Colloquia
Department of Mathematics and Statistics
Texas Tech University
The Hilbert matrix is an infinite matrix that is very simple to describe. Its action on the space of square summable sequences induces a bounded linear operator which is a most
typical example of a Hankel operator and whose norm can be determined by applying the well-known Hilbert inequality. It is possible to go well beyond the norm
computation: the spectrum of the Hilbert matrix on the space of square summable sequences is well understood. We will try to make the first part of this talk accessible
to a wider audience and cover briefly these early results (up to the late 1950s, approximately).
The Hilbert matrix also induces a bounded operator on various other sequence spaces and also on some spaces
of analytic functions (including certain Hardy and Bergman spaces), as was noticed from 2000 on. In the second
(and more specialized) part of the talk, we will review these more recent developments, including norm computations
on different spaces and some more recent results regarding the spectrum of the induced operator.
This Departmental Colloquium is held in conjunction with the Analysis seminar groug, and may be virtually attended via this Zoom link.
The first step in the development of a high-order accurate scheme for hyperbolic systems of conservation laws is the development of a robust first-order method supported by a rigorous mathematical basis. With that goal in mind, we develop a general framework of first-order fully-discrete numerical schemes that are guaranteed to preserve every convex invariant of the hyperbolic system and satisfy every entropy inequality.
We then proceed to present a new flux-limiting technique in order to recover second order (or higher) accuracy in space. This technique does not preserve or enforce pointwise bounds on conserved variables, but rather bounds on quasiconvex functionals of the conserved variables. This flux-limiting technique is suitable to preserve pointwise convex constraints of the numerical solution, such as: positivity of the internal energy and minimum principle of the specific entropy in the context of Euler’s equations. Catastrophic failure of the scheme is mathematically impossible. We have coined this technique “convex limiting’’.
Finally, we extend these developments to the case of compressible Navier-Stokes equations using operator-splitting in-time: hyperbolic terms are treated explicitly, parabolic terms are treated implicitly.
Operator-splitting is neither a new idea nor a widely adopted technique for compressible Navier-Stokes equation, most frequently received with skepticism. Contradicting current trends, we developed an operator-splitting scheme for which:
(i) Positivity of density and internal energy can be mathematically guaranteed.
(ii) Implicit stage uses primitive variables, but satisfies a total balance of mechanical energy. This is the key detail that is largely missing in most publications advocating the use of primitive variables and/or operator splitting techniques.
(iii) The scheme runs at the usual "hyperbolic CFL" dt <= O(h) dictated by Euler's subsystem, rather than the technically inapplicable "parabolic CFL" dt <= O(h^2).
The scheme is second-order accurate in space and time and exhibits remarkably robust behavior in the context of shock-viscous-layers interaction. We are not aware of any scheme in the market with comparable computational and mathematical credentials.
This Departmental Colloquium for invited speaker and job candidate Dr. Ignacio Tomás is held in conjunction with the Applied Math seminar group. Please virtually attend via this zoom link.
Variational methods have formed the foundation of classical mechanics for several hundred years. In this lecture, I will show how the applications of variational principles, coupled with some ideas from geometry, can solve a wide variety of seemingly disconnected problems using the same mathematical approach. After a general and gentle introduction, I will illustrate this method on the examples of modeling figure skating (a system with nonholonomic constraint), and fluid-structure interactions applied to porous media containing incompressible fluid (two media coupled through the incompressibility constraint). I will also discuss the limitations of these methods, i.e., what progress can be achieved by algorithmic thinking alone and at what point ingenuity and creativity must take over. Finally, I will also discuss the importance of teaching such universal, fundamental principles and methods for the successful employment of students and postdocs in the industry.
This is part of the W. Dayawansa Colloquium Series: Foundational and Applied Mathematics: Bridges to Industry, and is held in conjunction with the Applied Math seminar group. This event is hybrid, and available virtually via this Zoom link.
Higher Teichmuller theory studies representations of surface groups into Lie groups of higher rank, in contrast with the classical Teichmuller theory that concerns PSL(2,R). In this talk we will describe a scheme to find the analog of Thurston compactification for generalizations of Teichmuller space in the case where the Lie group is real split of rank 2 (SL(3,R), Sp(4,R), G2). For concreteness, we will mostly focus on SL(3,R), where the theory involves the study of convex real projective structures on surfaces, which in some sense extend the notion of hyperbolic structures.
