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 Events
Stanford University
Tue 2 Jan 2018, 12:30pm SPECIAL
Department Colloquium
MATH 102 (special time, lunch will be served)
Stability in the homology of configuration spaces
MATH 102 (special time, lunch will be served)
Tue 2 Jan 2018, 12:30pm-1:30pm

Abstract

This talk will illustrate some patterns in the homology of the configuration space F_k(M), the space of ordered k-tuples of distinct points in a manifold M. For a fixed manifold M, as k increases, we might expect the topology of these configuration spaces to become increasingly complicated. Church and others showed, however, that when M is connected and open, there is a representation-theoretic sense in which the homology groups of these spaces stabilize. In this talk I will explain these stability patterns, and describe higher-order stability phenomena -- relationships between unstable homology classes in different degrees -- established in recent work joint with Jeremy Miller. This project was inspired by work-in-progress of Galatius--Kupers--Randal-Williams.


Note for Attendees

Note special time. Lunch will be served.
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Stanford University
Mon 8 Jan 2018, 3:00pm
Department Colloquium
ESB 2012 (note special time)
Dynamics, geometry, and the moduli space of Riemann surfaces
ESB 2012 (note special time)
Mon 8 Jan 2018, 3:00pm-4:00pm

Abstract

The moduli space of Riemann surfaces of fixed genus is one of the hubs of modern mathematics and physics. We will tell the story of how simple sounding problems about polygons, some of which arose as toy models in physics, became intertwined with problems about the geometry of moduli space, and how the study of these problems in Teichmuller dynamics lead to connections with homogeneous spaces, algebraic geometry, dynamics, and other areas. The talk will mention joint works with Alex Eskin, Simion Filip, Curtis McMullen, Maryam Mirzakhani, and Ronen Mukamel.
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UC San Diego
Fri 12 Jan 2018, 3:00pm
Department Colloquium
ESB 2012
New and Improved Binary Embeddings of Data (and Quantization for Compressed Sensing with Structured Random Matrices)
ESB 2012
Fri 12 Jan 2018, 3:00pm-4:00pm

Abstract

We discuss two related problems that arise in the acquisition and processing of high-dimensional data. First, we consider distance-preserving fast binary embeddings. Here we propose fast methods to replace points from a set \mathcal{X} \subset \R^N with points in a lower-dimensional cube \{\pm 1\}^m, which we endow with an appropriate function to approximate Euclidean distances in the original space.
 
Second, we consider a problem in the quantization (i.e., digitization) of compressed sensing measurements. Here, we deal with measurements arising from the so-called bounded orthonormal systems and partial circulant ensembles, which arise naturally in compressed sensing applications. In both these problems we show state-of-the art error bounds, and to our knowledge, some of our results are the first of their kind.
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MIT
Mon 15 Jan 2018, 4:00pm SPECIAL
Department Colloquium
MATH 100 (note special time and place)
The symplectic topology of affine varieties
MATH 100 (note special time and place)
Mon 15 Jan 2018, 4:00pm-5:00pm

Abstract

In this talk we will study complex affine varieties via symplectic topology. First, I will explain how to describe their complex structures, up to deformation, using Legendrian knots. Second, we will focus on the study of these Legendrian knots and provide techniques to distinguish them or show they are isotopic. Then, we will apply them to obtain new results about complex affine manifolds. In particular, we will recover the mirror symmetry functor from the perspective of Legendrian knot theory.

Note for Attendees

Refreshments will be served before this colloquium in MATH 125, the Math Lounge area, at 3:45 p.m.
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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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Courant Institute, NYU
Fri 26 Jan 2018, 3:00pm
Department Colloquium
ESB 2012
Traveling Waves in Cell Populations
ESB 2012
Fri 26 Jan 2018, 3:00pm-4:00pm

Abstract

PDE models can be a powerful tool for understanding emerging structures and patterns, such as aggregates and traveling waves formed by large populations of cells. As a specific example, I will discuss myxobacteria, which, due to their co-operative nature, lie on the boundary between uni- and multicellular organisms. I will present a novel age-structured, continuous macroscopic model. The derivation is based on simple interaction rules and set within the SOH (Self-Organized Hydrodynamics) framework. The strength of this combined approach is that microscopic information can be incorporated into the particle model in a straight-forward manner, whilst the continuous model can be analyzed using mathematical tools, such as stability and asymptotic analysis.

It has been suggested that myxobacteria are not able to react to signals immediately after they have reversed their direction. Our analysis reveals that this insensitivity period is not necessary for wave formation, but is essential for wave synchronization. A more mathematical focus will be the existence and stability of such traveling waves moving in two opposing waves frames. Fascinatingly, while the wave profiles do not change, the wave composition does, and the fractions of reversible and non- reversible bacteria form waves traveling in the opposite direction. I will discuss the explicit construction of such waves and show simulation results.

This is joint work with Pierre Degond and Hui Yu.

Note for Attendees

Refreshments will be served in ESB 4133 (PIMS Lounge) at 2:45 p.m. before this colloquium.
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