| Date | SPEAKER | Host |
|---|---|---|
| January 23 | TITLE: ABSTRACT: |
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| January 30 | Ivan Kuznetsov, Novosibirsk State University (Virtual)
TITLE : Impulsive Kelvin-Voigt equations for active nematic liquid crystals ABSTRACT : The present report is devoted to impulsive Kelvin-Voigt equations for incompressible homogeneous active nematics On the right hand side there is an approximation of the Dirac delta-function $\delta_{(t=\tau)}$. This singular source term is linked with spontaneous reorientation $d$ and change of velocity $v$ in active nematic liquid crystals. We put $\tau=0$. The main challenge is to pass to the limit. On the infinitesimal initial layer, initial data $(v_0,d_0)$ is transformed into a new one. |
Torres |
| February 6 | Nathan Glatt-Holtz, Indiana University
TITLE : Statistical inference for high dimensional parameters from PDE constrained data: theoretical and computational developments ABSTRACT : The Bayesian approach to inverse problems provides a principled and flexible methodology for the estimation of high dimensional unknown parameters appearing in partial differential equations. This methodology therefore represents an important frontier for statistical inference from sparse and noise corrupted data arising in physics informed settings. This talk will overview this emerging field and survey some of our recent and ongoing work in this domain. Specifically we will (i) describe some new model PDE inference problems related to the measurement of fluid flow. (ii) Overview developments in Markov Chain Monte Carlo (MCMC) sampling methods which partially beat the curse of dimensionality and which are indispensable for resolving a wide variety of problems including the models in (i). (iii) Describe some results concerning consistency in the large data limit for certain `infinite dimensional' PDE-informed problems. This is joint work with Jeff Borggaard (Virginia Tech), Christian Frederiksen (Tulane), Andrew Holbrook (UCLA), Justin Krometis (Virginia Tech), and Cecilia Mondaini (Drexel). |
Novack |
| February 13 | Alex Misiats, VCU
TITLE: Linear Elastic Models of Shape Memory Alloys: Variational Perspective ABSTRACT: In this talk I will describe a class of minimization problems arising in modelling shape memory alloys. I will start with a shape memory material illustration, followed by a simple one dimensional model of it. Its extensions in 2D and 3D will help us understand the energetic mechanism behind the formation of twin patterns in physical experiments. By means of sharp upper and lower bounds, we show that certain experimentally observed structures provide optimal energy scaling law, and - in certain cases - are true energy minimizers. Despite the fact that the problem is highly nonconvex due to the presence of phase constraints and the singular perturbation, in my talk, I will describe a relaxation method which allows the use of the convex duality technique for the purpose of obtaining a sharp lower bound. I will also discuss the extension of the linear elastic models for two twins to model multiscale patterns, which appear in recent physical experiments and involve mixing of four twins | Yip |
| February 20 | Adrian Lam, Ohio State University TITLE : PDEs in the evolution of dispersal ABSTRACT : In this talk, we will discuss some recent results in competition model motivated by the evolution of dispersal. This question was first formulated by A. Hastings in 1983 concerning the outcome of competition among species which are identical except for their dispersal strategies. We will also mention the recent progress on the $N$-species model and conjecture Dockery et al. Next, we discuss a structured population model formulated by Perthame & Souganidis which is equivalent to the competition of a continuum of phenotype. For this latter model, we discuss the existence, uniqueness, and stability of equilibrium solutions and their relation to Nash equilibrium, as well as the corresponding time-dependent problem. This is joint work with Steve. Cantrell (Miami), Wenrui Hao (Penn State), Benoit Perthame (Sorbonne) and Yuan Lou (Shanghai Jiaotong) | Gao |
| February 27 | Jay Bang, WestLake University, China
TITLE: Self-Similar Solutions to the Stationary Navier-Stokes Equations in a Higher Dimensional Cone ABSTRACT : Self-similar solutions play an important role in understanding the regularity and asymptotic behavior of solutions to the Navier-Stokes equations. We recently showed that axisymmetric self-similar solutions to the stationary Navier-Stokes equations in an $n$-dimensional cone with the no-slip boundary condition except at the origin must be trivial when $n\geq 4$. It rules out this particular scenario of boundary singularity, which has finite Dirichlet energy when $n\geq 5$. The main idea is to apply ODE techniques along with a sign property of the head pressure. This is a joint work with Changfeng Gui, Hao Liu, Yun Wang and Chunjing Xie |
Wang |
| March 6 | Armin Schikorra, University of Pittsburgh
TITLE: On s-Stability of W^{s,n/s}-minimizing maps between spheres in homotopy classes ABSTRACT: We consider maps between spheres S^n to S^\ell that minimize the Sobolev-space energy W^{s,n/s} for some s \in (0,1) in a given homotopy class. The basic question is: in which homotopy class does a minimizer exist? This is a nontrivial question since the energy under consideration is conformally invariant and bubbles can form. Sacks-Uhlenbeck theory tells us that minimizers exist in a set of homotopy classes that generates the whole homotopy group \pi_{n}(\S^\ell). In some situations explicit examples are known if n/s = 2 or s=1. In our talk we are interested in the stability of the above question in dependence of s. We can show that as s varies locally, the set of homotopy classes in which minimizers exist can be chosen stable. We also discuss that the minimum W^{s,n/s}-energy in homotopy classes is continuously depending on s. Joint work with K. Mazowiecka (U Warsaw) | Wang |
