Events
Department of Mathematics and Statistics
Texas Tech University
We will introduce the theory of circle packings on Riemann surfaces. This talk is designed to be accessible to graduate students.
Mathematical models of cell migration in the context of wound healing, embryonic development, and cancer growth have been developed using a wide variety of frameworks, including reaction-diffusion equations, continuum mechanics, and agent-based models. However, studying model uncertainty or model selection in these settings is less common. We develop a method for studying the appropriateness of model equation components that combines approximate Bayesian computation (ABC) and sensitivity analysis (SA). We provide two case studies in cell migration where we apply this method to sparse experimental data sets of retina development in the eye and tumor-immune dynamics in the brain. We identify model components that can be removed via model reduction using ABC+SA and potential cancer treatment pathways.
Zoom link:
https://texastech.zoom.us/j/94471029838?pwd=ZlJXR2JhU0ZjUHhYOUlmVGN3VFFJUT09
We will discuss stochastic quantization of the Yang-Mills model on two and three dimensional torus. In stochastic quantization we consider the Langevin dynamic for the Yang-Mills model which is described by a stochastic PDE. We construct local solution to this SPDE and prove that the solution has a gauge invariant property in law, which then defines a Markov process on the space of gauge orbits. We will also describe the construction of this orbit space, on which we have well-defined holonomies and Wilson loop observables. Based on joint work with Ajay Chandra, Ilya Chevyrev, and Martin Hairer.
Watch online via this Zoom link.
Motivated by an open question of Dyson and Serre we define Jacobi
forms with CM analogously to modular forms with CM. We will mainly
focus on the history of this problem and some beautiful sum-to-product
identities that result from this new construction. As applications we
will discuss Jacobi forms associated to elliptic curves with CM and a
process to construct partition statistics which explain Ramanujan-type
congruences for integer partitions.
Follow the talk via this Zoom link
Applications of atmospheric re-entry and geophysical flows are characterized by a large variety of separate models for specific tasks. This model variety poses significant difficulties both for the analysis and for the numerical solution. We thus need to rethink mathematical modelling and model order reduction for future numerical simulations.
In this talk, I will introduce hierarchical moment models as a flexible way to derive hierarchies of models in fluid dynamics and other applications. The general derivation procedure of the reduced models results in structural similarities of the models, which facilitate physical insight, model adaptivity, and the development of suitable numerical methods. Based on kinetic equations and shallow flows, I will exemplify the hierarchical moment approach and highlight runtime and accuracy improvements.
Please attend this week's Applied Math seminar at 4 PM Wednesday via this Zoom link.
Meeting ID: 976 3095 1027
Passcode: applied
We develop a novel approach for the construction of quantile processes governing the stochastic
dynamics of quantiles in continuous time.
Two classes of quantile diffusions are identified: the first, which we largely focus on,
features a dynamic random quantile level and allows for direct interpretation of the resulting
quantile process characteristics such as location, scale, skewness and kurtosis, in terms of the
model parameters.
The second type are function-valued quantile diffusions and are driven by stochastic parameter processes,
which determine the entire quantile function at each point in time.
By the proposed innovative and simple -- yet powerful -- construction method, quantile processes
are obtained by transforming the marginals of a diffusion process under a composite map consisting
of a distribution and a quantile function.
Such maps, analogous to rank transmutation maps, produce the marginals of the resulting quantile process.
We discuss the relationship and differences between our approach and existing methods and
characterisations of quantile processes in discrete and continuous time.
As an example of an application of quantile diffusions, we show how probability measure distortions,
a form of dynamic tilting, can be induced.
Though particularly useful in financial mathematics and actuarial science, examples of which are
given in this work, measure distortions feature prominently across multiple research areas.
For instance, dynamic distributional approximations (statistics), non-parametric and asymptotic
analysis (mathematical statistics), dynamic risk measures (econometrics), behavioural economics,
decision making (operations research), signal processing (information theory), and not least in
general risk theory including applications thereof, for example in the context of climate change.