Georgia Institute of Technology College of Sciences

A translation surface is obtained by identifying edges of polygons in the plane to create a compact Riemann surface equipped with a nonzero holomorphic one-form. Every Riemann surface can be given as an algebraic curve via its Jacobian variety. We aim to construct explicitly the underlying algebraic curves from their translation surfaces, given as polygons in the plane. The key tools in our approach are discrete Riemann surfaces, which allow us to approximate the Riemann matrices, and then, via theta functions, the equations of the curves. In this talk, I will present our algorithm and numerical experiments. From the newly found Riemann matrices and equations of curves, we can then make several conjectures about the curves underlying the Jenkins-Strebel representatives, a family of examples that until now, lived squarely on the analytic side of the transcendental divide between Riemann surfaces and algebraic curves.

In this talk, we obtain new computational insights into two classical areas of statistics: generalization and sampling. In the first part, we study generalization: the performance of a learning algorithm on unseen data. We define a notion of generalization for non-converging training with local descent approaches via the stability of loss statistics. This notion yields generalization bounds in a similar manner to classical algorithmic stability. Then, we show that more information from the training dynamics provides clues to generalization performance.   

In the second part, we discuss a new method for constructing transport maps. Transport maps are transformations between the sample space of a source (which is generally easy to sample) and a target (typically non-Gaussian) probability distribution. The new construction arises from an infinite-dimensional generalization of a Newton method to find the zero of a "score operator". We define such a score operator that gives the difference of the score -- gradient of logarithm of density -- of a transported distribution from the target score. The new construction is iterative, enjoys fast convergence under smoothness assumptions, and does not make a parametric ansatz on the transport map.

The classical braid groups can be viewed from many different angles and admit generalizations in just as many directions. Surface braid groups are a topological generalization of the braid groups that have close connections with mapping class groups of surfaces. In this talk we review a recent result on minimal nonabelian finite quotients of braid groups. In considering the analogous problem for surface braid groups, we construct nilpotent nonabelian quotients by generalizing the Heisenberg group. These Heisenberg quotients do not arise as quotients of the braid group.

We will talk about our results on the elasticity and stability of the 
collision of two kinks with low speed v>0 for the nonlinear wave 
equation of dimension 1+1 known as the phi^6 model. We will show that 
the collision of the two solitons is "almost" elastic and that, after 
the collision, the size of the energy norm of the remainder and the size 
of the defect of the speed of each soliton can be, for any k>0, of the 
order of any monomial v^{k} if v is small enough.

References:
This talk is based on our current works:
On the collision problem of two kinks for the phi^6 model with low speed 
   [https://arxiv.org/abs/2211.09749]
Approximate kink-kink solutions for the phi^6 model in the low-speed 
limit [https://arxiv.org/abs/2211.09714]

We show that Erdős-Renyi random graph with constant density has correspondence chromatic number $O(n/\sqrt{\log n})$; this matches a prediction from linear Hadwiger’s conjecture for correspondence colouring. The proof follows from a sufficient condition for correspondence colourability in terms of the numbers of independent sets, following Bernshteyn's method. We conjecture the truth to be of order $O(n/\log n)$ as suggested by the random correspondence assignment. This is joint work with Zdenek Dvorak.

We study $L^p$ bounds on Nikodym maximal functions associated to spheres. In contrast to the spherical maximal functions studied by Stein and Bourgain, our maximal functions are uncentered: for each point in $\mathbb R^n$, we take the supremum over a family of spheres containing that point. This is joint work with Georgios Dosidis and Jongchon Kim.

When do commuting homeomorphisms of S^2 have a common fixed point? Christian Bonatti gave the first sufficient condition: Commuting diffeomorphisms sufficiently close to the identity in Diff^+(S^2) always admit a common fixed point. In this talk we present a result of Michael Handel that extends Bonatti's condition to a much larger class of commuting homeomorphisms. If time permits, we survey results for higher genus surfaces due to Michael Handel and Morris Hirsch, and connections to certain compact foliated 4-manifolds.

We describe a geometric framework to study Newton's
equations on infinite-dimensional configuration spaces of
diffeomorphisms and smooth probability densities. It turns out that
several important PDEs of hydrodynamical origin can be described in
this framework in a natural way. In particular, the so-called Madelung
transform between the Schrödinger-type equations on wave functions and
Newton's equations on densities turns out to be a Kähler map between
the corresponding phase spaces, equipped with the Fubini-Study and
Fisher-Rao information metrics. This is a joint work with G.Misiolek
and K.Modin.

 I will discuss the soliton resolution and asymptotic stability problems for the sine-Gordon equation. It is known that the obstruction to the asymptotic stability for the sine-Gordon equation in the energy space is the existence of small breathers which is also closely related to the emergence of wobbling kinks. Our stability analysis gives a criterion for the weight which is sharp up to the endpoint so that the asymptotic stability holds. This is joint work with Jiaqi Liu and Bingying Lu.

Zoom link to the talk: https://gatech.zoom.us/j/91558578481

In this talk, we will consider stochastic processes on (random) graphs. They arise naturally in epidemiology, statistical physics, computer science and engineering disciplines. In this set-up, the vertices are endowed with a local state (e.g., immunological status in case of an epidemic process, opinion about a social situation). The local state changes dynamically as the vertex interacts with its neighbours. The interaction rules and the graph structure depend on the application-specific context. We will discuss (non-equilibrium) approximation methods for those systems as the number of vertices grow large. In particular, we will discuss three different approximations in this talk: i) approximate lumpability of Markov processes based on local symmetries (local automorphisms) of the graph, ii) functional laws of large numbers in the form of ordinary and partial differential equations, and iii) functional central limit theorems in the form of Gaussian semi-martingales. We will also briefly discuss how those approximations could be used for practical purposes, such as parameter inference from real epidemic data (e.g., COVID-19 in Ohio), designing efficient simulation algorithms etc.

We will continue our discussion of the key ingredients of a multi-scale analysis, namely resolvent decay and the Wegner type estimate. After briefly discussing how the Wegner estimate is obtained in the regime of regular noise, we will discuss the strategy used in Bourgain-Kenig (2005) and Ding-Smart (2018) to obtain analogues thereof using some form of unique continuation principle.

From here, we'll examine the quantitative unique continuation principle used by Bourgain-Kenig, and the lack of any even qualitative analogue on the two-dimensional lattice. From here, we'll discuss the quantitative probabilistic unique continuation result used in Ding-Smart.