This Job Colloquium is held in tandom with the PDGMP seminar group. Please watch online via this Zoom link.A "geometric Cauchy problem" means a geometric formulation of the Cauchy problem for some geometric PDE. For instance the Björling problem from the 19th Century is to construct the (unique) minimal surface that contains a given space curve with the surface normal prescribed along the curve. In that case the Schwarz formula gives the solution from the Cauchy data via analytically extension and taking the real part. This is based on the Weierstrass representation of minimal surfaces in terms of holomorphic data.
More generally, harmonic maps into symmetric spaces also have solution methods based on holomorphic maps into infinite dimensional spaces. It turns out that an infinite dimensional analogue of Schwarz's formula can be applied to solve Cauchy problems for surfaces with harmonic Gauss maps. The geometric Cauchy problem for Willmore surfaces can be stated in different ways. One way is to prescribe both the surface and its dual surface along a curve, as well as the conformal Gauss map. From this data, we obtain a unique solution. This can be used, for example, to classify equivariant Willmore sufaces.
Please virtually attend Dr. Brander's online Colloquium at this Zoom link. Passcode: Brander
Since the emergence of the Calculus of Variations, obtaining equilibria for energies whose Lagrangians depend on geometric invariants have played an important role in Differential Geometry.
In this talk we will examine the history of these pioneer variational problems, their development toward the present, and some of their multiple applications, paying special attention to the speaker's own results.
This Job Colloquium is held in tandom with the PDGMP seminar group. Please watch online via this Zoom link.The relation between Hurwitz polynomials and some sequences of orthogonal polynomials is well known in the literature. More precisely, the even and odd parts of any Hurwitz polynomial can be related to an orthogonal polynomial and the associated second kind polynomial, respectively. In this talk we present several recent results that allow us to construct, by using perturbed sequences of orthogonal polynomials, families of Hurwitz polynomials (with one or more uncertain parameters) that are robustly stable. Some applications and open problems will be discussed.
This Job Colloquium is sponsored by the Applied Math seminar group. Please virtually attend via this Zoom link.
The development of probability theory was motivated by moral considerations in gambling without attempting to predict the outcomes. Making predictions with probability always assumes switching temporal and ensemble perspectives under the ergodic hypothesis. “No probability without ergodicity”, Nassim Taleb said. Famously, ergodicity is assumed in equilibrium statistical mechanics, which successfully describes the thermodynamic behavior of gases. However, in a wider context, many observables, such as the growth of personal fortune, don’t satisfy the ergodic hypothesis that makes probabilistic predictions nonsensical. The failure of the expected wealth model to describe actual human behavior is known as the St. Petersburg paradox and is treated in many textbooks on economics and probability theory. Following the recent works of Ole Peters, we address the question of non-ergodicity in the lottery and relate its degree of non-ergodicity with the degree of predictability studied by us. Finally, we present the recent numerical results of Veniamin Smirnov on non-ergodicity and predictability of the logistic map during its transition to chaos obtained with the use of supercomputing resources provided by the Texas Tech University's High Performance Computing Center.
Please virtually attend this talk Thursday the 9th at 3:30PM CDT via this Zoom link.
Weyl law describes the asymptotic behavior of eigenvalues. I will introduce the eigenvalue problem of the Schrodinger operators with singular potentials on compact manifolds. In recent works with Xiaoqi Huang (U. of Maryland), we proved the pointwise Weyl laws for the Schrodinger operators with singular potentials, and showed that they are sharp by constructing examples. This work extends the 3d results of Frank-Sabin (2020) to any dimensions.
This Job Candidate Colloquium is held in conjunction with the Analysis seminar group. Please virtually attend Monday the 13th at 4 PM CDT via this Zoom link.
