Georgia Institute of Technology College of Sciences

Let $r_2(k)$ denote the smallest integer $n$ such that every $2$-edge-colored complete graph $K_n$ has a monochromatic $k$-connected subgraph. In 1983, Matula established the bound $4(k-1)+1 \leq r_2(k) < (3+\sqrt{11/3})(k-1)+1$. Furthermore, In 2008, Bollobás and Gyárfás conjectured that for any $k, n \in \mathbb{Z}^+$ with $n > 4(k-1)$, every 2-edge-coloring of the complete graph on $n$ vertices 

leads to a $k$-connected monochromatic subgraph with at least $n-2k+2$ vertices. We find a counterexample with $n = \lfloor 5k-2.5-\sqrt{8k-\frac{31}{4}} \rfloor$ for $k\ge 6$, thus disproving the conjecture, 

and we show the conclusion holds for $n > 5k-2.5-\sqrt{8k-\frac{31}{4}}$ when $k \ge 16$. Additionally, we improve the upper bound of $r_2(k)$ to $\lceil (3+\frac{\sqrt{497}-1}{16})(k-1) \rceil$ for all $k \geq 4$.

Traditionally, chip-firing is a discrete dynamical system where poker chips move around the vertices of a graph. One fascinating result is that number of configurations of a fixed number of chips, modulo a firing equivalence relation, is the number of spanning trees of the graph. This relationship gives the set of spanning trees group-like properties.

In this talk, I will discuss how chip-firing ideas can be generalized from graphs to regular matroids, where bases play the role of spanning trees. This will lead to an overview of joint work with Ding, Tóthmérész, and Yuen on the consistency of the Backman-Baker-Yuen Sandpile Torsor. 

============(Below is the information on the pre-talk.)============

Title (pre-talk): Transforming Spanning Trees Using Mathematicians and Coffee Cups

Abstract (pre-talk): There is a fascinating structure to the set of spanning trees of a plane graph, which allows this set to behave much like a group. Perhaps most incredibly, there is a sense in which this structure is canonical.
In this talk, I will show you how spanning trees can be transformed after introducing mathematicians and coffee cups on some of the vertices. This is a variant of the rotor-routing process which takes advantage of a special property of plane graphs.

Dirac’s classical theorem asserts that, for $n\ge 3$, any $n$-vertex graph with minimum degree at least $n/2$ is Hamiltonian.  Furthermore, if we additionally assume that such graphs are regular, then, by the breakthrough work of Csaba, Kühn, Lo, Osthus, and Treglown, they admit a decomposition into Hamilton cycles and at most one perfect matching, solving the well-known Nash‑Williams conjecture. In the pseudorandom setting, it has long been conjectured that similar results hold in much sparser graphs.

We prove two overarching theorems for graphs that exclude excessively dense subgraphs, which yield nearly optimal resilience and Hamilton‑decomposition results in sparse pseudorandom graphs. In particular, we show that for every fixed $\gamma>0$, there exists a constant $C>0$ such that if $G$ is a spanning subgraph of an $(n,d,\lambda)$-graph satisfying $\delta(G)\ge\bigl(\tfrac12+\gamma\bigr)d$ and $ d/\lambda\ge C,$ then $G$ must contain a Hamilton cycle.

Secondly, we show that for every $\varepsilon>0$, there is $C>0$ so that any $(n,d,\lambda)$-graph with $d/\lambda\ge C$ contains at least $\bigl(\tfrac12-\varepsilon\bigr)d$ edge‑disjoint Hamilton cycles, and, finally, we prove that the entire edge set of $G$ can be covered by no more than $\bigl(\tfrac12+\varepsilon\bigr)d$ such cycles.

All bounds are asymptotically optimal and significantly improve earlier results on Hamiltonian resilience, packing, and covering in sparse pseudorandom graphs.

This is joint work with Nemanja Draganić, Jaehoon Kim, David Munhá Correia, Matías Pavez-Signé, and Benny Sudakov.

We develop a unified framework for improving numerical solvers with Neural Networks with Locally Converging Inputs (NNLCI). First, we applied NNLCI to 2D Maxwell’s equations with perfectly matched‐layer boundary conditions for light–PEC (perfect electric conductor) interactions. A network trained on local patches around specific PEC shapes successfully predicted solutions on globally different geometries. Next, we tested NNLCI on various ODEs: it failed for chaotic systems (e.g., double pendulum) but was effective for nonchaotic dynamics, and in simple cases can be interpreted as a well‐defined function of its inputs. Although originally formulated for hyperbolic conservation laws, NNLCI also performed well on parabolic and elliptic problems, as demonstrated in a 1D Poisson–Nernst–Planck ion‐channel model. Building on these results, we applied NNLCI to multi‐asset cash‐or‐nothing options under Black–Scholes. By correcting coarse‐ and fine‐mesh ADI solutions, NNLCI reduced RMSE by factors of 4–12 on test parameters, even when trained on a small fraction of the parameter grid. Careful treatment of far‐field boundary truncation was critical to maintain convergence far from the strike price. Finally, we demonstrate NNLCI’s first application to quantum algorithms by improving variational quantum‐algorithm (VQA) outputs for the 1D Poisson equation under realistic NISQ‐device noise. Although noisy VQA solutions deviate from classical finite‐difference references and do not converge to true solutions, NNLCI effectively maps these noisy outputs toward high‐accuracy references. We hypothesize that NNLCI implicitly composes the map from coarse quantum outputs to a noisy convergence space, then to the true solution. We discuss conditions for NNLCI to approximate a well‐defined inverse of the numerical scheme and contrast this with Monte Carlo methods, which lack deterministic intermediate states. These results establish NNLCI as a versatile, data‐efficient tool for accelerating solvers in classical and quantum settings.