Seminars and Colloquia by Series

Complexity of the pure spherical p-spin model

Series
Stochastics Seminar
Time
Thursday, March 12, 2020 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Julian GoldNorthwestern University

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

Series
Math Physics Seminar
Time
Thursday, March 12, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Wei LiLouisiana State University

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.

Cancelled

Series
School of Mathematics Colloquium
Time
Thursday, March 12, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Oscar BrunoCaltech, Computing and Mathematical Sciences

Robustly Clustering a Mixture of Gaussians

Series
High Dimensional Seminar
Time
Wednesday, March 11, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Santosh Vempala Georgia Tech

Please Note: We give an efficient algorithm for robustly clustering of a mixture of two arbitrary Gaussians, a central open problem in the theory of computationally efficient robust estimation, assuming only that the the means of the component Gaussians are well-separated or their covariances are well-separated. Our algorithm and analysis extend naturally to robustly clustering mixtures of well-separated logconcave distributions. The mean separation required is close to the smallest possible to guarantee that most of the measure of the component Gaussians can be separated by some hyperplane (for covariances, it is the same condition in the second degree polynomial kernel). Our main tools are a new identifiability criterion based on isotropic position, and a corresponding Sum-of-Squares convex programming relaxation. This is joint work with He Jia.

Noncollapsed Ricci limit spaces and the codimension 4 conjecture

Series
Geometry Topology Student Seminar
Time
Wednesday, March 11, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Xingyu ZhuGeorgia Tech

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.

Essentially Coercive Forms and asympotically compact semigroups

Series
Analysis Seminar
Time
Wednesday, March 11, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Isabelle Chalendar Université Paris-Est - Marne-la-Vallée

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. 

Cancelled

Series
Time
Wednesday, March 11, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Isabelle Chalendar Université Paris-Est - Marne-la-Vallée

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. 

CANCELLED - Quantitative modeling of protein RNA interactions

Series
Mathematical Biology Seminar
Time
Wednesday, March 11, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ralf BundschuhThe Ohio State University

The prediction of RNA secondary structures from sequence is a well developed task in computational RNA Biology. However, in a cellular environment RNA molecules are not isolated but rather interact with a multitude of proteins. RNA secondary structure affects those interactions with proteins and vice versa proteins binding the RNA affect its secondary structure.  We have extended the dynamic programming approaches traditionally used to quantify the ensemble of RNA secondary structures in solution to incorporate protein-RNA interactions and thus quantify these effects of protein-RNA interactions and RNA secondary structure on each other. Using this approach we demonstrate that taking into account RNA secondary structure improves predictions of protein affinities from RNA sequence, that RNA secondary structures mediate cooperativity between different proteins binding the same RNA molecule, and that sequence variations (such as Single Nucleotide Polymorphisms) can affect protein affinity at a distance mediated by RNA secondary structures.

Thesis Defense: The Maxwell-Pauli Equations

Series
PDE Seminar
Time
Tuesday, March 10, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Thomas KiefferGeorgia Tech

Energetic stability of matter in quantum mechanics, which refers to the question 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 works 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 situation changes dramatically 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 $Z > Z_c$ and the ground state energy exists if $Z < Z_c$.
 
Another notion of stability in quantum mechanics is that of dynamic stability. Dynamic stability 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 on the global well-posedness of the corresponding dynamical equations. In this regard,
we study the quantum mechanical many-body problem of $N$ non-relativistic electrons with spin interacting with their self-generated classical electromagnetic field and $K$ 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. If time permits, we
will discuss several open problems that remain.

The Maxwell-Pauli Equations

Series
Dissertation Defense
Time
Tuesday, March 10, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Forrest KiefferGeorgia Institute of Technology

Please Note: Thesis Defense

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).

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