Georgia Institute of Technology College of Sciences

 Recall that an excedance of a permutation $\pi$ is any position $i$
 such that $\pi_i > i$. Inspired by the work of Hopkins, McConville and
 Propp (arXiv:1612.06816) on sorting using toppling, we say that
 a permutation is toppleable if it gets sorted by a certain sequence of
 toppling moves. For the most part of the talk, we will explain the
 main ideas in showing that the number of toppleable permutations on n
 letters is the same as those for which excedances happen exactly at
 $\{1,\dots, \lfloor (n-1)/2 \rfloor\}$. Time permitting, we will give
 some ideas showing that this is also the number of acyclic
 orientations with unique sink (also known as the Ursell function) of the
 complete bipartite graph $K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor + 1}$.

 This is joint work with D. Hathcock (CMU) and P. Tetali (Georgia Tech).

(Joint work with Timo Seppäläinen) We establish estimates for the coalescence time of semi-infinite directed geodesics in the planar corner growth model with i.i.d. exponential weights. There are four estimates: upper and lower bounds on the probabilities of both fast and slow coalescence on the correct spatial scale with exponent 3/2. Our proofs utilize a geodesic duality introduced by Pimentel and properties of the increment-stationary last-passage percolation process. For fast coalescence our bounds are new and they have matching optimal exponential order of magnitude. For slow coalescence, we reproduce bounds proved earlier with integrable probability inputs, except that our upper bound misses the optimal order by a logarithmic factor.

For positive integers $d < k$ and $n$ divisible by $k$, let $m_d(k,n)$ be the minimum $d$-degree ensuring the existence of a perfect matching in a $k$-uniform hypergraph. In the graph case (where $k=2$), a classical theorem of Dirac says that $m_1(2,n) = \lceil n/2\rceil$. However, in general, our understanding of the values of $m_d(k,n)$ is still very limited, and it is an active topic of research to determine or approximate these values. In the first part of this talk, we discuss a new "transference" theorem for Dirac-type results relative to random hypergraphs. Specifically, we prove that a random $k$-uniform hypergraph $G$ with $n$ vertices and "not too small" edge probability $p$ typically has the property that every spanning subgraph with minimum $d$-degree at least $(1+\varepsilon)m_d(k,n)p$ has a perfect matching. One interesting aspect of our proof is a "non-constructive" application of the absorbing method, which allows us to prove a bound in terms of $m_d(k,n)$ without actually knowing its value.

The ideas in our work are quite powerful and can be applied to other problems: in the second part of this talk we highlight a recent application of these ideas to random designs, proving that a random Steiner triple system typically admits a decomposition of almost all its triples into perfect matchings (that is to say, it is almost resolvable).

Joint work with Asaf Ferber.

Many systems of nonlinear PDEs are arising from engineering and biology and have attracted research scientists to study the multiple solution structure such as pattern formation. In this talk, I will present several methods to compute the multiple solutions of nonlinear PDEs. In specific, I will introduce the homotopy continuation technique to compute the multiple steady states of nonlinear differential equations and also to explore the relationship between the number of steady-states and parameters. Then I will also introduce a randomized Newton's method to solve the nonlinear system arising from neural network discretization of the nonlinear PDEs. Several benchmark problems will be used to illustrate these ideas.