Georgia Institute of Technology College of Sciences

Intermittency is a property observed in the study of turbulence. Two of the most popular ways to measure it are based on the concept of flatness, one with structure functions in the physical space and the other one with high-pass filters in the frequency space. Experimental and numerical simulations suggest that the two approaches do not always give the same results. In this talk, we prove they are not analytically equivalent. For that, we first adapt them to a rigorous mathematical language, and we test them with generalizations of Riemann’s non-differentiable function. This work is motivated by the discovery of Riemann’s non-differentiable function as a trajectory of polygonal vortex filaments.

The seminar will be held on Zoom.  Here is the link

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

All 3-manifolds bound 4-manifolds, and many construction of 3-manifolds automatically come with a 4-manifold bounding it. Often times these 4-manifolds have definite intersection form. Using Heegaard Floer correction terms and an analysis of short characteristic covectors in bimodular lattices, we give an obstruction for a 3-manifold to bound a definite 4-manifold, and produce some concrete examples. This is joint work with Kyle Larson.

The main topics of this thesis concern two types of approximate Schauder frames for the Banach sequence space $\ell_1$. The first main topic pertains to finite-unit norm tight frames (FUNTFs) for the finite-dimensional real sequence space $\ell_1^n$. We prove that for any $N \geq n$, FUNTFs of length $N$ exist for real $\ell_1^n$. To show the existence of FUNTFs, specific examples are constructed for various lengths. These constructions involve repetitions of frame elements. However, a different method of frame constructions allows us to prove the existence of FUNTFs for real $\ell_1^n$ of lengths $2n-1$ and $2n-2$ that do not have repeated elements.

The second main topic of this thesis pertains to normalized unconditional Schauder frames for the sequence space $\ell_1$. A Schauder frame provides a reconstruction formula for elements in the space, but need not be associated with a frame inequality. Our main theorem on this topic establishes a set of conditions under which an $\ell_1$-type of frame inequality is applicable towards unconditional Schauder frames. A primary motivation for choosing this set of hypotheses involves appropriate modifications of the Rademacher system, a version of which we prove to be an unconditional Schauder frame that does not satisfy an $\ell_1$-type of frame inequality.

Bluejeans link to meeting: https://bluejeans.com/544995272
 

Consider the three body problem with masses $m_0,m_1,m_2>0$. Take units such that $m_0+m_1+m_2 = 1$. In 1922 Chazy classified the possible final motions of the three bodies, that is the behaviors the bodies may have when time tends to infinity. One of them are what is known as oscillatory motions, that is, solutions of the three body problem such that the positions of the bodies $q_0, q_1, q_2$ satisfy
\[
\liminf_{t\to\pm\infty}\sup_{i,j=0,1,2, i\neq j}\|q_i-q_j\|<+\infty \quad \text{ and }\quad 
\limsup_{t\to\pm\infty}\sup_{i,j=0,1,2, i\neq j}\|q_i-q_j\|=+\infty.
\] At the time of Chazy, all types of final motions were known, except the oscillatory ones. We prove that, if all three masses $m_0,m_1,m_2>0$ are not equal to $1/3$, such motions exist. In fact, we prove more, since our result is based on the construction of a hyperbolic invariant set whose dynamics is conjugated to the Bernouilli shift of infinite symbols, we prove (if all masses are not all three equals to $1/3$) 1) the existence of chaotic motions and positive topological entropy for the three body problem, 2) the existence of periodic orbits of arbitrarily large period in the 3BP. Reversing time, Chazy's classification describes ``starting'' motions and then, the question if starting and final motions need to coincide or may be different arises.  We also prove that one can construct solutions of the three body problem whose starting and final motions are of different type.