Spring 2020

January 23, 2020: Boris Pittel (OSU)

Title: On random stable matchings:  cyclic ones with strict preferences and two-sided ones with partially ordered preferences.

Abstract: Consider a cyclically ordered collection of r equi-numerous “agent” sets with strict

preferences of every agent over the agents from the next  set. A weakly stable cyclic matching is a partition of the set of agents into disjoint union of r-long cycles, one agent from each set per cycle, such that there are no destabilizing r-long cycles, i.e. cycles in which every agent strictly prefers its successor to its successor in the matching. Assuming that the preferences are uniformly random and independent, we show that the expected number of stable matchings grows with n (cardinality of each agent set) as (n\log n)^{r-1}.
Next we consider a bipartite stable matching problem where preference list of each agent
forms a partially ordered set. Each partial order is an intersection of several, k_i for side i, independent, uniformly random, strict orders. For k_1+k_2>2, the expected number of stable matchings is analyzed for three, progressively stronger, notions of stability. The expected number of weakly stable matchings is shown to grow super-exponentially fast. In contrast, for \min(k_1,k_2)>1, the fraction of instances with at least one strongly stable (super-stable) matching is super-exponentially small.
Some open problems are discussed.

 

January 30, 2020: Open

Title: Open

Abstract: Open

 

February 6, 2020: Open

Title: Open

Abstract: Open

 

February 13, 2020: Kevin Leder (UMN)

Title: Estimating Rare Event Probabilities in Reflecting Brownian Motion

Abstract:

Reflecting Brownian motion (RBM) is a stochastic process that behaves like a Brownian motion in the interior of its domain and is pushed into the interior whenever it reaches the boundary of its domain. RBM’s in the positive orthant were first introduced by Harrison and Reiman 1981, and arise naturally in a wide variety of settings, e.g., heavy traffic queueing networks. A difficult question that has received a significant amount of attention is identifying the asymptotics for tail probabilities associated with RBM in the positive orthant. In this work we focus on the specific tail probability that a stable RBM started near the origin exits a large box before returning to the origin. We develop particle based algorithms to estimate this probability.

Using results of Dean and Dupuis 2008 we are able to develop algorithms that efficiently estimate this tail probability in two dimensions. In three and higher dimensions, we are not able to construct an efficient estimator, but we do construct estimators that are provably superior (in an asymptotic sense) to standard Monte Carlo. Numerical results show the benefits of our algorithm to standard Monte Carlo. This is based on joint work with Xin Liu and Zicheng Wang.

 

February 20, 2020: Wai-Tong (Louis) Fan  (Indiana)

Title:Stochastic and deterministic spatial models for complex systems

Abstract:

Interacting particle models are often employed to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge, which is fundamental in any multi-scale modeling approach for complex systems, is how to simultaneously retain key information in microscopic models as well as efficiency and robustness of macroscopic models.
 In this talk, I will discuss how this challenge can be overcome by elucidating the probabilistic connections between particle models and PDE, in particular, why naively adding diffusion terms to ordinary differential equations might fail to account for spatial dynamics in population models. These connections also explain how stochastic partial differential equations (SPDE) arise naturally under a suitable choice of level of detail in modeling complex systems. I will also present some novel scaling limits including SPDE on graphs and coupled SPDE. These SPDE not only interpolate between particle models and PDE, but also quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and new duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of certain population dynamics.

 

February 27, 2020: Open, but our regular room is not available. May need to find another space.

Title: Open

Abstract: Open

 

March 5, 2020: Open

Title: Open

Abstract: Open

 

March 12, 2020: Spring Break (No seminar)

Title: (No Seminar)

Abstract: (No Seminar)

 

March 19, 2020: Open

Title: TBA

Abstract: TBA

 

March 26, 2020: Phillip Matchett Wood (UCB)

Title: TBA

Abstract: TBA

 

 

March 27, 2020 (special date & time): Andras Meszaros (CEU-Renyi Institute)

Title: TBA

Abstract: TBA

 

 

April 2, 2020:Brian Skinner (OSU, Physics)

Title: TBA

Abstract: TBA

 

April 9, 2020: Tom Trogdon (UW Seattle)

Title: Open

Abstract: Open

 

April 16, 2020: Timo Seppalainen speaking in Colloquium at 4:15-5:15pm

Title: TBA

Abstract: TBA

 

April 23, 2020: Open

Title: Open

Abstract: Open

 

April 30, 2020: Open

Title: Open

Abstract: Open