Georgia Institute of Technology College of Sciences

In this talk we will survey some of the developments of Cheeger and Colding’s conjecture on a sequence of n dimensional manifolds with uniform two sides Ricci Curvature bound, investigated by Anderson, Tian, Cheeger, Colding and Naber among others. The conjecture states that every Gromov-Hausdorff limit of the above-mentioned sequence, which is a metric space with singularities,  has the singular set with Hausdorff codimension at least 4. This conjecture was proved by Colding-Naber in 2014, where the ideas and techniques like \epsilon-regularity theory, almost splitting and quantitative stratification were extensively used. I will give an introduction of the background of the conjecture and talk about the idea of the part of the proof that deals with codimension 2 singularities.

Form methods are most efficient to prove generation theorems for semigroups but also for proving selfadjointness. So far those theorems are based on a coercivity notion which allows the use of the Lax-Milgram Lemma. Here we consider weaker "essential" versions of coerciveness which already suffice to obtain the generator of a semigroup S or a selfadjoint operator. We also show that one of these properties, namely essentially positive coerciveness implies a very special asymptotic behaviour of S, namely asymptotic compactness; i.e. that $\dist(S(t),{\mathcal K}(H))\to 0$ as $t\to\infty$, where ${\mathcal K}(H)$ denotes the space of all compact operators on the underlying Hilbert space. 

Energetic stability of matter in quantum mechanics, which refers to the ques-
tion of whether the ground state energy of a many-body quantum mechanical
system is finite, has long been a deep question of mathematical physics. For a
system of many non-relativistic electrons interacting with many nuclei in the
absence of electromagnetic fields this question traces back to the seminal work
of Tosio Kato in 1951 and Freeman Dyson and Andrew Lenard in 1967/1968.
In particular, Dyson and Lenard showed the ground state energy of the many-
body Schrödinger Hamiltonian is bounded below by a constant times the total
particle number, regardless of the size of the nuclear charges. This says such a
system is energetically stable (of the second kind). This situation changes dra-
matically when electromagnetic fields and spin interactions are present in the
problem. Even for a single electron with spin interacting with a single nucleus
of charge $Z > 0$ in an external magnetic field, Jurg Fröhlich, Elliot Lieb, and
Michael Loss in 1986 showed that there is no ground state energy if $Z$ exceeds
a critical charge $Z_c$ and the ground state energy exists if $Z < Z_c$ . In other
words, if the nuclear charge is too large, the one-electron atom is energetically
unstable.

Another notion of stability in quantum mechanics is that of dynamic stabil-
ity, which refers to the question of global well-posedness for a system of partial
differential equations that models the dynamics of many electrons coupled to
their self-generated electromagnetic field and interacting with many nuclei. The
central motivating question of our PhD thesis is whether energetic stability has
any influence over dynamic stability. Concerning this question, we study the
quantum mechanical many-body problem of $N \geq 1$ non-relativistic electrons with
spin interacting with their self-generated classical electromagnetic field and $K \geq 0$
static nuclei. We model the dynamics of the electrons and their self-generated
electromagnetic field using the so-called many-body Maxwell-Pauli equations.
The main result presented is the construction time global, finite-energy, weak
solutions to the many-body Maxwell-Pauli equations under the assumption that
the fine structure constant $\alpha$ and the nuclear charges are sufficiently small to
ensure energetic stability of this system. This result represents an initial step
towards understanding the relationship between energetic stability and dynamic
stability. If time permits, we will discuss several open problems that remain.

Committee members: Prof. Michael Loss (Advisor, School of Mathematics,
Georgia Tech), Prof. Brian Kennedy (School of Physics, Georgia Tech), Prof.
Evans Harrell (School of Mathematics, Georgia Tech), Prof. Federico Bonetto
(School of Mathematics, Georgia Tech), Prof. Chongchun Zeng (School of Math-
ematics, Georgia Tech).

The satellite construction, which associates to a pattern knot P in a solid torus and a companion knot K in the 3-sphere the so-called satellite knot P(K), features prominently in knot theory and low-dimensional topology.  Besides the intuition that P(K) is “more complicated” than either P or K, one can attempt to quantify how the complexity of a knot changes under the satellite operation. In this talk, I’ll discuss how several notions of complexity based on the minimal genus of an embedded surface change under satelliting. In the case of the classical Seifert genus of a knot, Schubert gives an exact formula. In the 4-dimensional context the situation is more complicated, and depends on whether we work in the smooth or topological category: the smooth category is sometimes asymptotically similar to the classical setting, but our main results show that the topological category is much weirder.  This talk is based on joint work with Peter Feller and Juanita Pinzón-Caicedo.