Georgia Institute of Technology College of Sciences

For a connected graph G, the set of G-parking functions are integer sequences counted by spanning trees that arise in the theory of chip-firing on G.  They can also be defined as the standard monomials of a `G-parking function ideal', whose homological properties have interesting combinatorial interpretations. We extend these constructions to the setting of hypergraphs, where edges can have multiple vertices. We study algebraic and combinatorial aspects of parking functions in this context, employing generalized notions of acyclic orientations and spanning trees. Minimal cellular resolutions of the underlying ideals can be understood in terms of certain generalized permutohedra. This is joint work with Ayah Almousa and Ben Smith, as well as an REU project with Timothy Blanton, Isabelle Hong, Suho Oh, and Zhan Zhan.

In this talk I will present both mathematical and numerical analysis as well as experiments to study a few basic computational issues in using neural network to approximate functions: (1) the stability and accuracy, (2) the learning dynamics and computation cost, and (3) structured and balanced approximation. These issues are investigated for both approximation and optimization in asymptotic and non-asymptotic regimes.

In this talk, we introduce contact invariants in bordered sutured Floer homology. Given a contact 3-manifold with convex boundary, we apply a result of Zarev to derive contact invariants in the bordered sutured modules BSA and BSD. We show that these invariants satisfy a pairing theorem, which is a bordered extension of the Honda-Kazez-Matic gluing map for sutured Floer homology. We also show that there is a correspondence between certain A-infinity operations in bordered modules and bypass attachment maps in sutured Floer homology. As an application, we characterize the Stipsicz-Vertesi map in terms of A-infinity action on CFA. If time permits, we will further discuss applications to contact surgery.

Pseudoholomorphic curves are pivotal in establishing uniqueness and finiteness results in the classification of symplectic manifolds. In a series of works, Wendl used punctured pseudoholomorphic foliations to classify symplectic fillings of contact three-manifolds supported by planar open books, turning it into a problem about monodromy factorizations. In a joint work with Hyunki Min and Agniva Roy, we build on the works of Lisi--Van Horn-Morris--Wendl in using spinal open books to further delve into the classification problem of symplectic fillings of higher genus open books. In particular, we provide the local model of the mysterious "exotic fibers" in a generalized version of Lefschetz fibrations, which captures a natural type of singularity at infinity. We will give some applications to classifying symplectic fillings via this new phenomenon.

Graph coloring is a fundamental problem in graph theory in which the goal is to properly color a graph with the minimum number of colors in terms of some parameters (such as maximum degree). We explore this problem from the perspective of three different types of graphs: graphs with forbidden bipartite subgraphs; planar graphs; and Borel graphs that are line graphs. Each can be seen as graphs with a forbidden list of subgraphs; despite this similarity, the techniques used to study each are as varied as the results themselves.

We start with studying $F$-free graphs. We say a graph is $F$-free if it contains no subgraph isomorphic to $F$ (not necessarily induced). A conjecture of Alon, Krivelevich, and Sudakov states that, for any graph $F$, there is a constant $c_F > 0$ such that if $G$ is an $F$-free graph of maximum degree $\Delta$, then $\chi(G) \leq c_F \Delta / \log\Delta$. Alon, Krivelevich, and Sudakov verified this conjecture for a class of graphs $F$ that includes all bipartite graphs. Moreover, it follows from recent work by Davies, Kang, Pirot, and Sereni that %this conjecture holds for $F$ bipartite; moreover, if $G$ is $K_{t,t}$-free, then $\chi(G) \leq (t + o(1)) \Delta / \log\Delta$ as $\Delta \to \infty$. We improve this bound to $(1+o(1)) \Delta/\log \Delta$, making the constant factor independent of $t$. This matches the best known bound for several other class of graphs $F$, such as triangles, fans, and cycles, and lowering this bound further for nontrivial graphs is considered extremely challenging. We further extend our result to the correspondence coloring setting (also known as DP-coloring), introduced by Dvo\v{r}\'ak and Postle.

Next we study defective coloring of planar graphs. Defective coloring (also  known as relaxed or improper coloring) is a generalization of proper coloring defined as follows: for $d \in \mathbb{N}$, a coloring of a graph is $d$-defective if every vertex is colored the same as at most $d$ of its neighbors. We investigate defective coloring of planar graphs in the context of correspondence coloring. First we show there exist planar graphs that are not $3$-defective $3$-correspondable, strengthening a recent result of Cho, Choi, Kim, Park, Shan, and Zhu. Then we construct a planar graph that is $1$-defective $3$-correspondable but not $4$-correspondable, thereby extending a recent result of Ma, Xu, and Zhu from list coloring to correspondence coloring. Finally we show all outerplanar graphs are $3$-defective 2-correspondence colorable, with $3$ defects being best possible.

Finally, we study Borel graphs. We characterize Borel line graphs in terms of 10 forbidden induced subgraphs, namely the 9 finite graphs from the classical result of Beineke together with a 10th infinite graph associated to the equivalence relation $\mathbb{E}_0$ on the Cantor space. As a corollary, we prove a partial converse to the Feldman--Moore theorem, which allows us to characterize all locally countable Borel line graphs in terms of their Borel chromatic numbers.

This includes work coauthored with Anton Bernshteyn and Abhishek Dhawan.