Time series often experience structural changes due to external events or internal systematic changes. Over the last two decades changepoint analysis has received considerable attention. Most changepoint methods assume IID model errors; however, time series data are typically autocorrelated. As a known difficult problem in changepoint analysis, the challenge comes from the fact that changepoints can mask as natural fluctuations in a serially dependent process and vice versa. My research aims to develop a gradient descent based PELT approach to detect the mean shifts in an AR(p) series. The proposed algorithm is supported by both proofs and simulations.
Dr. Shi's Job Colloquium talk is sponsored by the Statistics seminar group. Please virtually attend via this Zoom link. Passcode: b0UPyX
In this talk, we present two different approaches on overcoming the curse of dimensionality for the model reduction and the numerical schemes for high dimensional PDEs. For the first part, we introduce the framework on constructing structure-preserving machine learning (ML) moment closure models for kinetic. Most of the existing ML closure models are not able to guarantee the stability, which directly causes blow up in the long-time simulations. In our work, with carefully designed neural network architectures, the ML closure model can guarantee stability (or hyperbolicity). Moreover, other mathematical properties, such as entropy stability, Galilean invariance, and physical characteristic speeds, are also discussed. Extensive benchmark tests show the good accuracy, long-time stability, and good generalizability of our ML closure model.
For the second part, we introduce our work on adaptive sparse grid discontinuous Galerkin (DG) schemes for nonlinear PDEs in multidimensions. Due to the difficulty in efficiently projecting a nonlinear function onto sparse grid DG space, previous work mainly focuses on linear problems. To generalize the sparse grid DG schemes from linear problems to nonlinear cases, we first apply a class of interpolatory multiwavelets to treat nonlinear terms, then use fast wavelet transform to convert point values to the coefficients of the hierarchical wavelets, and lastly perform fast matrix-vector multiplication compute integrals over elements and edges in DG schemes. The resulting algorithm achieves similar computational complexity as linear equations and is successfully applied to several important classes of nonlinear PDEs.
This Job Colloquium is held in tandom with the Applied Math seminar group. Please virtually attend Wednesday the 15th at 4 PM CDT via this Zoom link.
Dynamical systems are present latently or explicitly in most phenomena around us, such as in physical systems, signal processing, control, or even in the mechanisms of numerical methods. Some classical questions in this field are forecasting, reconstruction, and the identification of various forms of coherence or patterns. However, there are no methods that execute these tasks equally effectively across all systems, due to the varied nature and complexity of the dynamics. In this talk, we take a closer look at these tasks and show how they are related to core concepts of the underlying dynamics, such as the spectral measure, geometry and fractal dimensions of its attractor, as well as global properties such as (nonuniform) hyperbolicity. Furthermore, several fundamental questions or tasks in the disparate fields of Harmonic Analysis, Statistics, and Geometry can be formulated in terms of a latent dynamical process. For example, the outcome of Fourier averaging or spectral tapering of signals is determined by the spectral measure; the limiting behavior of the graph Laplacian on data is determined by the geometry of the underlying fractal attractor; and the distribution of windowed statistics from data can be stated in terms of a skew-product dynamical system. These connections reveal a rich intersection of dynamics, signal processing and geometry. I explore various aspects of such connections, introduce some recent results in these directions, and present some intriguing open questions.
Dr. Das' Job Colloquium talk is sponsored by the PDGMP seminar group. Please virtually attend via this Zoom link. Passcode: Das
Abstract: We propose and implement tests for the existence of a common stochastic discount factor (SDF). Our tests are agnostic because they do not require macroeconomic data or preference assumptions; they depend only on observed asset returns. After examining test features and power with simulations, we apply the tests empirically to data on U.S. equities, bonds, currencies, commodities and real estate. The empirical evidence is consistent with a unique positive SDF that prices all assets and satisfies the Hansen/Jagannathan bounds.
Brief Bio: Prof. Roll is known in particular for his work on portfolio theory and asset pricing. He has extensive business experience, having been a consultant for a range of corporations, law firms, and government agencies in addition to serving on boards and founding several businesses himself. At Boeing the early 1960s, he worked on the Minuteman missile and the Saturn moon rocket. He was a vice president at Goldman, Sachs & Co. in the 1980s, where he founded and directed the mortgage securities group.
