Seminars and Colloquia by Series

Ordered groups and n-dimensional dynamics

Series
School of Mathematics Colloquium
Time
Friday, December 6, 2019 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dale RolfsenUBC

A group is said to be torsion-free if it has no elements of finite order.  An example is the group, under composition, of self-homeomorphisms (continuous maps with continuous inverses) of the interval I = [0, 1] fixed on the boundary {0, 1}.  In fact this group has the stronger property of being left-orderable, meaning that the elements of the group can be ordered in a way that is nvariant under left-multiplication.  If one restricts to piecewise-linear (PL) homeomorphisms, there exists a two-sided (bi-)ordering, an even stronger property of groups.

I will discuss joint work with Danny Calegari concerning groups of homeomorphisms of the cube [0, 1]^n fixed on the boundary.  In the PL category, this group is left-orderable, but not bi-orderable, for all n>1.  Also I will report on recent work of James Hyde showing that left-orderability fails for n>1 in the topological category.  

A solution to the Burr-Erdos problems on Ramsey completeness

Series
School of Mathematics Colloquium
Time
Thursday, November 21, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jacob FoxStanford University

A sequence A of positive integers is r-Ramsey complete if for every r-coloring of A, every sufficiently large integer can be written as a sum of the elements of a monochromatic subsequence. Burr and Erdos proposed several open problems in 1985 on how sparse can an r-Ramsey complete sequence be and which polynomial sequences are r-Ramsey complete. Erdos later offered cash prizes for two of these problems. We prove a result which solves the problems of Burr and Erdos on Ramsey complete sequences. The proof uses tools from probability, combinatorics, and number theory. 

Joint work with David Conlon.

An inverse problems approach to some questions arising in harmonic analysis

Series
School of Mathematics Colloquium
Time
Tuesday, November 12, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Betsy StovallUniversity of Wisconsin

 One strategy for developing a proof of a claimed theorem is to start by understanding what a counter-example should look like.  In this talk, we will discuss a few recent results in harmonic analysis that utilize a quantitative version of this approach.  A key step is the solution of an inverse problem with the following flavor.  Let $T:X \to Y$ be a bounded linear operator and let $0 < a \leq \|T\|$.  What can we say about those functions $f \in X$ obeying the reverse inequality $\|Tf\|_Y \geq a\|f\|_X$?  

New invariants of homology cobordism

Series
School of Mathematics Colloquium
Time
Thursday, October 31, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kristen HendricksRutgers

This is a talk about 3-manifolds and knots. We will begin by reviewing some basic constructions and motivations in low-dimensional topology, and will then introduce the homology cobordism group, the group of 3-manifolds with the same homology as the 3-dimensional sphere up to a reasonable notion of equivalence. We will discuss what is known about the structure of this group and its connection to higher dimensional topology. We will then discuss some existing invariants of the homology cobordism group coming from gauge theory and symplectic geometry, particularly Floer theory. Finally, we will introduce a new invariant of homology cobordism coming from an equivariant version of the computationally-friendly Floer-theoretic 3-manifold invariant Heegaard Floer homology, and use it to construct a new filtration on the homology cobordism group and derive some structural applications. Parts of this talk are joint work with C. Manolescu and I. Zemke; more recent parts of this talk are joint work with J. Hom and T. Lidman.

Total Curvature and the isoperimetric inequality: Proving the Cartan-Hadamard conjecture

Series
School of Mathematics Colloquium
Time
Thursday, October 3, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mohammad GhomiGeorgia Institute of Technology

The classical isoperimetric inequality states that in Euclidean space spheres provide unique enclosures of least perimeter for any given volume. In this talk we discuss how this inequality may be extended to spaces of nonpositive curvature, known as Cartan-Hadamard manifolds, as conjectured by Aubin, Gromov, Burago, and Zalgaller in 1970s and 80s. The proposed proof is based on a comparison formula for total curvature of level sets in Riemannian manifolds, and estimates for the smooth approximation of the signed distance function, via inf-convolution and Reilly type formulas among other techniques. Immediate applications include sharp extensions of Sobolev and Faber-Krahn inequalities to spaces of nonpositive curvature. This is joint work with Joel Spruck.

Hidden symmetries of the hydrogen atom

Series
School of Mathematics Colloquium
Time
Tuesday, April 2, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
John BaezUC Riverside
A classical particle moving in an inverse square central force, like a planet in the gravitational field of the Sun, moves in orbits that do not precess. This lack of precession, special to the inverse square force, indicates the presence of extra conserved quantities beyond the obvious ones. Thanks to Noether's theorem, these indicate the presence of extra symmetries. It turns out that not only rotations in 3 dimensions, but also in 4 dimensions, act as symmetries of this system. These extra symmetries are also present in the quantum version of the problem, where they explain some surprising features of the hydrogen atom. The quest to fully understand these symmetries leads to some fascinating mathematical adventures.

Remez inequalities for solutions of elliptic PDEs

Series
School of Mathematics Colloquium
Time
Thursday, March 28, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Eugenia MalinnikovaNorwegian University of Science and Technology
The Remez inequality for polynomials quantifies the way the maximum of a polynomial over an interval is controlled by its maximum over a subset of positive measure. The coefficient in the inequality depends on the degree of the polynomial; the result also holds in higher dimensions. We give a version of the Remez inequality for solutions of second order linear elliptic PDEs and their gradients. In this context, the degree of a polynomial is replaced by the Almgren frequency of a solution. We discuss other results on quantitative unique continuation for solutions of elliptic PDEs and their gradients and give some applications for the estimates of eigenfunctions for the Laplace-Beltrami operator. The talk is based on a joint work with A. Logunov.

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