The moments of the Riemann zeta-function were introduced more than 100 years ago by Hardy and Littlewood, who showed that the Lindelof hypothesis (which provides a strong upper bound for the Riemann zeta-function on the critical line) is equivalent to obtaining sharp bounds on all the positive, even integral moments. Since then, the moments of the Riemann zeta-function and of more general L-functions have become natural objects of study. In this talk, I will review some of the history of the problem of evaluating moments, and focus on three different lines of research: studying negative moments of L-functions (which have been much less studied over the years, but which have rich applications nevertheless), computing lower-order terms in the moment asymptotics and obtaining non-vanishing results for L-functions evaluated at special points.

We will talk about recent work establishing a quantitative nonlinear scattering theory for asymptotically de Sitter solutions of the Einstein vacuum equations in (n+1) dimensions with n ≥ 4 even, which are determined by small scattering data at future infinity and past infinity. We will also explain why the case of even spatial dimension n poses significant challenges compared to its odd counterpart and was left open by the previous works in the literature.

A basic question for any knot invariant asks which knots the invariant detects. For example, it is famously open whether the Jones polynomial detects the unknot. I'll focus in this talk on the detection question for knot invariants coming from Floer theory and the Khovanov--Rozansky link homology theories. I'll survey the progress made over the past twenty years, and will describe some of the topological ideas that go into my recent work with Sivek on these questions. Time permitting, I'll end with applications of these knot detection results to problems in Dehn surgery, explaining in particular how we use them to dramatically extend some of Gabai's celebrated results from the 80's.

 I will discuss recent work with Chantal David, Alexander Dunn, and Joshua Stucky, in which we prove that a positive proportion of Hecke L-functions associated to the cubic residue symbol modulo squarefree Eisenstein integers do not vanish at the central point. Our principal new contribution is the asymptotic evaluation of the mollified second moment with power saving error term. No such asymptotic formula was previously known for a cubic family (even over function fields). Our new approach makes crucial use of Patterson's evaluation of the Fourier coefficients of the cubic metaplectic theta function, Heath-Brown's cubic large sieve, and a Lindelöf-on-average upper bound for the second moment of cubic Dirichlet series that we establish. The significance of our result is that the (unitary) family considered does not satisfy a perfectly orthogonal large sieve bound. This is quite unlike other families of Dirichlet L-functions for which unconditional results are known (namely the symplectic family of quadratic characters and the unitary family of all Dirichlet characters modulo q). Consequently, our proof has fundamentally different features from the corresponding works of Soundararajan and of Iwaniec and Sarnak.

The totally non-negative Grassmannian is the set of points in a real Grassmannian such that all Plucker coordinates have the same sign (some can be zero). I will show how points in totally non-negative Grassmannians arise from the spaces of polynomials in one variable whose Wronskian has only real roots. Then I will discuss a similar result for the spaces of quasi-exponentials.

The main statements of this talk should be understandable to an undergraduate student. Somewhat surprisingly, the proofs use the theory of quantum integrable systems related to $GL(n)$. I will try to explain the logic of such proofs in a gentle way.

This talk is based on a joint work with S. Karp and V. Tarasov.

Houdré and Tetali defined a class of isoperimetric constants phi_p of graphs for 1/2 <= p <= 1.  When p=1, the classical Cheeger's inequality relates phi_1 to the second smallest eigenvalue of the normalized Laplacian matrix.  Houdré and Tetali conjectured that a similar Cheeger-type inequality holds for p=1/2, which if true would be a strengthening of Cheeger's inequality.  Morris and Peres proved the Houdré-Tetali conjecture up to an additional log factor, using techniques from evolving sets.  In this talk, we discuss the following results about this conjecture:

  - There is a family of counterexamples to the conjecture of Houdré and Tetali, showing that the logarithmic factor is needed.

  - Morris and Peres' result can be recovered using standard spectral arguments. 

  - The Houdré-Tetali conjecture is true for any constant p strictly bigger than 1/2, which is also a strengthening of Cheeger's inequality.

If time permits, we also discuss other strengthenings of Cheeger's inequality.  No background is assumed from the audience.

In this talk, for an ergodic probability preserving system $(X,\mathcal{B},m,T)$, we will discuss the existence of a function $f:X\to \mathbb{Z}^d$, whose corresponding cocycle satisfies the $d$-dimensional local central limit theorem.
As an application, we resolve a question of Huang, Shao and Ye, and Franzikinakis and Host regarding non-convergence in $L^2$ of polynomial multiple averages of non-commuting zero entropy transformations. If time allows, we will also discuss the first examples of failure of multiple recurrence for zero entropy transformations along polynomial iterates. This is joint work with Zemer Kosloff (arXiv:2409.05087). 

The arithmetic quantum unique ergodicity (AQUE) conjecture predicts that the L^2 mass of Hecke-Maass cusp forms on an arithmetic hyperbolic manifold becomes equidistributed as the Laplace eigenvalue grows. If the underlying manifold is non-compact, mass could “escape to infinity”. This possibility was ruled out by Soundararajan for arithmetic surfaces, which when combined with celebrated work of Lindenstrauss completed the proof of AQUE for surfaces.

We establish non-escape of mass for Hecke-Maass cusp forms on a congruence quotient of hyperbolic 4-space. Unlike in the setting of hyperbolic 2- or 3-manifolds (for which AQUE has been proved), the number of terms in the Hecke relations is unbounded, which prevents us from naively applying Cauchy-Schwarz. We instead view the isometry group as a group of quaternionic matrices, and rely on non-commutative unique factorization, along with certain structural features of the Hecke action. Joint work with Zvi Shem-Tov.