Previously, Richard has been on the faculty at UCLA, where he held the Joel Fried Chair in Applied Finance in the Anderson School of Management and is now a professor emeritus; at Carnegie Mellon University; the European Institute for Advanced Study of Management in Brussels; and the French business school Hautes Etudes Commerciales (HEC) near Paris. Roll holds a Ph.D. in economics, finance, and statistics from the University of Chicago, as well as an M.B.A. from the University of Washington and a B.A.E. from Auburn University. He has received honorary doctorates from universities in both France and Germany.
In addition to his published books and journal articles, he is or has been an associate editor of 11 different journals in finance and economics. Among his honors, he has won the Graham and Dodd Award for financial writing four times and the Leo Melamed Prize for financial research. In 2009, he was named Financial Engineer of the Year by the International Association of Financial Engineers and in 2015 was one of two recipients of the Onassis Prize in Finance. He is a past president of the American Finance Association and a fellow of the Econometric Society.The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. A simple weak Galerkin finite element method is introduced for second-order elliptic problems. First, we have proved that stabilizers are no longer needed for this WG element. Then we have proved the supercloseness of order two for the WG finite element solution. The numerical results confirm the theory.
This Job Colloquium is sponsored by the Applied Math seminar group. Please virtually attend Wednesday the 22nd at 4 PM CDT (UT-5) via this Zoom link.
This talk is based on a joint paper of Matan Mussel and Marshall Slemrod (Quarterly of Applied Mathematics, 2021) and provides two new applications of conservation laws in biology. The first is the application of the van der Waals fluid formalism for action potentials. The second is the application of the conservation laws of differential geometry (Gauss–Codazzi equations) to produce non-smooth surfaces representing Endoplasmic Reticulum.
Please virtually attend Dr. Slemrod's colloquium at noon (CDT, UT-5) on September 22nd via this Zoom link. Passcode: Slemrod
Capacity of sets and condensers in Euclidean spaces has been an object of study by researchers working in several areas of Analysis and Geometry due to its applications and the important role it plays in physics.
In the first part of this talk, we will present some basic facts about condenser capacities and their connection and applications to conformal mappings. We will also present our solution to the conjecture of J. Ferrand, G. Martin and M. Vuorinen from 1991.
In the second part we will introduce composition operators and we will examine their behavior in several Banach spaces of holomorphic functions in the unit disc. Our solution of a problem posed by J. Laitila in 2010 about isometric composition operators will be presented and we will describe the connection of the approximation numbers of an operator with condenser capacity.
Dr. Pouliasis' Third Year Review Colloquium may be virtually attended via this Zoom link. Passcode: PouliasisSeveral interesting PDEs from computer graphics and data science are discussed in the context of the speaker's own work. Topics to be surveyed include the simulation of p-Willmore flow, the computation of quasiconformal mappings between immersed surfaces, the uniqueness of planar mappings with prescribed curl and Jacobian determinant, and the approximation of unknown high-dimensional functions given limited data.
Dr. Gruber's Third Year Review talk may be virtually attended via this Zoom link. Passcode: Gruber
Abstract pdf
This Job Candidate Colloquium is held in conjunction with the Biomath seminar group, and may be virtually attended via this Zoom link.
Can mathematics understand diseases of the brain and the peripheral nervous system? The modern medical perspective on neurological diseases has evolved, slowly, since the 20th century but recent breakthroughs in medical imaging have quickly transformed medicine into a quantitative science. Today, mathematical modeling and scientific computing allow us to go farther than observation alone. With the help of numerical methods and high-performance computing, experimental and data-informed mathematical models are leading to new clinical insights into serious human pathologies, that affect the nervous system, such as oedema and Alzheimer's disease. In this talk, I will discuss my work in the construction, analysis and solution of data and clinically-driven mathematical models of pathologies affecting the human nervous system. Mathematical modeling and scientific computing are indeed indispensible for cultivating a data-informed understanding of the brain, serious human diseases and for developing effective treatment.
This Job Colloquium is sponsored by the Applied Math seminar group. Please virtually attend Wednesday the 6th at 4 PM CDT (UT-5) via this Zoom link.
