Georgia Institute of Technology College of Sciences

Shellability is a fundamental concept in combinatorial topology and algebraic combinatorics. Two foundational results are Bruggesser–Mani’s line shellings of polytopes and Björner’s theorem that the order complex of a geometric lattice is shellable. 

Inspired by Bruggesser–Mani’s line shellings of polytopes, we introduce line shellings for the lattice of flats of a matroid:  given a normal complex for a Bergman fan of a matroid induced by a building set, we show that the lexicographic order of the coordinates of its vertices is a shelling order.  This yields a new geometric proof of Björner’s classical result and establishes shellability for all nested set complexes for matroids.

This is joint work with Galen Dorpalen-Barry, Anastasia Nathanson, Ethan Partida, and Noah Prime.

In this talk, we will discuss the construction of exotic 4-manifolds using Lefschetz fibrations over S^2, which are obtained by finite order cyclic group actions on Σg. We will first apply various cyclic group actions on Σg for g>0, and then extend it diagonally to the product manifolds ΣgxΣg. These will give singular manifolds with cyclic quotient singularities. Then, by resolving the singularities, we will obtain families of Lefschetz fibrations over S^2. Following the resolution process, we will determine the configurations of the singular fibers and the monodromy of the total space. In some cases, deformations of the Lefschetz fibrations give rise to nice applications using the rational blow-down operation, which provides exotic examples. This is a joint work with A. Akhmedov and M. Bhupal.

A representation of the category of finite sets is a slightly unusual algebraic structure, consisting of a vector space for each finite set and a linear transformation between vector spaces for each map of sets.  (It is a functor from finite sets to vector spaces).  I will talk about how these representations arise in the homology of moduli spaces of curves, and how they can be used to study the asymptotic behavior of sequences of homology groups.   

We say a partially ordered set $P$ is a tree poset if its Hasse diagram, the graph drawn by joining $x$ with $y$ if there is no $z$ such that $x > z > y$, is a tree. In this talk, we will be discussing a tool for embedding tree posets $P$ into subsets of the Boolean lattice, and some applications of it to counting copies of $P$ in subsets of the Boolean lattice, counting $P$-free subsets of the Boolean lattice, and largest $P$-free subsets of the Boolean lattice. This talk is based on joint work with Tao Jiang, Sam Spiro, and Liana Yepremyan. 

Many problems in number theory boil down to bounding the size of a set contained in a certain set of residue classes mod $p$ for various sets of primes $p$; and then sieve methods are the primary tools for doing so. Motivated by the inverse Goldbach problem, Green–Harper, Helfgott–Venkatesh, Shao, and Walsh have explored the inverse sieve problem: if we let $S \subseteq [N]$ be a maximal set of integers in this interval where the residue classes mod $p$ occupied by $S$ have some particular pattern for many primes $p$, what can one say about the  structure of the set $S$ beyond just its size? In this talk, I will give a gentle introduction to inverse sieve problems, and present some progress we made when $S$ mod $p$ has rich additive structure for many primes $p$. In particular, in this setting, we provide several improvements on the larger sieve bound for $|S|$, parallel to the work of Green–Harper and Shao for improvements on the large sieve. Joint work with Ernie Croot and Junzhe Mao.

We study the almost sure convergence of the Stochastic Approximation algorithm to the fixed point $x^\star$ of a nonlinear operator under a negative drift condition and a general noise sequence with finite $p$-th moment for some $p > 1$. Classical almost sure convergence results of Stochastic Approximation are mostly analyzed for the square-integrable noise setting, and it is shown that any non-summable but square-summable step size sequence is sufficient to obtain almost sure convergence. However, such a limitation prevents wider algorithmic application. In particular, many applications in Machine Learning and Operations Research admit heavy-tailed noise with infinite variance, rendering such guarantees inapplicable. On the other hand, when a stronger condition on the noise is available, such guarantees on the step size would be too conservative, as practitioners would like to pick a larger step size for a more preferable convergence behavior. To this end, we show that any non-summable but $p$-th power summable step size sequence is sufficient to guarantee almost sure convergence, covering the gap in the literature.

Our guarantees are obtained using a universal Lyapunov drift argument. For the regime $p \in (1, 2)$, we show that using the Lyapunov function $\|x-x^\star\|^p$ and applying a Taylor-like bound suffice. For $p > 2$, such an approach is no longer applicable, and therefore, we introduce a novel iterate projection technique to control the nonlinear terms produced by high-moment bounds and multiplicative noise.  We believe our proof techniques and their implications could be of independent interest and pave the way for finite-time analysis of Stochastic Approximation under a general noise condition. This is a joint work with Quang D. T. Nguyen, Duc Anh Nguyen, and Prof. Siva Theja Maguluri.