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# Departmental Colloquia

Our department is proud to host weekly colloquium talks featuring research by leading mathematicians from around the world. Most colloquia are held on Fridays at 4pm in Parker Hall, Room 250 (unless otherwise advertised) with refreshments preceding at 3:30pm in Parker Hall, Room 244.

**DMS Colloquium: Dr. Amy Huang**

**Apr 15, 2022 04:00 PM**

Speaker: **Dr. Amy Huang **(Department of Mathematics, Texas A&M University)

Title: Tensor Ranks and Matrix Multiplication Complexity

Abstract: Tensors are just multi-dimensional arrays. Notions of rank and border rank abound in the literature. Tensor decomposition also has a lot of applications in data analysis, physics, and other areas of science. I will survey my recent two results about tensor rank and its application to matrix multiplication complexity. The first result relates different notions of tensor rank to polynomials of vanishing Hessian. The second one computes the border rank of 3 X 3 permanent, which is important in the theory of matrix multiplication complexity. I will also briefly discuss the newest technique we used to achieve our results: border apolarity. Furthermore, I will survey how this new technique helps us to compute/bound the border rank of a lot of tensors of interests that were considered to be inaccessible before.

Faculty Host: Ash Abebe

**DMS Colloquium: Dr. Shigui Ruan**

**Apr 01, 2022 04:00 PM**

Speaker: **Dr. Shigui Ruan** (Department of Mathematics; University of Miami)

Title: Modeling the Growth, Invasion and Competition of *Aedes*Mosquitoes

Abstract: The *Aedes* mosquitoes, in particular *Aedes aegypti* and *Aedes albopictus*, are the primary vectors that transmit several arboviral diseases, including chikungunya fever, dengue fever, yellow fever, and Zika. Recently, the world has been experiencing a series of major outbreaks of these vector-borne diseases (for example, the 2016 Zika outbreak in Florida, etc.). In order to study the transmission dynamics of these vector-borne diseases, it is very important and necessary to understand the population dynamics, current distributions and movements of *Aedes* mosquitoes for successful surveillance and control programs. In this talk, we will introduce some of our recent studies on modeling the population dynamics of *Aedes*mosquitoes, the invasion of *Aedes albopictus* mosquitoes, and the competition between *Aedes aegypti *and *Aedes Albopictus *mosquitoes in Florida, the United States. In particular, we propose a competition model with road-field diffusion in which the invasive population not only disperses in the interior of the spatial domain but also moves faster on the boundary of the domain. Both strong-weak and weak-weak competitions are discussed. It is shown that the asymptotic spreading speed of the wave fronts is increasing only if the road diffusion rate is greater than the field diffusion rate. Numerical simulations are presented to illustrate our analytical results and to explain the current estimated distributions of these two mosquito species in Florida.

Faculty host: Maggie Han

**DMS Colloquium: Peter E. Kloeden**

**Mar 28, 2022 04:00 PM**

Speaker: **Peter E. Kloeden** (Mathematisches Institut; Universitát Tübingen, Germany; kloeden@na.uni-tuebingen.de)

Title: Lattice Difference Equations

Abstract: Lattice difference equations are essentially difference equations on a Hilbert space of bi-infinite sequences. They are motivated by the discretisation of the spatial variable in integrodifference equations arising in theoretical ecology. It is shown here that under similar assumptions to those used for such integrodifference equations they have a global attractor, to which the global attractors of finite dimensional approximations converge upper semi continuously. Corresponding results are also shown for lattice difference equations when only a finite number of interconnection weights are nonzero and when the interconnection weights themselves vary and converge in an appropriate manner.

**DMS Colloquium: Guang Lin**

**Mar 18, 2022 04:00 PM**

Speaker: **Guang Lin** (Director, Data Science Consulting Service; Professor, Departments of Mathematics and Statistics and School of Mechanical Engineering, Purdue University; www.math.purdue.edu/~lin491/

Title: Towards Third Wave AI: Interpretable, Robust Trustworthy Machine Learning for Diverse Applications in Science and Engineering