In this talk, we’ll discuss two themes in geometry. The first is to study the connection between the geometry and the topology of a manifold. Here, the geometry is about the shape and the measurement while the topology is about properties that are preserved under continuous deformations. We’ll mention progress in the theory of Ricci flows and advancements towards a Hopf’s conjecture and other well-known problems. Particularly, a highlighted result is a differentiable sphere theorem that resolves a conjecture proposed by S. Nishikawa in 1986.
The second theme is the search for a hierarchy of stationary surfaces, which are critical points of a certain geometric functional. In Calculus, we use derivative tests to classify critical points and determine whether one is a minimum, a maximum, or a saddle. The abstract generalization is to study the index of an operator associated with the second variation. In this direction, we’ll develop a novel mechanism when dealing with surfaces with boundaries and introduce a surprise perspective when dealing with constraints. Specifically, there is a partial resolution of a conjecture proposed by R. Schoen and A. Fraser and closely related to the Willmore conjecture.
This Third Year Review Colloquium may be attended virtually via this zoom link.
Abstract pdf
This Job Colloquium is sponsored by the Applied Math seminar group. Please virtually attend Wednesday the 13th at 4 PM CDT (UT-5) via this Zoom link.
We extend the Markov Decision Process setup to the cases of MFG and MFC problems and we generalize the optimality Bellman equation for Q-learning. By introducing two learning rates, one for the Q-matrix and one for the population distribution, we are able to design a single algorithm which learns the optimal policies for the MFG or for the MFC depending on the ratio of these two rates. Applications to problems in finance are also discussed.
This is joint work with Andrea Angiuli and Mathieu Laurière. Please virtually attend at 3:30 PM CDT (UT-5) via this Zoom link.
Abstract pdf
This Job Colloquium is sponsored by the Applied Math seminar group. Please virtually attend Wednesday the 20th at 4 PM CDT (UT-5) via this Zoom link.
This is a general talk about recent breakthroughs in the fields of PDEs in Fluid and Stochastic PDEs in Mathematical Physics, representative of my research interests for the past several years. Concerning the first direction, whether a solution to the three-dimensional Navier-Stokes equations starting from a smooth initial data exists uniquely for all time with kinetic energy bounds remains an outstanding open problem in Analysis of PDEs. As a significant step forward, very recently, De Lellis, Szekelyhidi Jr., Buckmaster, Vicol et al. have obtained breakthroughs proving non-uniqueness if we start from non-smooth initial data. Concerning the second direction, while there exist many PDEs in physics literature forced by space-time white noise (e.g., Kardar-Parisi-Zhang equation, Phi4 model in quantum field theory, Yang-Mills), the singularity of such noise caused the non-linear terms therein to be ill-defined. Very recently, Hairer, Gubinelli, Imkeller, Perkowski et al. have established new uniform methods to attain solution theory for such singular PDEs. This talk is intended for a general audience to provide the ideas of these research directions.
This Third Year Review Colloquium is held in tandem with the PDGMP seminar group, and may be virtually attended on the 21st via this Zoom link.
Both climate and human systems are impacting our risk of mosquito-borne diseases. To be able to forecast risk in the short term and predict change in risk longer-term, we combine mathematical and statistical models with weather, satellite, demographic, and Google Health trends data. We focus on a few areas with high-fidelity data to show that accurate predictions are possible with the help of heterogeneous, time varying data. I will also address the challenges of transferability, including integration of earth systems, mosquito population, and disease models.
This Colloquium is sponsored by the TTU chapter of SIAM. Please virtually attend Friday the 22nd at 3:00PM CDT (UT-5) via this Zoom link.
Consider the motion of a Brownian particle in two or more dimensions, whose coordinate processes are standard Brownian motions with zero drift initially, and then at some random/unobservable time, one of the coordinate processes gets a non-zero drift governed by an independent finite-state Markov chain. Given that the position of the Brownian particle is being observed in real time, the problem is to detect the time at which a coordinate process gets the drift as accurately as possible. This is the so-called quickest real-time detection problem of a Markovian drift. The motivation for a Markovian drift stems from the consideration of a switching environment in practical problems in finance, engineering and other areas. To solve such problem, our main efforts are devoted to solving an equivalent optimal stopping problem with respect to a regime switching diffusion. Our result shows that the optimal stopping boundary can be represented as a unique solution to a non-linear integral equation in some admissible class. This is a joint work with Professor Philip Ernst at Rice University.
