Group extensions are a natural way of building complicated groups out of simpler ones. We will develop techniques used to study group extensions. Through these techniques, we will motivate and discuss connections to the cohomology of groups.
We study a family of dynamical systems obtained by coupling a chaotic (Anosov) map on the two-dimensional torus -- the chaotic variable -- with the identity map on the one-dimensional torus -- the neutral variable -- through a dissipative interaction. We show that the two systems synchronize, in the sense that the trajectories evolve toward an attracting invariant manifold, and that the full dynamics is conjugated to its linearization around the invariant manifold. When the interaction is small, the evolution of the neutral variable is very close to the identity and hence the neutral variable appears as a slow variable with respect to the fast chaotic variable: we show that, seen on a suitably long time scale, the slow variable effectively follows the solution of a deterministic differential equation obtained by averaging over the fast variable.
The seminar can also be accessed online via zoom link: Meeting ID: 961 2577 3408
The braid group has many applications throughout the world of math due to its simple yet rich structure. In this talk we will focus on the Burau representation of the braid group, which has important implications in knot theory. Most notably, the open problem of faithfulness of the Burau representation of the braid group on 4 strands is equivalent to whether or not the Jones polynomial can detect the unknot. The Burau representation has a topological interpretation that uses the mapping class definition of the braid group. We'll introduce the braid group first and then discuss the Burau representation. We will go through examples for small n and discuss the proof of nonfaithfulness for n > 4. Time permitting, we may introduce the Gassner representation.
We consider the 1+1 dimensional vector valued Principal Chiral Field model (PCF) obtained as a simplification of the Vacuum Einstein Field equations under the Belinski-Zakharov symmetry. PCF is an integrable model, but a rigorous description of its evolution is far from complete. Here we provide the existence of local solutions in a suitable chosen energy space, as well as small global smooth solutions under a certain non degeneracy condition. We also construct virial functionals which provide a clear description of decay of smooth global solutions inside the light cone. Finally, some applications are presented in the case of PCF solitons, a first step towards the study of its nonlinear stability.
Please Note: Speaker will be in person, but also livestreamed but not recorded at https://gatech.zoom.us/j/98355006347
Modern neural networks are usually over-parameterized—the number of parameters exceeds the number of training data. In this case the loss function tends to have many (or even infinite) global minima, which imposes a challenge of minima selection on optimization algorithms besides the convergence. Specifically, when training a neural network, the algorithm not only has to find a global minimum, but also needs to select minima with good generalization among many others. We study the mechanisms that facilitate global minima selection of optimization algorithms, as well as its connection with good generalization performance. First, with a linear stability theory, we show that stochastic gradient descent (SGD) favors global minima with flat and uniform landscape. Then, we build a theoretical connection of flatness and generalization performance based on a special multiplicative structure of neural networks. Connecting the two results, we develop generalization bounds for neural networks trained by SGD. Our bounds take the optimization process into consideration. Furthermore, we study the behavior of optimization algorithms around manifold of minima and reveal the exploration of algorithms from one minimum to another.
The fundamental groups of knot complements have lots of finite quotients. We give a criterion for a hyperbolic knot in the three-sphere to be distinguished (up to isotopy and mirroring) from every other knot in the three-sphere by the set of finite quotients of its fundamental group, and we use this criterion as well as recent work of Baldwin-Sivek to show that there are infinitely many hyperbolic knots distinguished (up to isotopy and mirroring) by finite quotients.
The finite index subgroups of a finitely presented group generate a topology on the group. We will discuss using examples how this relates to the organization of a group's finite quotients, and introduce the ideas of profinite rigidity and flexibility.
The Zariski closure of the central path (which interior point algorithms track in convex optimization problems such as linear and semidefinite programs) is an algebraic curve, called the central curve. Its degree has been studied in relation to the complexity of these interior point algorithms. We show that the degree of the central curve for generic semidefinite programs is equal to the maximum likelihood degree of linear concentration models. This is joint work with Serkan Hoşten and Angélica Torres.
We demonstrate for the first time that ill-conditioned, non-smooth, constrained distributions in very high dimensions, upwards of 100,000, can be sampled efficiently in practice. Our algorithm incorporates constraints into the Riemannian version of Hamiltonian Monte Carlo and maintains sparsity. This allows us to achieve a mixing rate independent of condition numbers. On benchmark data sets from systems biology and linear programming, our algorithm outperforms existing packages by orders of magnitude. In particular, we achieve a 1,000-fold speed-up for sampling from the largest published human metabolic network (RECON3D). Our package has been incorporated into the COBRA toolbox. This is joint work with Yin Tat Lee, Ruoqi Shen, and Santosh Vempala.
We will continue to discuss lower and upper estimates of Widom factors. We will also introduce Cantor-type sets, constructed as the intersection of the level domains for simple sequences of polynomials. Using these Cantor-type sets we will prove some results on growth of Widom factors.
