## Oliver Tough : The Fleming-Viot Particle System with McKean-Vlasov dynamics

- Probability,Uploaded Videos ( 1252 Views )Quasi-Stationary Distributions (QSDs) describe the long-time behaviour of killed Markov processes. The Fleming-Viot particle system provides a particle representation for the QSD of a Markov process killed upon contact with the boundary of its domain. Whereas previous work has dealt with killed Markov processes, we consider killed McKean-Vlasov processes. We show that the Fleming-Viot particle system with McKean-Vlasov dynamics provides a particle representation for the corresponding QSDs. Joint work with James Nolen.

## David Herzog : Supports of Degenerate Diffusion Processes: The Case of Polynomial Drift and Additive Noise

- Probability ( 212 Views )We discuss methods for computing supports of degenerate diffusion processes. We assume throughout that the diffusion satisfies a stochastic differential equation on R^{d} whose drift vector field X_{0} is ``polynomial'' and whose noise coefficients are constant. The case when each component of X_{0} is of odd degree is well understood. Hence we focus our efforts on X_{0} having at least one or more components of even degree. After developing methods to handle such cases, we shall apply them to specific examples, e.g. the Galerkin truncations of the Stochastic Navier-Stokes equation, to help establish ergodic properties of the resulting diffusion. One benefit to our approach is that, to prove such consequences, all we must do is compute certain
Lie brackets.

## Johan Brauer : The Stabilisation of Equilibria in Evolutionary Game Dynamics through Mutation

- Probability ( 194 Views )The multi-population replicator dynamics (RD) can be considered a dynamic approach to the study of multi-player games, where it was shown to be related to Cross-learning, as well as of systems of co-evolving populations. However, not all of its equilibria are Nash equilibria (NE) of the underlying game, and neither convergence to an NE nor convergence in general are guaranteed. Although interior equilibria are guaranteed to be NE, no interior equilibrium can be asymptotically stable in the multi-population RD, resulting, e.g., in cyclic orbits around a single interior NE. We report on our investigation of a new notion of equilibria of RD, called mutation limits, which is based on the inclusion of a naturally arising, simple form of mutation, but is invariant under the specific choice of mutation parameters. We prove the existence of such mutation limits for a large range of games, and consider an interesting subclass, that of attracting mutation limits. Attracting mutation limits are approximated by asymptotically stable equilibria of the (mutation-)perturbed RD, and hence, offer an approximate dynamic solution of the underlying game, especially if the original dynamic has no asymptotically stable equilibria. Therefore, the presence of mutation will indeed stabilise the system in certain cases and make attracting mutation limits near-attainable. Furthermore, the relevance of attracting mutation limits as a game theoretic equilibrium concept is emphasised by the relation of (mutation-)perturbed RD to the Q-learning algorithm in the context of multi-agent reinforcement learning. However, in contrast to the guaranteed existence of mutation limits, attracting mutation limits do not exist in all games, raising the question of their characterization.

## Alex Blumenthal : Chaotic regimes for random dynamical systems

- Probability ( 154 Views )It is anticipated that chaotic regimes (e.g., strange attractors) arise in a wide variety of dynamical systems, including those arising from the study of ensembles of gas particles and fluid mechanics. However, in most cases the problem of rigorously verifying asymptotic chaotic regimes is notoriously difficult. For volume-preserving systems (e.g., incompressible fluid flow or Hamiltonian systems), these issues are exemplified by coexistence phenomena: even in quite simple models which should be chaotic, e.g. the Chirikov standard map, completely opposite dynamical regimes (elliptic islands vs. hyperbolic sets) can be tangled together in phase space in a convoluted way. Recent developments have indicated, however, that verifying chaos is tractable for systems subjected to a small amount of noise— from the perspective of modeling, this is not so unnatural, as the real world is inherently noisy. In this talk, I will discuss two recent results: (1) a large positive Lyapunov exponent for (extremely small) random perturbations of the Chirikov standard map, and (2) a positive Lyapunov exponent for the Lagrangian flow corresponding to various incompressible stochastic fluids models, including stochastic 2D Navier-Stokes and 3D hyperviscous Navier-Stokes on the periodic box. The work in this talk is joint with Jacob Bedrossian, Samuel Punshon-Smith, Jinxin Xue and Lai-Sang Young.

