Colloquium: Skyler Simmons (SUU)

Title: Stability of Collision-Based Periodic n-Body Problems

Abstract: The Newtonian n-body problem describes the motion of n point masses interacting with each other with gravitational force. The physical laws governing this motion were codified by Newton in his Principia in 1686. Under Newton’s laws, two colliding bodies’ velocities increase without bound as they approach collision. Levi-Civita demonstrated that through a suitable change of variables, certain collisions could be regularized, allowing the orbit to be continued past the collision. Variations on Levi-Civita’s work continue to be used in the study of collision-based orbits today. In this talk, I will present a few periodic orbits featuring collisions, describe the methods used to study their stability, and present known results about a few orbits.

Date: Thursday, February 22, 2018
Time: 4:00 PM
Room: 135 TMCB

Colloquium: Robert Snellman (University of California San Diego)

Title: Special Values of L-functions and Fitting Ideals

Abstract: The Main Conjecture in Iwasawa theory provides a deep connection between a certain p-adic zeta function and arithmetic data associated to class groups in p-power cyclotomic extensions. Proved by Mazur and Wiles, later generalized and proved by Wiles for totally real fields, the Main Conjecture can be interpreted in terms of a Fitting ideal. I will give an overview of the importance of higher Fitting ideals in the structure theory of modules, and provide a conjecture giving a relationship between higher Fitting ideals of Iwasawa modules and special values of L-functions.

Date: Thursday, February 15, 2018
Time: 4:00 PM
Room: 135 TMCB

Colloquium: Russell Ricks (Binghampton University)

Title: A Rank Rigidity Result for Certain Nonpositively Curved Spaces via Spherical Geometry

Abstract: To understand the geometry of nonpositively curved (NPC) spaces, it is natural to classify the various types of spaces that can occur. The Rank Rigidity Theorem for closed NPC manifolds separates the class of closed NPC manifolds into three very distinct types, and proves that nothing else can exist.

A version of Rank Rigidity has been conjectured for more general NPC spaces (CAT(0) spaces). In this talk, we discuss some progress toward the general conjecture, by reducing the problem to looking at patterns on spheres in the boundary at infinity. In particular, we can prove the conjecture for certain NPC spaces with one-dimensional boundary. In contrast to previous rank rigidity results, we do not assume any additional structure (such as a polyhedral or manifold structure) on the space.

Date: Tuesday, February 13
Time: 4:00 PM
Room: 135 TMCB

Colloquium: Tuan Pham (University of Minnesota)

Speaker: Tuan Pham

Title: Minimal blowup data for potential Navier-Stokes singularities in the half space

Abstract: It is known that the mild solutions to the incompressible 3D Navier-Stokes Equations (NSE), either in the whole space or a smooth domain with nonslip boundary condition, exist locally in time. Provided that a blowup solution exists, then it has been shown that a so-called minimal blowup data for NSE in the whole space exists. In this talk, I will explain how the presence of the boundary might influence the existence of minimal blowup data. This is joint work with Vladimir Sverak. Our results are motivated by a theorem of Gregory Seregin on boundary regularity.

Date: Thursday, February 8, 2018
Room: 135 TMCB

Colloquium: Dane Skabelund

Title: Algebraic Curves with Many Points over Finite Fields

Abstract: Algebraic curves with many points over finite fields have proven useful for creating good error-correcting codes and designing efficient algorithms for multiplication in finite fields. In this talk, I will discuss these applications, and describe the construction of two families of curves which meet the Hasse-Weil bound.

Date and time: Thursday, January 18 at 4:00 in room 135 TMCB.

Colloquium: Domingo Toledo

Title: Inequalities in Topology Motivated by the Schwarz Lemma

Abstract: We will discuss a number of inequalities that can be stated in purely topological terms, but their proofs may involve other ideas. The protoptype is Knesers inequality for the degree of a map of Riemann surfaces, published in 1930: if X, Y are Riemann surfaces of genus (gX,gY) > 1 and f : XY is a continuous map, then |degree(f)| ≤ (gX − 1)/(gY − 1). If X, Y have complex structures and f is holomorphic, then the inequality would be an immediate consequence of the Schwarz Lemma: holomorphic maps of the unit disk do not increase length in the Poincaré metric. We will discuss proofs of this inequality and related ones by various methods: bounded cohomology, harmonic maps, Higgs bundles. We will also indicate, as time permits, how these inequalities have motivated much work on the structure of the space of representations of the fundamental group of a surface in various Lie groups. This will be an exposition of work of Milnor, Wood, Dupont, Goldman, Gromov, Hitchin, and many others. It should be mentioned that some of the current work on this subject uses Ahlfors generalization of the Schwarz Lemma in a very essential way.

Date and time: Tuesday, January 16 at 4:00 in room 135 TMCB.