Georgia Institute of Technology College of Sciences

Random and irregular growth is all around us. We see it in the form of cancer growth, bacterial infection, fluid flow through porous rock, and propagating flame fronts. In this talk, I will introduce several different models for random growth and the different shapes that can arise from them. Then I will talk in more detail about one model, first-passage percolation, and some of the main questions that researchers study about it.

Taking the double branched cover of $S^3$ over a knot $K$ is natural way to associate $K$ with a 3-manifold, and to study the double branched cover, we often want a Dehn surgery description for it. The Montesinos trick gives a systematic way to get such a description. In this talk, we will go over the broad statement of this trick: that a rational tangle replacement on the knot corresponds to Dehn surgery on the double branched cover. This gives particularly nice descriptions for some satellites of $K$ as surgery on $K \mathrel\# K^r$. We will also discuss an application of the trick which characterizes the 2-bridge knots with unknotting number 1.

The first meeting of our club/seminar will feature a brief introduction to the three CAS (computer algebra systems): Macaulay2, OSCAR, and SageMath. All of these are open-source software and are used by research mathematicians for algebraic computation. 

Everyone is welcome to the club! The only requirement is being optimistic about using computer algebra to (potentially) help your research.

A wide array of network growth models have been proposed across various domains as test beds to understand questions such as the effect of network change point (when a shock to the network changes the probabilistic rules of its evolution) or the role of attributes in driving the emergence of network structure and subsequent centrality measures in real world systems. 

The goal of this talk will be to describe three specific settings where continuous time branching processes give mathematical insight into asymptotic properties of such models. In the first setting, a natural network change point model can be directly embedded into continuous time thus leading to an understanding of long range dependence of the initial network system on subsequent properties imply the difficulty in understanding and estimating network change point. In the second application, we will describe a notion of resolvability where convergence of a simple macroscopic functional in a model of networks with vertex attributes, coupled with stochastic approximation techniques implies local weak convergence of a standard model of nodal attribute driven network evolution to a limit infinite random structure driven by a multitype continuous time branching process. In the second setting, continuous time branching processes only emerge in the limit. In the final setting we will describe network evolution models with delay where once again such processes arise only in the limit. 

This series will tie together algebraic, complex analytic, symplectic, and contact geometries together in one coherent story. This will be done via the study of a series of couplets from different fields of geometry:

Algebraic manifolds:
Affine and quasi-projective varieties (non-compact models)
Projective varieties (compact models)

Complex manifolds:
Stein manifolds
Stein compactifications

Symplectic manifolds:
Liouville/ Weinstein geometry
Compact Kahler manifolds 

Depending on how long it takes to discuss these items, I will also attempt to include discussions on:

• Biran-Giroux decompositions of symplectic manifolds • Boothby-Wang bundles and contact plumbings of these • Milnor's fibration theorem for isolated singularities and connections to open book decompositions and Lefschetz fibrations • Open questions and interesting avenues of research

Most of our discussion will, as a side effect, outline the topological structure behind Type IIA String theory (the "topological A-model") which requires a 6-dimensional Calabi-Yau (Kahler) background.

Given a feasible degree sequence D, we consider the uniform distribution over all graphs with degree sequence D. In 1995, Molloy and Reed gave a criterion for determining the existence of a giant (i.e. linear in n) component for degree sequences satisfying certain technical conditions; in 2018, Joos, Perarnau, Rautenbach, and Reed gave a refined result that applies to essentially all feasible D. In this talk, we work in the "supercritical" regime and uncover the precise structure of the giant component when it exists, obtaining bounds on the diameter and mixing time of the random walk on the giant which are tight up to polylogarithmic factors. Our techniques involve a variation of core-kernel reduction and analysis of the switch Markov chain. Joint work with Louigi Addario-Berry and Bruce Reed.

Cislunar space—the region between Earth and the Moon—has reemerged as a critical area for space exploration. From a mathematical perspective, this region is governed by multi-body dynamics that give rise to rich structures, including invariant manifolds, resonant orbits, and homoclinic chaos. This talk will introduce classical and modern tools from celestial mechanics to analyze motion in the Earth–Moon system, with an emphasis on restricted 3- and 4-body problems. We will discuss how perturbative methods (normal forms) and invariant manifold theory (parameterization method) reveal the underlying organization of the phase space. Particular attention will be placed on connecting the perturbative regime, where classical methods apply, with the realistic system, which often lies far outside that regime, using computer-assisted techniques. Our ultimate goal is to establish rigorous results for the real solar system while enhancing the engineering capabilities needed to design and fly missions, highlighting how mathematics contributes both to theory and to the practical challenges of contemporary space exploration.

No prior knowledge is needed; the talk will be self-contained and accessible. Undergraduates are encouraged to attend.