## Dave Rose : The EilenbergMazur swindle

- Graduate/Faculty Seminar ( 134 Views )At some point in every mathematician's life they have seen the paradoxical 'proof' that 1=0 obtained by different groupings of the infinite sum 1-1+1-1+... As we learn, the issue is that this series does not converge. The Eilenberg-Mazur swindle is a twist on this argument which shows that A+B+A+B+... = 0 implies that A=0=B in certain situations where we can make sense of the infinite sum. In this talk, we will explore these swindles, touching on many interesting areas of mathematics along the way.

## Jason Mireles-James : Adaptive Set-Oriented Algorithms for Conservative Systems

- Presentations ( 130 Views )We describe an automatic chaos verification scheme based on set oriented numerical methods, which is especially well suited to the study of area and volume preserving diffeomorphisms. The novel feature of the scheme is an iterative algorithm for approximating connecting orbits between collections of hyperbolic fixed and periodic points with greater and greater accuracy. The algorithm is geometric rather than graph theoretic in nature and, unlike existing methods, does not require the computation of chain recurrent sets. We give several example computations in dimension two and three.

## Lillian Pierce : Class numbers of quadratic number fields: a few highlights on the timeline from Gauss to today

- Graduate/Faculty Seminar ( 129 Views )Each number field (finite extension of the rational numbers) has an invariant associated to it called the class number (the cardinality of the class group of the field). Class numbers pop up throughout number theory, and over the last two hundred years people have been considering questions about the growth and divisibility properties of class numbers. Well focus on class numbers of quadratic extensions of the rationals, surveying some key results in the two centuries since the pioneering work of Gauss, and then turning to very recent joint work of the speaker with Roger Heath-Brown on averages and moments associated to class numbers of imaginary quadratic fields.

## Steven Baer : Multiscale Modeling of Neural Subcircuits and Feedback Mechanisms in the Outer Plexiform Layer of the Retina

- Mathematical Biology ( 127 Views )Visual processing begins in the outer plexiform layer of the retina, where

bipolar, horizontal, and photoreceptor cells interact. In vertebrates, the

onset of dim backgrounds can enhance small spot flicker responses of

retinal horizontal cells. This flicker response is called background-

induced flicker enhancement. The underlying mechanism for the feedback

is unclear but competing hypotheses have been proposed. One is the GABA

hypothesis, which states that the inhibitory neurotransmitter GABA,

released from horizontal cells, mediates the feedback by blocking calcium

channels. Another is the ephaptic hypothesis, which contends that calcium

entry is regulated by changes in the electrical potential within the

intersynaptic space between cones and horizontal cells. In this study, a

continuum spine model of cone-horizontal cell synaptic circuitry is

formulated. The model captures two spatial scales - the scale of an

individual synapse and the scale of the receptive field involving hundreds

to thousands of synapses. We show that the ephaptic mechanism produces

reasonable qualitative agreement with the temporal dynamics exhibited by

flicker enhancement experiments. We find that although GABA produces

enhancement, this mechanism alone is insufficient to reproduce the

experimental results. We view this multiscale continuum approach as a

first step in formulating a multi-layer mathematical model of retinal

circuitry, which would include the other brain nuclei within the retina:

the inner plexiform layer where bipolar, amacrine, interplexiform, and

ganglion cells interact.

## Freydoon Shahidi : Local Langlands correspondence and the exterior and symmetric square root numbers for GL(n)

- Number Theory ( 124 Views )We will discuss the notion of Artin root numbers attached to an n-dimensional continuous Frobenius-semisimple complex representation of the Weil-Deligne group and show their equalities with those defined by Langlands-Shahidi method through local Langlands correspondence for GL(n) and the exterior and symmetric square representation of the L-group GL(n,C) of GL(n). The proof is a robust deformation argument using local-global techniques, complemented with suitable asymptotic expansions for partial Bessel functions inspired by certain generalized Shalika germ expansions of Jacquet and Ye. This is a joint work with J. Cogdell and T.-L. Tsai.

