Georgia Institute of Technology College of Sciences

We present a mean field model of electroencephalographic activity in the brain, which is composed of a system of coupled ODEs and PDEs. We show the existence and uniqueness of weak and strong solutions of this model and investigate the regularity of the solutions. We establish biophysically plausible semidynamical system frameworks and show that the semigroups of weak and strong solution operators possess bounded absorbing sets. We show that there exist parameter values for which the semidynamical systems do not possess a global attractor due to the lack of the compactness property. In this case, the internal dynamics of the ODE components of the solutions can create asymptotic spatial discontinuities in the solutions, regardless of the smoothness of the initial values and forcing terms.

Weighted direct samplers, sometimes also called importance samplers, are Monte Carlo algorithms for generating independent, weighted samples from a given target probability distribution. They are used in, e.g., data assimilation, state estimation for dynamical systems, and computational statistical mechanics. One challenge in designing weighted samplers is to ensure the variance of the weights, and that of the resulting estimator, are well-behaved. Recently, Chorin, Tu, Morzfeld, and coworkers have introduced a class of novel weighted samplers called implicit samplers, which possess a number of nice empirical properties. In this talk, I will summarize an asymptotic analysis of implicit samplers in the small-noise limit and describe a simple method to obtain a higher-order accuracy. I will also discuss extensions to stochastic differential equatons. This is joint work with Jonathan Goodman, Andrew Leach, and Matthias Morzfeld.

Peter Lambert-Cole: Mutant knots are notoriously hard to distinguish. Many, but not all, knot invariants take the same value on mutant pairs. Khovanov homology with coefficients in Z/2Z is known to be mutation-invariant, while the bigraded knot Floer homology groups can distinguish mutants such as the famous Kinoshita-Terasaka and Conway pair. However, Baldwin and Levine conjectured that delta-graded knot Floer homology, a singly-graded reduction of the full invariant, is preserved by mutation. In this talk, I will give a new proof that Khovanov homology mod 2 is mutation-invariant. The same strategy can be applied to delta-graded knot Floer homology and proves the Baldwin-Levine conjecture for mutations on a large class of tangles. -----------------------------------------------------------------------------------------------------------------------------------------------Alex Zupan: Generally speaking, given a type of manifold decomposition, a natural problem is to determine the structure of all decompositions for a fixed manifold. In particular, it is interesting to understand the space of decompositions for the simplest objects. For example, Waldhausen's Theorem asserts that up to isotopy, the 3-sphere has a unique Heegaard splitting in every genus, and Otal proved an analogous result for classical bridge splittings of the unknot. In both cases, we say that these decompositions are "standard," since they can be viewed as generic modifications of a minimal splitting. In this talk, we examine a similar question in dimension four, proving that -- unlike the situation in dimension three -- the unknotted 2-sphere in the 4-sphere admits a non-standard bridge trisection. This is joint work with Jeffrey Meier.

In this talk we will discuss the following question: When does there exist a curve of degree d and genus g passing through n general points in P^r? We will focus primarily on what is known in the case of space curves (r=3).

Existence of ballistic transport for Schr ̈odinger operator with a quasi- periodic potential in dimension two is discussed. Considerations are based on the following properties of the operator: the spectrum of the operator contains a semiaxis of absolutely continuous spectrum and there are generalized eigenfunctions being close to plane waves ei⟨⃗k,⃗x⟩ (as |⃗k| → ∞) at every point of this semiaxis. The isoenergetic curves in the space of momenta ⃗k corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure).

Almost all biological activities involve transport and distribution of ions and charged particles. The complicated coupling and competition between different ionic solutions in various biological environments give the intricate specificity and selectivity in these systems. In this talk, I will introduce several extended general diffusion systems motivated by the study of ion channels and ionic solutions in biological cells. In particular, I will focus on the interactions between different species, the boundary effects and in many cases, the thermal effects.

