Date: Wednesday, January 10th, 2024

Registration link:

Host: Professional Master鈥檚 Program Director Brian Zaharatos and Associate Director Judith Law. 

Description of the event:

CU 麻豆影院鈥檚 graduate program in Applied Mathematics is consistently ranked in the top 15 programs in the nation by U.S. News & World Report. Our Professional Master鈥檚 degree joins real world collaboration with rigorous coursework administered by Applied Mathematics faculty renowned in their specializations. The program is designed for those with a background in mathematics who would like to take their skill to the next level.

Join us to learn more about pursuing a Master鈥檚 Degree in Applied Mathematics at CU 麻豆影院! Each information session will feature a 30 minute presentation by the Directors of the program followed by Q&A.

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Date: Friday, March 24th, 2023; 9:00am - 10:00am.

Registration link: 

Host: Professional Master鈥檚 Program Director Brian Zaharatos and Associate Director Judith Law. 

Description of the event:

Our Professional Master鈥檚 degree joins real world collaboration with rigorous coursework administered by Applied Mathematics faculty renowned in their specializations. The program is designed for those with a background in mathematics who would like to take their skill to the next level.

Join us to learn more about pursuing a master鈥檚 degree in Applied Mathematics at CU 麻豆影院! Each information session will feature a 30-minute presentation by the Directors of the program followed by Q&A. 

Consideration for fall admissions:

• Early Decision Application Deadline:  Feb. 1 for domestic applicants
• Early Decision Application Deadline:  Jan. 15 for international applicants
• Official Fall Application Deadline:  April 15

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Title: Masters Information Session

Date/time: Wednesday, November 3, 2021 at 11:00 AM until 12:30 PM

The drop-down details 鈥淛oin CU鈥檚 Office of Admissions on November 3rd for virtual 鈥淭ed-Talk鈥� style presentations from several departments on campus including APPM! Listen in as Associate Director, Dr. Judith Law will present our Master鈥檚 program offering specialization in Statistics and Data Science. Sign up .

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Jeremy Hoskins, Department of Mathematics, Yale University

Elliptic PDEs on regions with corners

Many of the boundary value problems frequently encountered in the simulation of physical problems (electrostatics, wave propagation, fluid dynamics in small devices, etc.) can be solved by reformulating them as boundary integral equations. This approach reduces the dimensionality of the problem and enables high-order accuracy in complicated geometries. Unfortunately, in domains with sharp corners, the solution to both the original governing equations as well as the corresponding boundary integral equations develop singularities at the corners. This poses significant challenges to many existing integral equation methods, typically requiring the introduction of many additional degrees of freedom. In this talk, I show that the solutions to the Laplace, Helmholtz, and biharmonic equations in the vicinity of corners can be represented by a series of elementary functions. Knowledge of these representations can be leveraged to construct accurate and efficient Nystr枚m discretizations for solving the resulting integral equations. I illustrate the performance of this method with several numerical examples.

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Dan Larremore, Department of Computer Science, 麻豆影院

Complex Networks, Math, and Malaria: From Evolution to Epidemiology

Progress in the global battle for malaria elimination has flatlined since 2015, with the single-cell P. falciparum parasite killing one child for every minute of the year. Some of this plateau can be attributed to the usual suspects鈥攄rug-resistant parasites, insecticide-resilient mosquitos, and counterfeit pharmaceuticals. But fundamentally, malaria persists because parasites prolong infections, evade immune systems, and adapt to individuals and populations. In this talk, I will describe ongoing work to analyze malaria parasites' adaptation strategies. In particular, we will focus on analyzing the parasites' var genes, whose expression allows parasites to evade human immune responses indefinitely. Importantly, due to their complex structure, var genes defy traditional phylogenetic analysis methods and have therefore inspired the development of new and more sophisticated mathematical tools. I will introduce two of those tools, focusing on recent advances in (1) methods for community detection in bipartite networks and (2) Bayes-optimal inference of parasite relatedness. We'll then explore how they can be used to answer open questions in malaria's evolution and genetic epidemiology.

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Justin Cole, Department of Applied Mathematics, 麻豆影院

Soliton Dynamics in the Korteweg-de Vries Equation with Nonzero Boundary Conditions

Inspired by recent experiments, the Korteweg-de Vries equation with nonzero Dirichlet boundary conditions is considered. Two types of boundary data are examined: step up (which generates a rarefaction wave) and step down (which creates a dispersive shock wave). Soliton dynamics are analytically studied via inverse scattering transform and a small dispersion asymptotic approximation. Depending on it's initial position and amplitude, an incident soliton will either transmit through or become trapped inside a step-induced wave. A formula for the transmitted soliton and it's phase shift is derived. The asymptotic approximation provides a description of the trapped soliton dynamics. Finally, direct numerics are shown to agree well with the analytical results.