## Gregory Herschlag : Optimal reservoir conditions for material extraction across pumping and porous channels

- Mathematical Biology ( 113 Views )In this talk, I will discuss a new result in fluid flows through channels with permeable membranes with simple pumping dynamics. Fluid will be exchanged and metabolized in a simple reservoir and I will demonstrate the existence of optimal reservoir properties that may either maximize or minimized the amount of fluid being extracted across the channel walls. The biological relevance of this work may be seen by noting that all living organisms of a sufficient size rely on complex systems of tubular networks to efficiently collect, transport and distribute nutrients or waste. These networks exchange material with the interstitium via embedded channels leading to effective permeabilities across the wall separating the channel interior from the interstitium. In many invertebrates, for example, respiratory systems are made of complex tracheal systems that branch out through the entire body allowing for passive exchange of oxygen and carbon dioxide. In many of these systems, certain animals utilize various pumping mechanisms that alter the flow of the air or fluid being transported. Although the net effect of pumping of the averaged rates of fluid flow through the channel is typically well understood, it is still a largely open problem to understand how, and in what circumstances, pumping enables and enhances the exchange of material across channel walls. It has been demonstrated experimentally, for example, that when certain insects flap their wings, compression of the trachea allow for more efficient oxygen extraction, however it is unclear if this pumping is optimized for flight, oxygen uptake or neither, and understanding this problem quantitatively will shed insight on this biological process. Many of these interesting scenarios occur at low Reynolds number and this regime will be the focus of the presentation.

## Hubert Bray : Voting Rules for Democracy without Institutionalized Parties

- Graduate/Faculty Seminar ( 108 Views )This talk will be a fun discussion of the mathematical aspects of preferential ballot elections (in which voters are allowed to express their rankings of all of the candidates). After describing how single vote ballots can lead to an institutionalized two party system by discouraging third party candidates, we will then discuss the various vote counting methods for preferential ballot elections and the characteristics, both good and bad, that these various methods have. We will also touch on Arrow's Paradox, one of the most over-rated "paradoxes" in mathematics, and explain how it is much less relevant to discussions of vote counting methods than is sometimes believed.

## Richard Bertram : GPUfit: A Tool for Real-Time Model Calibration and Prediction Testing

- Mathematical Biology ( 100 Views )

Mathematical modeling has become a widely-used tool for integrating

biological data, designing experiments, and ultimately understanding

biological systems. In recent years two important challenges for the

successful use of mathematical models have become apparent. One is that

models contain parameters that determine the behavior of the model, and

the values of these parameters are often hard to determine from the

available biological data. The other challenge is that many biological

systems exhibit a great deal of heterogeneity in behavior, so even if the

model parameters could be perfectly calibrated by pooling cell behaviors

to produce an average cell model, this model may not provide a good

description of any single cell in the population. In this seminar I will

describe a technique that we are using to integrate mathematical modeling

into experimental studies in a way that addresses both of these challenges.

We study endocrine pituitary cells that release a variety of hormones into

the blood, and our aim is to develop an approach for modeling the

behaviors of these cells with enough accuracy so that we can use the

models to make and test predictions in real time.

## Tobias Ekholm : Wrapped Floer cohomology and Legendrian surgery

- Geometry and Topology ( 94 Views )We first review the relation between wrapped Floer cohomology of co-core disks after Lagrangian handle attachment and the Legendrian DGA of the corresponding attaching spheres. Then we discuss a generalization of this result to the partially wrapped setting where the Legendrian dga should be enriched with loop space coefficients, and describe several cases when explicit calculations are possible via parallel copies or local coefficient systems. We also discuss applications of these ideas to the topology of Lagrangian fillings of Legendrian submanifolds. The talk reports on joint work with Y. Lekili.

## Kirill B. Skouibine : The Role of Cardiac Tissue Structure in Reentry Induction: A Modeling Study

- Applied Math and Analysis ( 14 Views )Most dangerous cardiac arrhythmia, ventricular fibrillation (VF), is characterized by chaotic electrical behavior of the tissue. At the onset of the first, more organized stage of VF waves of electric activity in the heart become reentrant leading to fast irregular contraction. Better understanding of the mechanisms underlying early VF events will lead to more efficient treatment. Reentry induction has been performed in several experiments. We devise a model of cardiac tissue and use it to obtain a close match to the experimental results. The model combines macroscopic and microscopic properties of cardiac tissue.

## B. Scott Gaudi : Microlensing and the Search for Extrasolar Planets

- Applied Math and Analysis ( 14 Views )The PLANET collaboration has monitored nearly 100 microlensing events of which more than 20 have the sensitivity required to detect perturbations due to a planetary companion to the primary lens. No planets have been detected. These null results indicate that Jupiter mass planets with separations from 1-5 AU are not common -- the first such limits for extrasolar planets at these separations by any technique. While interpretation of null results is not trivial, interpretation of future detections will be substantially more difficult, due to degeneracies among the planetary fit parameters and degeneracies with perturbations due to other, non-planetary phenomena. The analysis is further complicated by the unusual situation that observational strategies are altered real-time when perturbations are detected. I discuss these difficulties and present methods to cope with them. Finally, I discuss future prospects for microlensing planet searches.