In this talk I will introduce a Tb Theorem that characterizes all Calderón-Zygmund operators that extend compactly on L^p(R^n) by means of testing functions as general as possible. In the classical theory for boundedness, the testing functions satisfy a non-degeneracy property called accretivity, which essentially implies the existence of a positive lower bound for the absolute value of the averages of the testing functions over all dyadic cubes. However, in the setting of compact operators, due to their better properties, the hypothesis of accretivity can be relaxed to a large extend. As a by-product, the results also describe those Calderón-Zygmund operators whose boundedness can be checked with non-accretive testing functions.

The Lickorish Wallace Theorem states that any closed 3-manifold is the result of a +/- 1-surgery on a link in S^3. I shall discuss the relevant definitions, and present the proof as outlined in Rolfsen's text 'Knots and Links' and Lickorish's 'Introduction to Knot Theory'.

Let G be a graph containing 5 different vertices a0, a1, a2, b1 and b2. We say that (G, a0, a1, a2, b1, b2) is feasible if G contains disjoint connected subgraphs G1, G2, such that {a0, a1, a2}⊆V(G1) and {b1, b2}⊆V(G2). In this talk, we will prove the existence of 5-edge configurations in (G, a0, a1, a2, b1, b2). Joint work with Changong Li, Robin Thomas, and Xingxing Yu.

We study an online algorithm for making a well—equidistributed random set of points in an interval, in the spirit of "power of choice" methods. Suppose finitely many distinct points are placed on an interval in any arbitrary configuration. This configuration of points subdivides the circle into a finite number of intervals. At each time step, two points are sampled uniformly from the interval. Each of these points lands within some pair of intervals formed by the previous configuration. Add the point that falls in the larger interval to the existing configuration of points, discard the other, and then repeat this process. We then study this point configuration in the sense of its largest interval, and discuss other "power of choice" type modifications. Joint work with Pascal Maillard.

We study an online algorithm for making a well—equidistributed random set of points in an interval, in the spirit of "power of choice" methods. Suppose finitely many distinct points are placed on an interval in any arbitrary configuration. This configuration of points subdivides the circle into a finite number of intervals. At each time step, two points are sampled uniformly from the interval. Each of these points lands within some pair of intervals formed by the previous configuration. Add the point that falls in the larger interval to the existing configuration of points, discard the other, and then repeat this process. We then study this point configuration in the sense of its largest interval, and discuss other "power of choice" type modifications. Joint work with Pascal Maillard.

In this series of talks I will introduce branched coverings of manifolds and sketch proofs of most the known results in low dimensions (such as every 3 manifold is a 3-fold branched cover over a knot in the 3-sphere and the existence of universal knots). This week we continue discussing branched covers of 3-manifolds and prove universal links exist.

We consider Hamiltonian systems with normally hyperbolic manifold with a homoclinic connection. The systems are of the form H_0(I, phi, x,y) = h(I) + P(x,y) ,where P is a one dimensional system with a homoclinic intersection. The above Hamiltonian is a standard normal form for near integrable Hamiltonians close to a resonance. We consider perturbations that are time dependent and may be not Hamiltonian. We derive explicit formulas for the first order effects on the stable/unstable manifolds. In particular, we give sufficient conditions for the existence of homoclinic intersections to the normally hyperbolic manifold. Previous treatments in the literature specify the types of the unperturbed orbits considered (periodic or quasiperiodic) and are restricted to periodic or quasi-periodic perturbations. We do not need to distinguish on the perturbed orbits and we allow rather general dependence on the time (periodic, quasiperiodic or random). The effects are expressed by very fast converging improper integrals. This is joint work with M. Gidea. https://arxiv.org/abs/1710.01849

There is an increasing trend for robots to serve as networked mobile sensing platforms that are able to collect data and interact with humans in various types of environment in unprecedented ways. The need for undisturbed operation posts higher goals for autonomy. This talk reviews recent developments in autonomous collective foraging in a complex environment that explicitly integrates insights from biology with models and provable strategies from control theory and robotics. The methods are rigorously developed and tightly integrated with experimental effort with promising results achieved.

Join us for a discussion of making professional mathematics diagrams and illustrations with free vector graphics editing software Inkscape. We'll discuss and tinker with Bezier curves, TexTex, and vectorization of scanned images.