Please Note: Mathapalooza! is the biggest math event of the Atlanta Science Festival
Mathapalooza! is back at this year's Atlanta Science Festival! Come join us on Saturday, March 18, for an afternoon of mathematical fun beginning at 1:00pm at the Paideia School. There will be interactive puzzles and games, artwork, music, stage acts, and mathematics in motion.
The Deligne category of symmetric groups is the additive Karoubi closure of the partition category. The partition category may be interpreted, following Comes, via a particular linearization of the category of two-dimensional oriented cobordisms. In this talk we will use a generalization of this approach to the Deligne category coupled with the universal construction of two-dimensional topological theories to construct their multi-parameter monoidal generalizations, one for each rational function in one variable. This talk is based on joint work with M. Khovanov.
https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09
G. W. Hill made major contributions to Celestial Mechanics. One of them is to develop his lunar theory as an alternative approach for the study of the motion of the Moon around the Earth, which is the classical Lunar Hill problem. The mathematical model we study is one of the extensions of the classical Hill approximation of the restricted three-body problem. Considering a restricted four body problem, with a hierarchy between the bodies: two larger bodies, a smaller one and a fourth infinitesimal body, we encounter the shapes of the three heavy bodies via oblateness. We first find that the triangular central configurations of the three heavy bodies is a scalene triangle. Through the application of the Hill approximation, we obtain the limiting Hamiltonian that describes the dynamics of the infinitesimal body in a neighborhood of the smaller body. As a motivating example, we identify the three heavy bodies with the Sun, Jupiter and the Jupiter’s Trojan asteroid Hektor.
A link of an isolated complex surface singularity is the intersection of the surface with a small sphere centered at the singular point. The link is a smooth 3-manifold that carries a natural contact structure (given by complex tangencies); one might then want to study its symplectic or Stein fillings. A special family of Stein fillings, called Milnor fillings, can be obtained by smoothing the singular point of the original complex surface. We will discuss some properties and constructions of Milnor fillings and general Stein fillings, and ways to detect whether the link of singularity has Stein fillings that do not arise from smoothings.
We study the minimum number of paths needed to express the edge set of a given graph as the symmetric difference of the edge sets of the paths. This can be seen as a weakening of Gallai’s path decomposition problem, and a variant of the “odd cover” problem of Babai and Frankl which asks for the minimum number of complete bipartite graphs whose symmetric difference gives the complete graph. We relate this “path odd-cover” number of a graph to other known graph parameters and prove some bounds. Joint work with Steffen Borgwardt, Calum Buchanan, Eric Culver, Bryce Frederickson, and Puck Rombach.
A well-known conjecture of Dennis Sullivan asserts that a hyperbolic n-manifold with n>2 cannot admit a complex structure. This conjecture is known to be true in dimension four but little is known in higher dimensions. In this talk, I will outline a new proof of the fact that a hyperbolic 4-manifold cannot support a complex structure. This new proof has some nice features, and it generalizes to show that all extended graph 4-manifolds with positive Euler number cannot support a complex structure. This is joint work with M. Albanese.
We will prove the key lemma underlying the probabilistic unique continuation result of Ding-Smart, namely that for "thin" tilted rectangles, boundedness on all of one of the long edges and on a 1-\varepsilon proportion of the opposite long edge implies a bound (in terms of the dimensions of the rectangle) on the whole rectangle (with high probability).
Please Note: Note the unusual time!
In this talk we explore a class of $\lambda$-convex bodies, i.e., convex bodies with curvature at each point of their boundary bounded below by some $\lambda >0$. For such bodies, we solve two reverse isoperimetric problems.
In $\mathbb{R}^3$, we show that the intersection of two balls of radius $1/\lambda$ (a $\lambda$-convex lens) is the unique volume minimizer among all $\lambda$-convex bodies of given surface area. We also show a reverse inradius inequality in arbitrary dimension which says that the $\lambda$-convex lens has the smallest inscribed ball among all $\lambda$-convex bodies of given surface area.
This is a joint work with Kostiantyn Drach.
The estimation of distributions of complex objects from high-dimensional data with low-dimensional structures is an important topic in statistics and machine learning. Deep generative models achieve this by encoding and decoding data to generate synthetic realistic images and texts. A key aspect of these models is the extraction of low-dimensional latent features, assuming data lies on a low-dimensional manifold. We study this by developing a minimax framework for distribution estimation on unknown submanifolds with smoothness assumptions on the target distribution and the manifold. The framework highlights how problem characteristics, such as intrinsic dimensionality and smoothness, impact the limits of high-dimensional distribution estimation. Our estimator, which is a mixture of locally fitted generative models, is motivated by differential geometry techniques and covers cases where the data manifold lacks a global parametrization.
