $\mathrm{GL}_4$ Rapoport-Zink space has the most fundamental moduli description, yet relatively little of its specific geometry has been explored. In this talk, I will give a full description of the geometry of the $\mathrm{GL}_4$ Rapoport-Zink space, including the connected components, irreducible components, and intersection behavior of the irreducible components. As an application of the main result, I will also give a description of the supersingular locus of the Shimura variety for the group $\mathrm{GU}(2,2)$ over a prime split in the corresponding imaginary quadratic field.
A trip to the zoo: examples of open 3-manifolds
In this talk, we will give examples of open 3-manifolds that describe some of the main difference with compact 3-manifolds. In particular, we will be interested in hyperbolization results for a specific class of these 3-manifolds.
Representations of the Braid Group
Braids are an important mathematical object with connections to other subjects including knot theory and mapping class groups. Braids on n strands form a group, and investigating the algebraic properties of these braid groups has useful applications to other fields. For example: faithful representations of the braid group can be used to find faithful representations of other mapping class groups. In this talk, we will examine representations of the braid group and their relationship to knot polynomials. We will also explore a topological interpretation of the Burau representation and how similar topological ideas can be used to produce faithful representations of the braid group.
Polynomial dynamics on the Riemann sphere
If we start with a polynomial f defined on the Riemann sphere S^2, we can look at the itinerary of different points under repeated application of f. In general, different points will exhibit different behavior: some will remain fixed, others might accumulate at a point or be repulsed, and still others might travel so erratically as to be unpredictable. Our hope is to get a handle on this situation, and since we’re working in just about the nicest setting possible, we can actually say quite a lot. What exactly? You’ll have to come to the talk to find out! There will be tons of pretty pictures and hands-on examples to motivate and illustrate the talk.
Generic Graphs
Erdös and Renyi in 1963 showed that any random construction of a countable graph results in a unique graph, almost surely. This graph, which we will discuss, is huge and messy with each vertex adjacent to infinitely many others. So it is a bit difficult to visualize.
What happens if we restrict our attention to graphs where each vertex only has finitely many neighbors? What does a ‘generic’ graph look like? We will introduce a measure on the space of graphs to answer these questions.
Informal introduction to how Langlands program evolved over function fields.
A Crude Classification of Surface Groups
I will discuss metrics on finitely generated groups, the fundamental theorem of geometric group theory, and – time permitting – give a crude classification of surface groups.
Root Systems: An Introduction via Finite Reflection Groups
Recall that for a set of commuting diagonalizable matrices, there exists a basis of simultaneous eigenvectors. For each of these eigenvectors, we consider the function whose inputs are the commuting diagonalizable matrices and whose output is the eigenvalue for these matrices corresponding to the simultaneous eigenvector. Roots play an integral role in the subject of Representation theory, but unfortunately, but the precise definition is hidden behind a wall of semisimple Lie algebras. In my talk I will try to give a brief introduction to root systems from the point of view of finite groups, bypassing all the hardcore linear algebra.
Spanning Surfaces for (mainly Torus) Knots
Generally speaking, given a knot K inside the 3-sphere S^3, a spanning surface for K is a surface S in either S^3 or B^4 (which is bounded by S^3) such that the boundary of S is precisely K. By restricting to certain classes of spanning surfaces, one can define various notions of "the genus of a knot".
I will first discuss the two most common such notions, namely the Seifert genus and the (orientable) 4-genus. After that, I will talk about the nonorientable 4-genus and present an open conjecture about this genus of torus knots.
Grothendieck Topology and the Kummer Sequence
Let M be a compact oriented manifold and let f:M --> M be a smooth map. Then in algebraic topology, the Lefschetz fixed point formula relates the fixed points of f to the singular cohomology of M. The notion of fixed points of a map, however, is of immense importance in number theory: heuristically, the number of points of a (nice) variety over F_p is related to the number of fixed points of the (suitable) Frobenius.
To find the correct analogue of Lefschetz's formula in algebraic geometry, one has to look into better cohomology theories, and from this the Grothendieck topology arises.
I will not go into details of the cohomologies involved, but rather I will motivate using the Kummer sequence and explain why the Zariski topology is not good enough, and finally I will talk about how the étale and fppf topologies make the Kummer sequence exact. If time permits (though this will likely not happen), I will talk about something else such as the étale cohomology or fppf descent.