Georgia Institute of Technology College of Sciences

A surface of genus $g$ has many symmetries. These form the surface’s mapping class group $Mod(S_g)$, which is finitely generated. The most commonly used generating sets for $Mod(S_g)$ are comprised of infinite order elements called Dehn twists; however, a number of authors have shown that torsion generating sets are also possible. For example, Brendle and Farb showed that $Mod(S_g)$ is generated by six involutions for $g \geq 3$. We will discuss our extension of these results to elements of arbitrary order: for $k > 5$ and $g$ sufficiently large, $Mod(S_g)$ is generated by three elements of order $k$. Generalizing this idea, in joint work with Margalit we showed that for $g \geq 3$ every nontrivial periodic element that is not a hyperelliptic involution normally generates $Mod(S_g)$. This result raises a question: does there exist an $N$, independent of $g$, so that if $f$ is a periodic normal generator of $Mod(S_g)$, then $Mod(S_g)$ is generated by $N$ conjugates of $f$? We show that in general there does not exist such an $N$, but that there does exist such a universal bound for the class of non-involution normal generators.

On June 10, 2000, the Millennium Bridge in London opened to the public. As people crossed the bridge, it wobbled. The sway of the bridge was large enough that prompted many on the bridge to hold on to the rails. Three days later, the bridge closed. It reopened only after modifications to prevent the wobbling were made, eighteen months later. We develop and study a model motivated by this event

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. 

Merris and Watkins interpreted results of Littlewood to give generating functions for symmetric group characters induced from one-dimensional characters of Young subgroups.  Beginning with an n by n matrix X of formal variables, one obtains induced sign and trivial characters by expanding sums of products of certain determinants and permanents, respectively. We will look at a new analogous result which holds for hyperoctahedral group characters induced from the four one-dimensional characters of its Young subgroups.  This requires a 2n by 2n matrix of formal variables and four combinations of determinants and permanents.  This is joint work with Jongwon Kim.

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).

Energetic stability of matter in quantum mechanics, which refers to the question of whether the ground state energy of a

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. 

Abstract: 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. 

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.

The Neumann-Poincaré (NP) operator arises in boundary value problems, and plays an important role in material design, signal amplification, particle detection, etc. The spectrum of the NP operator on domains with corners was studied by Carleman before tools for rigorous discussion were created, and received a lot of attention in the past ten years. In this talk, I will present our discovery and verification of eigenvalues embedded in the continuous spectrum of this operator. The main ideas are decoupling of spaces by symmetry and construction of approximate eigenvalues. This is based on two works with Stephen Shipman and Karl-Mikael Perfekt.

The pure spherical p-spin model is a Gaussian random polynomial H of degree p on an N-dimensional sphere, with N large. The sphere is viewed as the state space of a physical system with many degrees of freedom, and the random function H is interpreted as a smooth assignment of energy to each state, i.e. as an energy landscape. 

In 2012, Auffinger, Ben Arous and Cerny used the Kac-Rice formula to count the average number of critical points of H having a given index, and with energy below a given value. This number is exponentially large in N for p > 2, and the rate of growth itself is a function of the index chosen and of the energy cutoff. This function, called the complexity, reveals interesting topological information about the landscape H: it was shown that below an energy threshold marking the bottom of the landscape, all critical points are local minima or saddles with an index not diverging with N. It was shown that these finite-index saddles have an interesting nested structure, despite their number being exponentially dominated by minima up to the energy threshold. The total complexity (considering critical points of any index) was shown to be positive at energies close to the lowest. Thus, at least from the perspective of the average number of critical points, these random landscapes are very non-convex. The high-dimensional and rugged aspects of these landscapes make them relevant to the folding of large molecules and the performance of neural nets. 

Subag made a remarkable contribution in 2017, when he used a second-moment approach to show that the total number of critical points concentrates around its mean. In light of the above, when considering critical points near the bottom of the landscape, we can view Subag's result as a statement about the concentration of the number of local minima. His result demonstrated that the typical behavior of the minima reflects their average behavior. We complete the picture for the bottom of the landscape by showing that the number of critical points of any finite index concentrates around its mean. This information is important to studying associated dynamics, for instance navigation between local minima. Joint work with Antonio Auffinger and Yi Gu at Northwestern. 

Cancelled due to COVID-19

An afternoon of public engagement of mathematics through puzzles, games, and the arts, including:  magic (by Matt Baker), juggling and other circus arts, music, dance, an art gallery, and a live construction of a Fibonacci-based sculpture (by Akio Hizume).  It is free and open to the public, but our partner the Julia Robinson Mathematics Festival recommends registering at https://jrmf.org/event-details/mathapalooza .  If you want to get involved, please contact Evans Harrell directly.