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Amir Ajalloeian; Department of Electrical, Computer, and Energy Engineering; 麻豆影院

Inexact Online Proximal-gradient Method for Time-varying Convex Optimization

This paper considers an online proximal-gradient method to track the minimizers of a composite convex function that may continuously evolve over time. The online proximal-gradient method  is "inexact,'' in the sense that: (i) it relies on an approximate first-order information of the smooth component of the cost; and, (ii)~the proximal operator (with respect to the non-smooth term) may be computed only up to a certain precision. Under suitable  assumptions, convergence of the error iterates is established for strongly convex cost functions. On the other hand, the dynamic regret is investigated when the cost is not strongly convex, under the additional assumption that the problem includes feasibility sets that are compact. Bounds are expressed in terms of the cumulative error and the path length of the optimal solutions. This suggests how to allocate resources to strike a balance between performance and  precision  in the gradient computation and in the proximal operator. 

Maddela Avinash; Department of Electrical, Computer, and Energy Engineering; 麻豆影院

Semidefinite Relaxation technique to solve Optimal power flow Problem

I would like to discuss about using the convex relaxation technique to find the  optimal solution for  cost function of a power distribution system. Conventional optimal power flow problem is a nonconvex problem. Traditional Newton-Rpahson method has a convergence issue when the system reaches its limit. Semi definite programming makes an approximation  to the power flow constraints  by increasing the boundaries of the feasible set to make it a convex problem.  This convex problem can therefore be solved to minimize the total cost of generation, transmission and distribution of Electric Power. 

Ayoub Ghriss, Department of Computer Science, 麻豆影院

Hierarchical Deep Reinforcement Learning through Mutual Information Maximization

As it鈥檚 the case of the human learning, biological organisms can master tasks from extremely small samples. This suggests that acquiring new skills is done in a hierarchical fashion starting with simpler tasks that allow the abstraction of newly seen samples. While reinforcement learning is rooted in Neuroscience and Psychology, Hierarchical Reinforcement Learning (HRL) was developed in the machine learning field by adding the abstraction of either the states or the actions. Temporally abstract actions, our main focus, consists of top-level/manager policy and a set of temporally extended policies (low-level/workers). At each step, a policy from this set is picked by the manager and continues to run until a set of specified termination states is reached. The decision making in this hierarchy of policies starts by top-level policy that assigns low-level policies to different domains of the state space. These low-level policies operate as any other monolithic policy on the assigned domain. In this talk, we introduce HRL and present our method to learn the hierarchy: we use Information Maximization to learn the top-level policies with on-policy method (Trust Region Policy Optimization) to learn the low-level policies.

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Dan Messenger, Department of Applied Mathematics, 麻豆影院

Aggregation-Diffusion Phenomena in Domains with Boundaries

This talk relates to a class of mathematical models for the collective behaviour of autonomous agents, or particles, in general spatial domains, where particles exhibit pair-wise interactions and may be subject to environmental forces. Such models have been shown to exhibit non-trivial behaviour due to interactions with the boundary of the domain. More specifically, when there is a boundary, it has been observed that the swarm of particles readily evolves into unstable states. Given this behaviour, we investigate the regularizing effect of adding noise to the system in the form of Brownian motion at the particle level, which produces linear diffusion in the continuum limit. To investigate the effect of linear diffusion and interactions with the boundary on swarm equilibria, we analyze critical points of the associated energy functional, establishing conditions under which global minimizers exist. Through this process we uncover a new metastability phenomenon which necessitates the use of external forces to confine the swarm. We then introduce numerical methods for computing critical points of the energy, along with examples to motivate further research. Finally, we consider the short-time dynamics of the stochastic particle system as the noise strength approaches zero.

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Leonid Berlyand, Department of Mathematics, Penn State University

PDE models of Active Matter

In this talk we attempt to demonstrate how mathematical analysis could be helpful in the study of active matter, with the focus on active gels and cell motility.     

We first discuss mathematical challenges and developments of novel mathematical tools due to out-of-equilibrium state of active matter (e.g., active cytoskeleton gels,  bacterial suspensions, etc.). 

Next we present three minimal PDE models of active gels: (i) phase-filed model (ii) mean curvature type free boundary model and (iii) Hele-Shaw type free boundary model.  These models are designed to capture key biophysical phenomena in cell motility such as persistent & turning  motion, symmetry breaking, and viscous fingering while having  minimal set of  parameters and variables. 

Our goal is to provide theoretical understanding of cell polarity phenomenon via mathematical analysis of stability/instability  and bifurcation from steady states to traveling waves.  This is done by identification of key mathematical structures behind the models such as gradient coupling in Phase-Field model, Liouville type equation, Keller-Segel cross-diffusion, and nonlinearity due to the  free boundary. We employ mathematical techniques of (i)  sharp interface limit via asymptotic analysis,  (ii) construction of steady states and traveling waves via Crandall-Rabinowitz bifurcation theory and  (iii) topological methods such as Lerey-Schauder degree theory. 

These are joint works with V. Rybalko (ILTPE, Kharkiv, Ukraine), J. Fuhrman (PSU & Mainz, Germany),  M. Potomkin (PSU, USA).

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Thomas Hagstrom, Department of Mathematics, Southern Methodist University Title and Abstract Pending https://calendar.colorado.edu/event/applied_math_colloquium_-_thomas_hagstrom

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