Mathematics Dept.
  Events
University of Auckland
Wed 17 Jan 2018, 3:10pm
Probability Seminar
LSK 460
The gaps left by a Brownian motion
LSK 460
Wed 17 Jan 2018, 3:10pm-4:10pm

Abstract


Run a Brownian motion on a torus for a long time.  How large are the

random gaps left behind when the path is removed?

 

In three (or more) dimensions, we find that there is a deterministic spatial

scale common to all the large gaps anywhere in the torus.  Moreover, we can

identify whether a gap of a given shape is likely to exist on this scale, in

terms of a single parameter, the classical (Newtonian) capacity.  I will

describe why this allows us to identify a well-defined "component" structure in

our random porous set.


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Oxford University
Thu 18 Jan 2018, 11:00am SPECIAL
Mathematical Biology Seminar / Probability Seminar
Math 126
Modelling mutations: mechanisms and evolutionary consequences
Math 126
Thu 18 Jan 2018, 11:00am-12:00pm

Abstract

 As the source of new genetic variation, mutations constitute a fundamental process in evolution. While most mutations lower fitness, rare beneficial mutations are essential for adaptation to changing environments. Thus, understanding the effects of mutations and estimating their rate is of strong interest in evolutionary biology. The necessity to treat rare mutational events stochastically has also stimulated a rich mathematical literature. Typically, mutations are modelled simply as an instantaneous change of type, occurring at a fixed rate. However, the underlying biology is more complex. I will present two recent projects delving deeper into mutational mechanisms and their consequences. Firstly, mutations can exhibit a multi-generational delay in phenotypic expression. Secondly, individuals within a population can vary in their propensity to mutate. Through analytical and simulation methods, we investigated the impact of these biological complexities on (a) population fitness and capacity to evolve, and (b) our ability to accurately infer mutation rates from data. I will conclude by discussing some future directions to incorporate these insights into more realistic models and to quantify the distribution of mutation rate empirically.

Note for Attendees

 Math 126 is behind a locked glass door. Latecomers without access should knock loudly!
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Kornelia Hera
Eotvos Lorand University, Budapest
Fri 19 Jan 2018, 2:00pm
Harmonic Analysis Seminar
MATH 126
Furstenberg-type estimates for unions of affine subspaces
MATH 126
Fri 19 Jan 2018, 2:00pm-3:00pm

Abstract

A plane set is called a t-Furstenberg set for some t in (0,1), if it has an at least t-dimensional intersection with some line in each direction (here and in the sequel dimension refers to Hausdorff dimension).  Classical results are that every t-Furstenberg set has dimension at least 2t, and at least t + 1/2.

We generalize these estimates for families of affine subspaces. As a result, we prove that the union of any s-dimensional family of k-dimensional affine subspaces is at least k + s/(k+1) -dimensional, and is exactly k + s -dimensional if s is at most 1.

Based on joint work with Tamas Keleti and Andras Mathe.
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Oxford University
Fri 19 Jan 2018, 3:00pm SPECIAL
Department Colloquium
ESB 2012
Stochastic population dynamic models with applications to pathogen evolution
ESB 2012
Fri 19 Jan 2018, 3:00pm-4:00pm

Abstract

Biological populations facing severe environmental change must adapt in order to avoid extinction. This so-called “evolutionary rescue” scenario is relevant to many applied problems, including pathogen evolution of drug resistance during the treatment of infectious diseases. Understanding what drives the rescue process gives rise to interesting mathematical modelling challenges arising from two key features: demographic and evolutionary processes occur on the same timescale, and stochasticity is inherent in the emergence of rare well-adapted mutants. In this talk, I will present recent work on population dynamics in changing environments, merging biological realism with tractable stochastic models. Firstly, I will describe a model of drug resistance evolution in chronic viral infections, which serves as a case study for a novel mathematical approach yielding analytical approximations for the probability of rescue. Secondly, I will present a combined theoretical and experimental investigation of the classical problem of establishment (non-extinction) of new lineages, using antibiotic-resistant bacteria as a model system. Finally, I will discuss some future directions in modelling antibiotic treatment to predict optimal dosing strategies, and in developing a general theoretical framework for evolutionary rescue that unites approaches to distinct applied problems.

