A brief meeting to discuss the plan for the semester, followed by an informal discussion over lunch (most likely at Ferst Place).
Great News Everyone! The cartoon series Futurama is packed with science jokes. Adopting my Professor Farnsworth Alterego, I will explain some of these mathematical jokes with stills and clips from the series.
Residential crime is one of the toughest issues in modern society. A quantitative, informative, and applicable model of criminal behavior is needed to assist law enforcement. We have made progress to the pioneering statistical agent-based model of residential burglary (Short et al., Math. Models Methods Appl., 2008) in two ways. (1) In one space dimension, we assume that the movement patterns of the criminals involve truncated Lévy distributions for the jump length, other than classical random walks (Short et al., Math. Models Methods Appl., 2008) or Lévy flights without truncation (Chaturapruek et al., SIAM J. Appl. Math, 2013). This is the first time that truncated Lévy flights have been applied in crime modeling. Furthermore (2), in two space dimensions, we used the Poisson clocks to govern the time steps of the evolution of the model, rather than a discrete time Markov chain with deterministic time increments used in the previous works. Poisson clocks are particularly suitable to model the times at which arrivals enter a system. Introduction of the Poisson clock not only produces similar simulation output, but also brings in theoretically the mathematical framework of the Markov pure jump processes, e.g., a martingale approach. The martingale formula leads to a continuum equation that coincides with a well-known mean-field continuum limit. Moreover, the martingale formulation together with statistics quantifying the relevant pattern formation leads to a theoretical explanation of the finite size effects. Our conjecture is supported by numerical simulations.
Please Note: Please note different day and room.
In this talk, I will describe joint work with Maximilien Péroux on understanding Koszul duality in ∞-topoi. An ∞-topos is a particularly well behaved higher category that behaves like the category of compactly generated spaces. Particularly interesting examples of ∞-topoi are categories of simplicial sheaves on Grothendieck topologies. The main theorem of this work is that given a group object G of an ∞-topos, there is an equivalence of categories between the category of G-modules in that topos and the category of BG-comodules, where BG is the classifying object for G-torsors. In particular, given any pointed space X, the space of loops on X, denoted ΩX, can be lifted to a group object of any ∞-topos, so if X is in addition a connected space, there is an equivalence between objects of any ∞-topos with an ΩX-action, and objects with an X-coaction (where X is a coalgebra via the usual diagonal map). This is a generalization of the classical equivalence between G-spaces and spaces over BG for G a topological group.
The study of LIn, the length of the longest increasing subsequences, and of LCIn, the length of the longest common and increasing subsequences in random words is classical in computer science and bioinformatics, and has been well explored over the last few decades. This dissertation studies a generalization of LCIn for two binary random words, namely, it analyzes the asymptotic behavior of LCbBn, the length of the longest common subsequences containing a fixed number, b, of blocks. We first prove that after proper centerings and scalings, LCbBn, for two sequences of i.i.d. Bernoulli random variables with possibly two different parameters, converges in law towards limits we identify. This dissertation also includes an alternative approach to the one-sequence LbBn problem, and Monte-Carlo simulations on the asymptotics of LCbBn and on the growth order of the limiting functional, as well as several extensions of the LCbBn problem to the Markov context and some connection with percolation theory.
In this talk we will review compactness results and singularity theorems related to harmonic maps. We first talk about maps from Riemann surfaces with tension fields bounded in a local Hardy space, then talk about stationary harmonic maps from higher dimensional manifolds, finally talk about heat flow of harmonic maps.
We investigate aspects of Kauffman bracket skein algebras of surfaces and modules of 3-manifolds using quantum torus methods. These methods come in two flavors: embedding the skein algebra into a quantum torus related to quantum Teichmuller space, or filtering the algebra and obtaining an associated graded algebra that is a monomial subalgebra of a quantum torus. We utilize the former method to generalize the Chebyshev homomorphism of Bonahon and Wong between skein algebras of surfaces to a Chebyshev-Frobenius homomorphism between skein modules of marked 3-manifolds, in the course of which we define a surgery theory, and whose image we show is either transparent or (skew)-transparent. The latter method is used to show that skein algebras of surfaces are maximal orders, which implies a refined unicity theorem, shows that SL_2C-character varieties are normal, and suggests a conjecture on how this result may be utilized for topological quantum compiling.
We say that trees with common root are (edge-)independent if, for any vertex in their intersection, the paths to the root induced by each tree are internally (edge-)disjoint. The relationship between graph (edge-)connectivity and the existence of (edge-)independent spanning trees is explored. The (Edge-)Independent Spanning Tree Conjecture states that every k-(edge-)connected graph has k-(edge-)independent spanning trees with arbitrary root.
We prove the case k=4 of the Edge-Independent Spanning Tree Conjecture using a graph decomposition similar to an ear decomposition, and give polynomial-time algorithms to construct the decomposition and the trees. We provide alternate geometric proofs for the cases k=3 of both the Independent Spanning Tree Conjecture and Edge-Independent Spanning Tree Conjecture by embedding the vertices or edges in a 2-simplex, and conjecture higher-dimension generalizations. We provide a partial result towards a generalization of the Independent Spanning Tree Conjecture, in which local connectivity between the root and a vertex set S implies the existence of trees whose independence properties hold only in S. Finally, we prove and generalize a theorem of Györi and Lovász on partitioning a k-connected graph, and give polynomial-time algorithms for the cases k=2,3,4 using the graph decompositions used to prove the corresponding cases of the Independent Spanning Tree Conjecture.
The first half of this dissertation concerns the following problem: Given a lattice in $\mathbf{R}^d$ which refines the integer lattice $\mathbf{Z}^d$, what can be said about the distribution of the lattice points inside of the half-open unit cube $[0,1)^d$? This question is of interest in discrete geometry, especially integral polytopes and Ehrhart theory. We observe a combinatorial description of the linear span of these points, and give a formula for the dimension of this span. The proofs of these results use methods from classical multiplicative number theory.
In the second half of the dissertation, we investigate oriented matroids from the point of view of tropical geometry. Given an oriented matroid, we describe, in detail, a polyhedral complex which plays the role of the Bergman complex for ordinary matroids. We show how this complex can be used to give a new proof of the celebrated Bohne-Dress theorem on tilings of zonotopes by zonotopes with an approach which relies on a novel interpretation of the chirotope of an oriented matroid.
The pedestrian-induced lateral oscillation of London's Millennium bridge on the day it opened in 2000 has become a much cited paradigm of an instability caused by phase synchronization of coupled oscillators. However, a closer examination of subsequent theoretical studies and experimental observations have brought this interpretation into question.
To elucidate the true cause of instability, we study a model in which each pedestrian is represented by a simplified biomechanically-inspired two-legged inverted pendulum. The key finding is that synchronization between individual pedestrians is not a necessary ingredient of instability onset. Instead, the side-to-side pedestrian motion should on average lag that of the bridge oscillation by a fraction of a cycle. Using a multi-scale asymptotic analysis, we derive a mathematically rigorous general criterion for bridge instability based on the notion of effective negative damping. This criterion suggests that the initiation of wobbling is not accompanied by crowd synchrony and crowd synchrony is a consequence but not the cause of bridge instability.
