Title: Fixed-point subgroups and Galois descent
Abstract: Let $\theta$ be an automorphism of $SL_n(k)$. In this talk we will investigate the structure of the subgroup $SL_n(k)^\theta=\{x \in SL_n(k) \mid \theta x = x\}$$. We will also discuss the relation between such subgroups and the Galois descent of algebraic groups.
Title: From the Upper Half Plane to Shimura Varieties
Abstract: Number theorists often consider quotients of the upper half plane by arithmetic or congruence subgroups, and the modular curve(s) is one such quotient, which, though defined from analytic constructions, can miraculously be realized as an algebraic variety. Shimura and Deligne generalized this in the study of Shimura varieties, which are parameter spaces of certain types of Hodge structures, just as the modular curve(s) parametrizes elliptic curves with certain additional features. In light of this, I will start off by talking about some results in Hodge theory and in the theory of algebraic groups, and streamline certain behaviours/properties of the upper half plane to motivate a "concrete" example of a Shimura variety. Unfortunately (or fortunately, for optimists) due to the technicality of the subject and time constraints, I will not be able to talk about definitions or results with extreme rigor, but I hope the talk suffices to paint a clear picture of an introduction to the topic.
Title: The Class and Unit Groups of a Number Ring
Abstract: Two fundamental aspects of number rings are their group of units and the ideal class group. I will present proofs about the structure of these groups using lattice techniques.
Title: $\Z \oplus \Z$ filtered complexes and piecewise linear functions
Abstract: When defining his reinterpretation of $\Upsilon_K(t)$, Livingston gives the idea of a general construction for defining piecewise linear function from a $\Z \oplus \Z$ filtered complex. Since his reinterpretation, multiple groups of low dimensional topologists have applied Livingston's construction to generate invariants that take the form of piecewise linear functions. We will examine Livingston's construction in order to describe how the structure of the $\Z \oplus \Z$ filtered complexes used to define these invariants provides strong restrictions on the values for the invariants.
Title: Congruences for Coefficients of Weakly Holomorphic Modular Forms in Level 2
Abstract: Modular forms are functions on the upper half-plane that have many symmetries. These functions have many ties to interesting parts of mathematics and especially number theory. Modular forms are typically written as a Fourier series, and the coefficients of this series can be interesting by themselves. For example, the coefficients can satisfy congruences mod n for interesting n. We study congruences for a basis of a space of weakly holomorphic modular forms of level 2. This partially answers a question posed by Andersen and Jenkins on their work for a related spaces of weakly holomorphic modular forms. This was joint work with Paul Jenkins and Ryan Keck at Brigham Young University.
Title: Dynamics: From one dimension to two and back again
Abstract: In this talk we explore the basic results and theory in one-dimensional discrete dynamics, including the central role played by a special class of algebraic integers called weak Perron numbers. We then use these ideas to build 2-dimensional systems with interesting behavior, and conclude by examining a sequence of particularly nice pseudo-Anosov maps constructed in this manner by the speaker.
Pictures and examples will be provided!
Title: Mission Inadmissible: Constructing Rigid Analytic Spaces in 50 Minutes
Abstract: The admissible goal of this talk is to write down the construction of rigid analytic spaces. These are the non-archimedean counterparts of complex analytic spaces. The talk will start with the analytification functor and Serre's GAGA in the complex setting, then gradually shifts to a discussion of Tate algebras and affinoid algebras, which are the local models for rigid analytic spaces. To globalize, we will go through the notion of $G$-topological spaces and put the Tate topology on affinoid algebras, which is not a topology in the usual sense. Time admitting, we will see the concept of rigid analytification and the statement of rigid analytic GAGA.
Title: Conics in the Projective Plane
Abstract: I will try to convince you today that there are 3264 comics in the projective plane tangent to 5 given conics. The proof is completely elementary!
Title: Galois groups of enumerative geometry
Abstract: Whenever you find an enumerative problem on an algebraic variety, you may ask if you can find the solutions via radicals using the coefficients on your variety. This question may be equivalent to determining the solvability of a Galois group. It will be explained how to associate a Galois group to an enumerative problem. These tools will be used for determining if we can find the solutions for the 28 bitangents on a plane quartic and the 27 lines on a cubic surface.
Title: Partially extending powers of pseudo-Anosovs to 3-manifolds
Abstract: We will look at interesting examples of surface homeomorphism that partially extend. Then we will construct a family of pseudo-Anosov homeomorphisms that require power to partially extend to the 3-manifold . There will be lots of pictures and no background in 3-manifolds is assumed.
Title: Exotic structures on \R^4
Abstract: In dimension three the smooth and topological categories of manifolds are equivalent. In dimension four, however, they are radically different. In particular there exist uncountably many distinct smooth structures on \R^4. In this talk we will show how to construct one of these exotic structures.
Title: Contact Structures and Foliations
Abstract: Tools from contact topology have recently proven extraordinarily useful in low-dimensional topology, most notably in results about Dehn surgery and Heegaard Floer homology. Many of these results rely on the interaction between contact structures and foliations on 3-manifolds. In this talk, I'll give an introduction to contact topology from the ground up, with the theory of foliations in mind. You can expect little to no background requirements and lots of pictures!
Title: Connectedness and local connectedness of the space of pointed hyperbolic surfaces
Abstract:
In this talk, I will give a meaningful way to topologize the space of all (pointed) hyperbolic surfaces—including both compact and non-compact, both finite- and infinite-type surfaces.This natural Gromov-Hausdorff topology allows us to make sense of convergence of surfaces and of discrete groups of PSL_2(R). I will quickly review surface decompositions and the Fenchel-Nielsen coordinates on the Teichmüller space before sketching a proof of the connectedness and local connectedness of this space. No prior familiarity with hyperbolic geometry is assumed.