Note for Attendees

Refreshments will be served at 2:45 p.m. in ESB 4133, the PIMS Lounge.
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Jay Newby
University of North Carolina, Chapel Hill
Mon 22 Jan 2018, 12:00pm SPECIAL
ESB 4133 (PIMS Lounge)
Seminar Talk in Math Biology, Applied Stochastics: How molecular crowding is changing our understanding of spatial patterning in living cells
ESB 4133 (PIMS Lounge)
Mon 22 Jan 2018, 12:00pm-1:00pm

Details


Molecular crowding has recognized consequences for biological function. However, there are also circumstances in which un-crowding is important that is, when molecules must evacuate from a region before a given process can occur. One example is offered by the T cell, where large surface molecules must evacuate from a region to allow for the T cell to interact with its target, thereby facilitating immune function. Evacuation is fundamentally stochastic and spatial, since diffusion is a major driver. Studies of molecular evacuation present mathematical and computational challenges. For example, in some scenarios, it is a "rare event", making straightforward simulation unfeasible. To obtain a complete picture of diffusional evacuation, we use a combination of perturbation theory and numerical simulation. I will also show evidence of persistent un-crowding in living fungal cells. Based on our understanding of diffusional evacuation, we know that diffusion alone cannot explain these observations. I will discuss our current efforts to quantify and resolve how fungal cells control un-crowding.

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Cornell Statistical Science and Biological Statistics & Computational Biology
Mon 22 Jan 2018, 3:00pm SPECIAL
Institute of Applied Mathematics
ESB 2012
An ODE to Statistics: Inference about Nonlinear Dynamics
ESB 2012
Mon 22 Jan 2018, 3:00pm-4:00pm

Abstract

Ordinary differential equation models are used extensively within mathematics as descriptions of processes in the real world. However, they are rarely employed by statisticians and there is a paucity of methods for combining differential equation models with data. This talk provides a survey of recently developed statistical methods for estimating parameters from data, conducting model criticism and improvement for differential equation models in the light of data, and designing experiments that yield optimal estimates of parameters. It ends with some perspectives on the current state of the field and open problems.

Note for Attendees

Reception before the talk in ESB 4133 (the PIMS lounge). This is in the IAM/PIMS distinguished speaker series.
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Saskatchewan
Mon 22 Jan 2018, 4:00pm
Algebraic Geometry Seminar
MATH 126
TBA
MATH 126
Mon 22 Jan 2018, 4:00pm-5:00pm

Abstract

 TBA
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Jay Newby
University of North Carolina, Chapel Hill
Tue 23 Jan 2018, 3:45pm SPECIAL
Department Colloquium
ESB 2012 (PIMS)
Weaker is better: how weak transient molecular interactions give rise to robust, dynamic immune protection
ESB 2012 (PIMS)
Tue 23 Jan 2018, 3:45pm-4:45pm

Abstract

The longstanding view in chemistry and biology is that high-affinity, tight-binding interactions are optimal for many essential functions, such as receptor-ligand interactions. Yet, an increasing number of biological systems are emerging that challenge this view, finding instead that low-affinity, rapidly unbinding dynamics can be essential for optimal function. These mechanisms have been poorly understood in the past due to the inability to directly observe such fleeting interactions and the lack of a theoretical framework to mechanistically understand how they work. In fact, it is only by tracking the motion of effector nanoprobes that afford detection of multiple such interactions simultaneously, coupled with inferences by stochastic modeling, Bayesian statistics, and bioimaging tools, that we recently begin to obtain definitive evidence substantiating the consequences of these interactions. A common theme has begun to emerge: rapidly diffusing third-party molecular anchors with weak, short-lived affinities play a major role for self organization of micron-scale living systems. My talk will demonstrate how these ideas can answer a longstanding question: how mucosal barriers selectively impede transport of pathogens and toxic particles, while allowing diffusion of nutrients.
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