# Seminars

## UPCOMING

Lorenz Knots and their Geometric Classification

**Speaker:**

*Thiago de Paiva Souza*- Monash University / IMPA

**Differential Geometry**

**Abstract: **Lorenz knots are periodic orbits of the Lorenz system, which is a system of three ordinary differential equations in $\mathbb{R}^3$. This family of knots piqued the interest of mathematicians as it appears in this unexpected setting. Thurston proved that a non-trivial knot in $S^{3}$ is either a torus, a satellite, or a hyperbolic knot, which we call the geometric type of a knot. The geometric type of a knot is the information that helps to classify it. In this talk we will summarize what we have done and what we should do to complete the geometric classification of Lorenz links.

Lie algebras associated with labeled directed graphs

**Speaker:**

*Mauricio Godoy-Molina*- Universidad de La Frontera, Temuco, Chile

**Algebra**

**Abstract: **We construct 2-step nilpotent Lie algebras using labeled directed simple graphs and give a criterion to detect certain ideals and subalgebras by finding special subgraphs. We prove that if a label occurs only once, then reversing its orientation leads to an isomorphic algebra. As a consequence, if every edge is labeled differently, the Lie algebra depends only on the underlying undirected graph. In addition, we construct the graphs of all 2-step nilpotent Lie algebras of dimension $\leq6$ and compute the algebra of strata preserving derivations of the Lie algebra associated with the complete bipartite graph $K_{m,n}$ with two different labelings. This is a joint work with Diego Lagos.

Symmetric rigidity and complex manifold structures on circle dynamics

**Speaker:**

*Yunping Jiang*- The City University of New York

**Ergodic Theory**

**Abstract: **Consider two $C^{1+\alpha}$ expanding circle endomorphisms $f$ and $g$ of the same degree. Starting from Shub’s work in the 1960s, we know that there is a circle homeomorphism $h$ such that $h\circ f=g\circ h$. Starting from Sullivan’s work in the 1980s, we can prove that $h$ is not only homeomorphic but also quasisymmetric.
Thus we can consider a Teichmüller distance between them using the boundary quasiconformal dilatation of $h$. It gives a pseudo-distance on the space of all $C^{1+\alpha}$ expanding circle endomorphisms for a fixed degree and induces a distance on the space of smooth conjugacy classes. Every smooth conjugacy class has a unique representation which preserves the Lebesgue measure. The space of smooth conjugacy classes under this distance is not complete. The completion is the space of all symmetric conjugacy classes of uniformly symmetric circle endomorphisms. Every symmetric conjugacy class has a representation preserving the Lebesgue measure, but the uniqueness has been a long-time question. We solved this uniqueness in a recent joint work with John Adamski, Yunchun Hu, and Zhe Wang. We actually proved a more general symmetric rigidity result that if $f$ and $g$ are two topological circle endomorphisms with bounded geometry and preserving the Lebesgue measure and suppose $h$ is a symmetric homeomorphism such that $h\circ f=g\circ h$, then $h$ is the identity. This symmetric rigidity result allows us to finish our project about the complex manifold structures on the Teichmüller space of uniformly symmetric circle endomorphisms and the Teichmüller space of uniformly quasisymmetric circle endomorphisms preserving the Lebesgue measure.

Hodge, Poisson and Ramanujan in the context of chiral homology

**Speaker:**

*Jethro van Ekeren*- IMPA

**Geometry, Arithmetic and Differential Equations of Periods (GADEPs)**

**Abstract: **The theory of chiral homology assigns homology groups to algebraic curves with coefficients in a vertex algebra. Although chiral homology itself is difficult to compute in general, and vertex algebras are somewhat exotic objects, I aim to demonstrate that concrete examples are related in interesting ways to classical themes such as Hodge theory, Poisson geometry and partition counting identities such as those of Rogers-Ramanujan. (Joint work with G. Andrews and R. Heluani)

This lectu will be hybrid. Use the link meet.google.com/hzq-vads-ntx for the online presentation.

On projective stochastic-gradient type methods for solving large scale systems of nonlinear ill-posed equations: Applications to machine learning

**Speaker:**

*Antonio Leitão*- Universidade Federal de Santa Catarina - Dept. de Matemática

**Optimization**

**Abstract: **A distinctive feature of our method resides in the *a posteriori* choice of the stepsize, which promotes a relaxed orthogonal projection of the current iterate onto a conveniently chosen convex set. This characteristic distinguish our method from other SGD type methods in the literature (where the stepsize is typically chosen *a priori*) and accounts for the faster convergence observed in the numerical experiments conducted in this manuscript.

The convergence analysis discussed here includes: monotonicity and mean square convergence of the iteration error (exact data case), stability and semi-convergence (noisy data case). In the later case, our method is coupled with an *a priori* stopping rule.

Numerical experiments are presented for two large scale nonlinear inverse problems in machine learning (both with real data): (i) we address, using neural networks, the big data problem of CO-concentration prediction considered in the above cited article; (ii) we tackle the classification problem for the MNIST database (http://yann.lecun.com/exdb/mnist/). The obtained numerical results demonstrate the efficiency of the proposed method.

Numerics of miscible displacement in porous media: heterogeneous permeability and intermediate concentration

**Speaker:**

*Sergey Tikhomirov*- PUC-Rio

**Applied and Computational Mathematics**

**Abstract: **We study the motion of viscous, miscible liquids in porous media. Injection of a less viscous fluid to a more viscous one leads to the growth of an instability often refereed as viscous fingering. The main goal is finding sharp estimates for the size of the mixing zone containing the instabilities.We perform two series of numerical experiments.

The first one devoted to study size of mixing zone in heterogeneous environments. We observe the non-monotonic nature of the dependence of the front end of the mixing zone on the correlation length of the permeability of the reservoir. Interestigly, while for small values of correlation length the pattern of viscous fingers looks stochastic, for intermediate values of correlation lenght it is almost periodic.

In the second we showed an important role of intermediate concentrations (different from minimum and maximum values) on the speed of the front and back end of mixing zones. As a result we demonstrate a potential for improvement of existing estimates on the size of the mixing zone and suggest a theoretical approach (work in progress) for this.

On short wave-long wave interactions in the relativistic context: Application to the Relativistic Euler Equations

**Speaker:**

*Hermano Frid*- FFCLRP-USP

**Analysis and Partial Differential Equations**

**Abstract: **This talk is about recent works on short wave-long wave interactions in the relativistic context. In particular, we introduce a model of relativistic short wave-long wave interaction where the short waves are described by the massless $1+3$-dimensional Thirring model of nonlinear Dirac equation and the long waves are described by the $1+3$-dimensional relativistic Euler equations. The interaction coupling terms are modeled by a potential proportional to the relativistic specific volume in the Dirac equation and an external force proportional to the square modulus of the Dirac wave function in the relativistic Euler equation. An important feature of the model is that the Dirac equations are based on the Lagrangian coordinates of the relativistic fluid flow. In particular, an important contribution of this paper is a clear formulation of the relativistic Lagrangian transformation. This is done by means of the introduction of natural auxiliary dependent variables, rendering the discussion totally similar to the non-relativistic case. As far as the authors know the definition of the Lagrangian transformation given in this paper is new. Finally, we establish the short-time existence and uniqueness of a smooth solution of the Cauchy problem for the regularized model. This follows through the symmetrization of the relativistic Euler equation introduced by Makino and Ukai (1995) and requires a slight extension of a well known theorem of T.~Kato (1975) on quasi-linear symmetric hyperbolic systems. This is a joint work with \textsc{João Paulo Dias}.