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PDE and Differential Geometry Seminar

Title: A uniqueness theorem for asymptotically cylindrical shrinking Ricci solitons

Speaker: Brett Kotschwar (Arizona State University)

Time: Monday, October 23, 2017 at 2:30 pm

Place: MONT 214Abstract: Shrinking Ricci solitons are generalized fixed points of the Ricci flow equation and models for the geometry of solutions to the flow in the neighborhood of a developing singularity. It is conjectured that every four-dimensional complete noncompact shrinking soliton is smoothly asymptotic to either a cone or a standard cylinder at infinity . I will discuss recent joint work with Lu Wang related to this conjecture in which we prove that a shrinking Ricci soliton which is asymptotic to infinite order along some end to one of the standard cylinders $S^k\times {\mathbb{R}}^{n-k}$ for $k\geq 2$ must actually be isometric to the cylinder on that end.

Title: Riesz transform on Riemannian manifolds and beyond III

Speaker: Li Chen (University of Connecticut)

Time: Tuesday, October 24, 2017 at 3:30 pm

Place: MONT 314

Title: Images of Iterated Polynomials over Finite Fields

Speaker: Jamie Juul (Amherst College)

Time: Wednesday, October 25, 2017 at 11:15 am

Place: MONT 313Abstract: We discuss how to bound the size of the image of the $n$-th iterate of a polynomial over a finite field using results about arboreal Galois representations. The main term in this bound involves the fixed point proportion of the Galois group of the field extension of $\mathbb{F}_q(t)$ obtained by adjoining all pre-images of the transcendental $t$ under the $n$-th iterate of the polynomial. We give explicit bounds on the fixed point proportion of the group in a general case.

Title: Hausdorff Dimension

Speaker: Lisa Naples (University of Connecticut)

Time: Wednesday, October 25, 2017 at 5:45 pm

Place: MONT 314Abstract: The size of simple shapes are measured in different ways depending on their dimension. For 1-dimensional shapes, like a line or curve, we use length. For 2-dimensional shapes, like rectangles and triangles, we use area. For 3-dimensional shapes, like a solid cube or cylinder, we use volume. In calculus we compute length, area, and volume using limits of Riemann sums: ordinary integrals, double integrals, and triple integrals. Each type of integral depends on the dimension of the shape. It turns out there are some shapes that are so wild their dimension is no longer a number like 1, 2, or 3 but something in between. How do we determine the dimension of such a set and what is a good measure of its size adapted to its dimension? In this talk we will meet the Cantor set, a classical example of a fractal, and discuss one way to show its dimension is $\log_3 2$ and measure its size using a method designed for shapes with that dimension. We also discuss how we can apply the techniques in this example to other fractals.Comments: Free pizza and drinks!

Title: Vaught's conjecture in computability theory

Speaker: Antonio Montalbán (UC Berkeley)

Time: Thursday, October 26, 2017 at 4:00 pm

Place: MONT 214Abstract: We'll describe Vaught's conjecture, which is one of the most well-known and longest-standing open questions in logic. The conjecture essentially says that the continuum hypothesis holds when restricted to counting the number of models of a theory. We'll mention the author's result that this conjecture is equivalent to a computability-theoretic statement.

Title: Industry Job Panel

Speaker: Several speakers (University of Connecticut)

Time: Friday, October 27, 2017 at 12:20 pm

Place: MONT 111Abstract: TBA

Title: Quiver representations and theta functions

Speaker: Mandy Cheung (Harvard University)

Time: Friday, October 27, 2017 at 3:00 pm

Place: MONT 313Abstract: Scattering diagrams theta functions and broken lines were developed in order to describe toric degenerations of Calabi-Yau varieties and construct mirror pairs. Later, Gross-Hacking-Keel-Kontsevich unravel the relation of those objects with cluster algebras. In the talk, we will discuss how we can combine the representation theory with these objects. We will also see how the broken lines on scattering diagram give a stratification of quiver Grassmannians using this setting.

Title: TBA

Speaker: Gianmarco Molino (University of Connecticut)

Time: Friday, October 27, 2017 at 3:30 pm

Place: MONT 314Abstract: TBA