This Job Candidate Colloquium is sponsored by the Statistics seminar group, and you are invited to attend Monday the 25th at 4 PM CDT (UT-5) via this Zoom link.
To slow the spread of COVID-19, many countries initially implemented interventions against the disease, such as limiting contacts and mask-wearing. However, approximately 7% of the population have a disability requiring assistance from home care aides to assist with activities of daily living and thus contact-limiting applies differently to these groups. In this talk, we look into an agent-based network model of COVID-19 and the effects of various interventions upon the disabled and care aide communities. Our model accounts for multiple disease compartments, including allowing for asymptomatic transmission; different types of contacts and associated risks; and different contact distributions based on an individual’s occupation. Our work suggests that care aides and disabled people are strongly affected by global intervention strategies and that care aides may be one of the most influential groups in spreading the illness to disabled people and many members of the general population.
Dr. Lindstrom's Job Candidate Colloquium is sponsored by the Applied Math and BioMath seminar groups. Please virtually attend Wednesday the 27th at 3 PM CDT (UT-5) via this Zoom link.
We introduce the flexibility program originally proposed by A. Katok and discuss first results and open questions. Roughly speaking, the aim of this program is to study natural classes of smooth dynamical systems and to find "constructive tools" to freely manipulate dynamical and geometric data inside a fixed class. We will illuminate the connections between flexibility and rigidity phenomena.
Dr. Erchenko's Job Candidate Colloquium is sponsored by the PDGMP seminar group, and you are invited to attend Wednesday the 27th at 4:00 PM CDT (UT-5) via this Zoom link.
Abstract pdf
This Job Candidate Colloquium is sponsored by the Applied Math seminar group, and may be attended Thursday the 28th at 3:15 PM CDT (UT-5) via this Zoom link.
I present a survey of some of my recent works. They include (1) stochastic systems and applications to mathematical biology, (2) large deviations analysis and applications to mathematical physics, (3) stability of regime-switching diffusions, and (4) statistical and recursive estimation and applications to machine learning. I will present general pictures but leave out the technical details. The topics to be presented are connected through stochastic analysis; each of them has its own distinct features, however. The selected representative published papers can be found below.
[1] D. Nguyen, N. Nguyen, and G. Yin, Stochastic functional Kolmogorov equations I: Persistence, Stochastic Process. Appl., 142 (2021), 319--364.
[2] D. Nguyen, N. Nguyen, and G. Yin, Stochastic functional Kolmogorov equations II: Extinction, J. Differential Equations, 294 (2021), 1--39.
[3] N. Nguyen and G. Yin, Stochastic Lotka-Volterra competitive reaction-diffusion systems perturbed by space-time white noise: Modeling and analysis, J. Differential Equations, 282 (2021), 184--232.
[4] D. Nguyen, N. Nguyen, and G. Yin, General nonlinear stochastic systems motivated by chemostat models: Complete characterization of long-time behavior, optimal controls, and applications to wastewater treatment, Stochastic Process. Appl., 130 (2020) 4608--4642.
[5] N. Du, N. Nguyen, Permanence and Extinction for the Stochastic SIR Epidemic Model, Journal of Differential Equations, Vol. 269 (2020), 9619-9652.
[6] N. Nguyen and G. Yin, A class of Langevin equations with Markov switching involving strong damping and fast switching, J. Math. Phys., 61 (2020), no. 6, 063301, 18 pp.
[7] N. Nguyen and G. Yin, Large deviations principles for Langevin equations in random environment and applications, J. Math. Phys. 62 (2021), 083301, 26 pp.
[8] N.H. Du, D. Nguyen, N. Nguyen, and G. Yin, Stability of stochastic functional differential equations with random switching and applications, Automatica, 125 (2021) 109410, 6 pp.
[9] J. Kirkby, D. Nguyen, Duy Nguyen, and Nhu Nguyen, Inversion-free Subsampling Newton's Method For Large Sample Logistic Regression, Statistical Papers, to appear, 21 pp.
