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We will discuss funding opportunities in pure and applied mathematics, and strategies for obtaining grants, fellowships, and so on. We will focus on NSERC grants in Canada and National Science Foundation grants in the U.S.

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Breathing is an interesting and essential life-sustaining behavior for humans and all mammals. Like many rhythmic motor behaviors, breathing movements originate due to neural rhythms that emanate from a central pattern generator (CPG) network. CPGs produce neural-motor rhythms that often depend on specialized pacemaker neurons or alternating synaptic inhibition. But conventional models cannot explain rhythmogenesis in the respiratory preBotzinger Complex (preBotzC), the principal central pattern generator for inspiratory breathing movements, in which rhythms persist under experimental blockade of synaptic inhibition and of intrinsic pacemaker currents. Using mathematical models and experimental tests, here we demonstrate an unconventional mechanism in which metabotropic synapses and synaptic disfacilitation play key rhythmogenic roles: recurrent excitation triggers Ca2+-activated nonspecific cation current (ICAN), which initiates the inspiratory burst. Robust depolarization due to ICAN also causes voltage-dependent spike inactivation, which diminishes recurrent excitation, allowing outward currents such as Na/K ATPase pumps and K+ channels to terminate the burst and cause a transient quiescent state in the network. After a recovery period, sporadic spiking activity rekindles excitatory interactions and thus starts a new cycle. Because synaptic inputs gate postsynaptic burst-generating conductances, this rhythm-generating mechanism represents a new paradigm in which the basic rhythmogenic unit encompasses a fully inter-dependent ensemble of synaptic and intrinsic components.

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The aim of this talk is to highlight the usefulness of continuous time branching process theory in understanding refined asymptotics about various random network models. We shall exhibit their usefulness in two different contexts:

(1) First passage percolation: Consider a connected network and suppose each edge in the network has a random positive edge weight. Understanding the structure and weight of the shortest path between nodes in the network is one of the most fundamental problems studied in modern probability theory. In the modern context these problems take an additional significance with the minimal weight measuring the cost of sending information while the number of edges on the optimal path (hopcount) representing the actual time for messages to get between vertices in the network. In the context of the configuration model of random networks we shall show how branching processes allow us to find the limiting distribution of the minimal weight path as well as establishing a general central limit theorem for the hopcount with matching means and variances.

(2) Spectral distribution of random trees: Many models of random trees (including general models embedded in continuous time branching processes) satisfy a general form of convergence locally to limiting infinite objects. In this context we find via soft arguments, the convergence of the spectral distribution of the adjacency matrix to a limiting (model dependent) non random distribution. For any \gamma we also find a sufficient condition for there to be a positive mass at \gamma in the limit.

Joint work with Remco van der Hoftsad, Gerard Hooghiemstra, Steve Evans and Arnab Sen.

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In this talk I show that the space of almost commuting elements in a compact Lie group G splits after one suspension.

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Social learning is an important ability seen in a wide range of animals. Especially, humans developed the advanced social learning ability such as language, which triggered rapid cultural evolution. On the other hand, many species, such as viruses, rely on genetic evolution to adapt to environmental fluctuations. Here we propose an evolutionary game model of competition among three strategies; social learning, individual learning, and genetic determination of behavior. We identify the condition for learning strategies to evolve.

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This is joint work with Eric Friedlander, Julia Pevtsova and Andrei Suslin.  We consider modules over an elementary abelian group on which every element in the radical, but not the square of the radical, has the same Jordan canonical form.  Such modules can be used to define bundles on projective spaces and Grassmanians.  They have many interesting properties.  We can get them as submodules of any module of the group algebra.  In this talk I will discuss some of the constructions and their generalizations.

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The basic reproductive number, R_0, which is generally defined as the expected number of secondary infections per primary case in a totally susceptible population, is an important epidemiological quantity. It helps us to understand the possible outcome of an initial infection seeding in a social setting: whether it leads to a small outbreak, or it evolves into a large-scale epidemic. The basic reproductive number encapsulates the information about the biology of disease transmission as well as the structure of human social contacts. We use concepts from network theory to present a novel method for estimating the value of the basic reproductive number during the early stage of an outbreak. This approach will greatly enhance our ability to reliably estimate the level of threat caused by an emerging infectious disease.

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In this talk, designed for a broad mathematical audience, we will describe what is known (and unknown) about the classification of smooth 4-manifolds. In particular we will uncover a new mechanism that makes progress towards the conjecture that every simply-connected closed 4-manifold has either zero or infinitely many distinct smooth structures.

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TBA

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We introduce and discuss some new stochastic models of optimal portfolio selection with reduced impact of forecast errors. In particular, we found some new examples of optimal myopic strategies, including some discrete time models with serial correlations. In addition, we found some new cases when the strategies that don't leave the efficient frontier even if there is an error in the forecast. It may happen for non-myopic strategies if the required information about the future is limited.