| March 13 | Zhiyuan Geng, Purdue University
TITLE: Singular set of the eigenframe for the Q-tensor model ABSTRACT: In the simplified Landau-de Gennes model for nematic liquid crystals, minimizing configurations of the order parameter Q are smooth. As a result, the defects are often interpreted as discontinuities in the eigenframe of Q. In particular, near the negative uniaxial state, where the eigenvalues of Q are r, r, -2r for some r >0, the leading eigenvector is discontinuous and form a “ring defect” along a disclination line. In this talk, I will first review some recent developments on the study of the ring defect profile for the Landau-de Gennes model. Then I will present joint work with Changyou Wang on the defect structure near the singular set of the eigenframe of Q. We consider essentially a minimizing harmonic map from a 3D domain into S^4, and show that the singular set of the eigenframe is countably 1-rectifiable. Moreover, by employing techniques from the study of singular sets in elliptic equations, we classify the blow-up profile of the leading eigenvector near these singularities. |
Wang |
| March 20 | No Seminar, Spring Break TITLE: ABSTRACT |
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| March 27 | Kuan-Ting Yeh, Purdue University TITLE : Structure of Measures for which Ehrhard Symmetrization is Perimeter Decreasing ABSTRACT : In this talk, we show that isotropic Gaussian functions are characterized by a rearrangement inequality for weighted perimeter in dimensions $n \geq 2$ within a broad class of positive weights. More specifically, we prove that within this class, generalized Ehrhard symmetrization is perimeter decreasing for all measurable sets in all directions if and only if the distribution function is an isotropic Gaussian. | Wang |
| April 3 | Phuc Nguyen, LSU TITLE: Some new capacitary inequalities and related function spaces ABSTRACT : We answer question posed by D. R. Adams on a capacitary strong type inequality that generalizes the classical capacitary strong type inequality of V. G. Maz'ya. As a result, we characterize related function spaces as K\"othe duals to a class of Sobolev multiplier type spaces. The boundedness of the Hardy-Littlewood maximal function and the spherical maximal function on related Choquet spaces are also discussed. This talk is based on joint work with Keng H. Ooi. |
Torres |
| April 10 | Trevor Leslie, IIT TITLE: Limiting configurations of solutions to the Euler Alignment system ABSTRACT : In the Euler Alignment system from collective dynamics, a typical long-time behavior is convergence of the density profile to a traveling wave solution (a "flocking state"). When one restricts attention to a single spatial dimension, it is often possible to describe features of the limiting density profile quite explicitly, in terms of the initial data and the communication protocol only. In this talk, we will provide an introduction to the Cucker-Smale and Euler Alignment systems and then discuss the flocking states in 1D. Most of the talk will be based on joint work with Changhui Tan. |
Novack |
| April 17 | Tuoc Phan, UTK
TITLE: On Krylov-Safonov Harnack inequality for a class of parabolic equations with singular degenerate coefficients ABSTRACT : In this talk, we discuss a class of linear parabolic equations in non-divergence form in which the leading coefficients are measurable, and they can be singular or degenerate as a weight that belongs to a class of Muckenhoupt weights. Krylov-Safonov Harnack inequality for solutions is established under some smallness assumption on a weighted mean oscillation of the weight. As corollaries, Holder regularity estimates of solutions with respect to a quasi-distance, and a Liouville type theorem are obtained in the paper. To prove the main result, we introduce a class of generic weighted parabolic cylinders and the smallness condition on the weighted mean oscillation of the weight through which several growth lemmas are established. Additionally, a perturbation method is used and the parabolic Aleksandrov-Bakelman-Pucci type maximum principle is applied to suitable barrier functions to control the solutions. The talk is based on the joint work with S. Cho (Gwangju National University of Education), and J. Fang (University of Tennessee). If time allows, we will also discuss some other recent related regularity results in Sobolev spaces for the class of the equations. |
Wang |
| April 24, 11-11:59 am (Zoom) | Zhengjiang Lin, MIT TITLE:Quantitative Long-Time Clustering in Mean-Field Transformer Models ABSTRACT : The introduction of Transformer models in 2017 revolutionized natural language processing and deep learning. Central to their success is the self-attention mechanism, which distinguishes Transformers from traditional architectures and plays a key role in their superior performance. In this talk, I will present a mathematical framework that views self-attention as an interacting particle system. I will then investigate the phenomenon of long-time clustering in mean-field Transformer models, drawing analogies to the synchronization observed in Kuramoto models. Specifically, I will connect this clustering behavior to the dynamics of Wasserstein gradient flow and establish quantitative exponential rates of contraction to a Dirac delta distribution for any sufficiently regular initialization. | Geng |
| May 1 |
Conglong Xu, George Washington University
TITLE:Weighted Low-rank Approximation via Stochastic Gradient Descents on Manifolds with Semi-Adaptive or Adaptive Learning Rates ABSTRACT : We solve a regularized weighted low-rank approximation problem by stochastic gradient descent on a manifold. To guarantee the convergence of our stochastic gradient descent, we establish convergence theorems on manifolds for retraction-based stochastic gradient descents admitting confinements with semi-adaptive or adaptive learning rates. On sample data from the Netflix Prize training dataset, our algorithm outperforms the existing stochastic gradient descent on Euclidean spaces. |
Yeh |