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Monday, August 26, 2019 - 12:45 ,
Location: Skiles 006 ,
Jasmine Powell ,
University of Michigan ,
jtpowell@umich.edu ,
Organizer: Justin Lanier

The field of complex dynamics melds a number of disciplines, including complex analysis, geometry and topology. I will focus on the influences from the latter, introducing some important concepts and questions in complex dynamics, with an emphasis on how the concepts tie into and can be enhanced by a topological viewpoint.

Series: CDSNS Colloquium

In this talk, we consider a quasi-periodically forced system arising from the problem of chemical reactions. For we demonstrate efficient algorithms to calculate the normally hyperbolic invariant

manifolds and their stable/unstable manifolds using parameterization method. When a random noise is added, we use similar ideas to give a streamlined proof of the existence of the stochastic

invriant manifolds.

Series: Stochastics Seminar

A random array indexed by the paths of an infinitely-branching rooted tree of finite depth is hierarchically exchangeable if its joint distribution is invariant under rearrangements that preserve the tree structure underlying the index set. Austin and Panchenko (2014) prove that such arrays have de Finetti-type representations, and moreover, that an array indexed by a finite collection of such trees has an Aldous-Hoover-type representation.

Motivated by problems in Bayesian nonparametrics and probabilistic programming discussed in Staton et al. (2018), we generalize hierarchical exchangeability to a new kind of partial exchangeability for random arrays which we call DAG-exchangeability. In our setting a random array is indexed by N^{|V|} for some DAG G=(V,E), and its exchangeability structure is governed by the edge set E. We prove a representation theorem for such arrays which generalizes the Aldous-Hoover representation theorem, and for which the Austin-Panchenko representation is a special case.

Series: Mathematical Biology Seminar

A brief meeting to discuss the plan for the semester, followed by an informal discussion over lunch (most likely at Ferst Place).

Monday, August 19, 2019 - 13:50 ,
Location: Skiles 005 ,
Chuntian Wang ,
The University of Alabama ,
cwang27@ua.edu ,
Organizer: Martin Short

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.

Series: Geometry Topology Seminar

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.

Series: Dissertation Defense

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.

Series: PDE Seminar

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.

Series: Dissertation Defense

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.

Series: Dissertation Defense

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.