Abstract: This talk aims to close the gap by developing new theories and scalable numerical algorithms for complex dynamical systems that can be realistically predicted and validated. We are creating new technologies that can be translated into more secure and reliable new trustworthy AI systems that can be deployed for real-time complex dynamical system prediction, surveillance, and defense applications to improve the stability and efficiency of complex dynamical systems and national security of the United States. We will present a novel neural homogenization-based physics-informed neural network (NN) for multiscale problems. We will also introduce new NNs that learn functionals and nonlinear operators from functions with simultaneous uncertainty estimates. In particular, we present a probabilistic neural operator network training procedure for solving partial differential equations with inhomogeneous boundary conditions. Using a light-weight extension of deep operator network (DeepONet) architecture, the trained networks are designed to provide rapid predictions along with simultaneous uncertainty estimates to help identify potential inaccuracies in the network predictions. DeepONet consists of an NN for encoding the discrete input function space (branch net) and another NN for encoding the domain of the output functions (trunk net). In particular, the predictive uncertainty of the network is calibrated to anticipate network errors by implementing a loss function that interprets the network prediction as a probability distribution as opposed to a single point estimate. The proposed technique is also capable of solving problems on irregular, non-rectangular domains, and a series of experiments are presented to evaluate the network accuracy as well as the quality of the predictive uncertainty estimates. We demonstrate that the novel probabilistic DeepONet can learn various explicit operators with predictive uncertainties.

Short Bio of Guang Lin

Guang Lin is a Full Professor in the School of Mechanical Engineering and Department of Mathematics at Purdue University. Lin is the Director of Data Science Consulting Service that performs cutting-edge research on data science and provides hands-on consulting support for data analysis and business analytics. He is also the Chair of the Initiative for Data Science and Engineering Applications at the College of Engineering. Lin received his Ph.D. from Brown University in 2007 and worked as a Research Scientist at DOE Pacific Northwest National Laboratory before joining Purdue in 2014. He has received various awards, such as the NSF CAREER Award, Mid-Career Sigma Xi Award, University Faculty Scholar, Mathematical Biosciences Institute Early Career Award, and Ronald L. Brodzinski Award for Early Career Exception Achievement.

**DMS Colloquium: Yaozhong Hu**

**Feb 18, 2022 04:00 PM**

Speaker: **Yaozhong Hu** (Centennial Professor, University of Alberta at Edmonton, Canada; https://sites.ualberta.ca/~yaozhong/)

Title: Asymptotics of the density of parabolic Anderson random fields

Abstract: We investigate the shape of the density \(\rho(t,x; y)\) of the solution \(u(t,x)\) to stochastic partial differential equation \(frac{\partial}{\partial t} u(t,x)=\frac12 \Delta u(t,x)+u\diamond \dot W(t,x)\), where \(\dot W\) is a general Gaussian noise and \(\diamond\) denotes the Wick product. We are mainly concerned with the asymptotic behavior of \(\rho(t,x; y)\) when \(y\rightarrow \infty\) or when \(t\to 0+\). Both upper and lower bounds are obtained and these two bounds match each other modulo some multiplicative constants. If the initial condition is positive, then \(\rho(t,x;y)\) is supported on the positive half line \(y\in [0, \infty)\) and, in this case, we show that \(\rho(t,x; 0+)=0\) and obtain an upper bound for \(\rho(t,x; y)\) when \(y\rightarrow 0+\). Our tool is Malliavin calculus and I will also present a very brief introduction.

This is a joint work with Khoa Le.

Faculty host: Le Chen

**DMS Colloquium: Mark Walker**

**Dec 03, 2021 04:00 PM**

Speaker: **Mark Walker** (Willa Cather Professor, University of Nebraska--Lincoln)

Title: The Total and Toral Rank Conjectures

Abstract: Assume \(X\) is a nice topological space (a compact \(CW\) complex) that admits a fixed-point free action by a \(d\)-dimensional torus \(T\). For example, \(X\) could be \(T\) acting on itself in the canonical way. The Toral Rank Conjecture, due to Halperin, predicts that the sum of the (topological) Betti numbers of \(X\) must be at least \(2^d\). Put more crudely, this conjecture predicts that it takes at least \(2^d\) cells to build such a space \(X\) by gluing them together.

Now suppose \(M\) is a module over the polynomial ring \(k[x_1, \dots, x_d]\) that is finite dimensional as a \(k\)-vector space. The Total Rank Conjecture, due to Avramov, predicts that the sum of (algebraic) Betti numbers of \(M\) must be at least \(2^d\). Here, the algebraic Betti numbers refer to the ranks of the free modules occurring in the minimal free resolution of \(M\).

In this talk, I will discuss the relationship between these conjectures and recent progress toward settling them.

Faculty host: Michael Brown

Last Updated: 09/11/2015