Mr. Nguyen's Job Colloquium talk is sponsored by the Statistics seminar group. Please virtually attend via this Zoom link.
This week's Job Colloquium is sponsored by the Applied Math seminar group, details available at this pdf. Please virtually attend via this Zoom link.
In this joint work with Rodrigo Treviño we consider the Lorentz gas model of category A (that is, with no corners and of finite horizon) on aperiodic repetitive tilings of $\mathbb{R}^2$ of finite local complexity. We show that the compact factor of the collision map has the K property, from which we derive mixing for pattern-equivariant functions as well as the planar ergodicity of the Lorentz gas flow.
Dr. Zelerowicz' Job Candidate Colloquium is sponsored by the PDGMP seminar group, and you are invited to attend Wednesday the 10th at 1:30 PM CST (UT-6) via this Zoom link.
Abstract details available at this pdf
This Job Colloquium is sponsored by the Applied Math seminar group. Please virtually attend Wednesday the 10th at 4 PM CST (UT-6) via this Zoom link.
The Navier-Stokes and Euler equations are the fundamental models for describing viscous and inviscid fluids, respectively. Based on ideas which date back to Kolmogorov and Onsager, solutions to these equations are expected to dissipate energy, which in turn suggests that such solutions are somewhat rough and thus only weak solutions. At these low regularity levels, however, one may construct wild weak solutions using convex integration methods. In this talk, I will discuss the motivation and methodology behind joint work with Tristan Buckmaster, Nader Masmoudi, and Vlad Vicol in which we construct wild solutions to the Euler equations which deviate from the predictions of Kolmogorov's classical K41 phenomenological theory.
Dr. Novack's Job Colloquium is sponsored by the Applied Math seminar group. Please virtually attend via this Zoom link.
Although basketball is a dynamic process sport, with 5 plus 5 players competing on both offense and defense simultaneously, learning some static information is predominant for professional players, coaches and team mangers. In order to have a deep understanding of field goal attempts among different players, we proposed two different approaches to learn the shooting habits of different players over the court and the heterogeneity among them. First approach is a mixture of finite mixtures (MFM) model to capture the heterogeneity of shot selection among different players based on Log Gaussian Cox process (LGCP). Second approach is a zero inflated Poisson model with clustered regression coefficients. Our proposed method can simultaneously estimate the number of groups and group configurations. We apply both proposed model to the National Basketball Association (NBA), for learning players’ shooting habits and heterogeneity among different players over the 2017–2018 regular season.
Dr. Yang's Job Colloquium talk is sponsored by the Statistics seminar group. Please virtually attend via this Zoom link.
We study the biharmonic equation with Navier boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the 4th-order problem as a system of Poisson equations. Our method differs from the usual mixed method that leads to two Poisson problems but only applies to convex domains; our decomposition involves a third Poisson equation to confine the solution in the correct function space, and therefore can be used in both convex and non-convex domains. A $C^0$ finite element algorithm is in turn proposed to solve the resulting system. In addition, we derive optimal error estimates for the numerical solution on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings. This is joint work with Hengguang Li and Zhimin Zhang.
This Job Candidate Colloquium is sponsored by the Applied Math seminar group, and you are invited to attend Wednesday the 17th at 4 PM CST (UT-6) via this Zoom link.
We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane. This is joint work with P. Kokocki and T. Cieślak.
Dr. Ożański's Job Colloquium talk is sponsored by the PDGMP seminar group. Please virtually attend via this Zoom link.
Tensors are just multi-dimensional arrays. Notions of ranks and border rank abound in the literature. Tensor decompositions also have a lot of application in data analysis, physics, and other areas of science. I will give a colloquium-style talk surveying my recent two results about tensor ranks and their application to matrix multiplication complexity. The first result relates different notion of tensor ranks to polynomials of vanishing Hessian. The second one computes the border rank of 3 X 3 permanent. I will also briefly discuss the newest technique we used to achieve our results: border apolarity. This talk assumes little background in geometry or algebra.
This Colloquium is sponsored by the Algebra and Number Theory seminar group. Please virtually attend Wednesday the 1st at 4 PM CST (UT-6) via this Zoom link. Meeting ID 959 8597 7889 Passcode Huang