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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: Saeed Nasseh**

**Oct 18, 2019 04:00 PM**

Speaker: **Saeed Nasseh**, Georgia Southern University

Title: Modules over ﬁnite dimensional algebras with an application to a conjecture in commutative algebra

Abstract: When \(F\) is a ﬁeld and \(R\) an \(F\)-algebra, it is often important to classify the \(R\)-module structures carried by some ﬁxed \(F\)-vector space \(V\). We will describe a classical approach to that problem, which involves some algebraic geometry and homological algebra. We will then sketch new developments of these techniques and their application to the proof of a result in commutative algebra, conjectured by Vasconcelos in 1974.

Biographical sketch

Saeed Nasseh is an associate professor in the Department of Mathematical Sciences at Georgia Southern University. Prior to that, he was a postdoctoral fellow at the University of Nebraska-Lincoln. His research areas are commutative algebra and homological algebra with some overlaps with algebraic geometry, algebraic topology, and representation theory. A major part of his recent research focuses on developing and applying differential graded (DG) homological techniques to solve problems on the vanishing of homology and cohomology over commutative rings. For instance, in a joint work with Sather-Wagstaff, using the DG methods and geometric aspects of representation theory, they provided a complete solution to a long-standing conjecture posed by Vasconcelos in 1974. In other recent (in-progress) joint works with Avramov, Iyengar, and Sather-Wagstaff again using the DG techniques, they introduce several classes of commutative noetherian local rings that satisfy another major conjecture from 1975, known as the Auslander-Reiten Conjecture (which originates from representation theory of Artin algebras). He is currently working (in separate projects with other collaborators) on developing DG methods in order to completely solve this conjecture. His past and recent results on the vanishing of (co)homology, ring homomorphisms, and homological dimensions are contained in 16 research papers to date, most of which have been published in top journals.

Faculty host: Overtoun Jenda

**DMS Colloquium: Dr. Robert Lipton**

**Oct 25, 2019 04:00 PM**

Speaker: **Prof. Robert Lipton**, LSU

Title: Manipulating light with photonic crystals

Abstract: Photonic crystals are patterned materials whose electromagnetic properties are controlled by their internal structure. Destructive interference can occur giving rise to frequency intervals where no waves can propagate inside the material (a band gap). These are the well-known photonic band gap crystals; their effects include the structural coloration of butterfly wings and have been used in ingenious ways for optical communication. In this lecture we provide a brief history of photonic band gap crystals and a window into the design of maximal band gaps using high contrast patterned dielectric media. Of particular interest is how the quasi-static resonance spectra of the structure plays a role in obtaining explicit formulas for band gaps.

Faculty host: Junshan Lin

**DMS Colloquium: Italo Lima**

**Nov 08, 2019 04:00 PM**

**Italo Lima**, Data Scientist at Facebook

**DMS Colloquium: Charles E. Chidume**

**Oct 04, 2019 04:00 PM**

Speaker: **Charles E. Chidume**, FTWAS, FAS, FNMS

Title: On the Strong Convergence of the Proximal Point Algorithm of Martinet and Rockafellar

Abstract: Rockafellar proved that the sequence of the celebrated proximal point algorithm introduced by Martinet for approximating a zero of a maximal monotone operator on a real Hilbert Space converges weakly. He then posed the following question: Does the sequence converge strongly? This was resolved in the negative by Guder. Consequently, several authors over the years, proposed various involved modifications of the proximal point algorithm to obtain strong convergence. In this talk, we present our new iterative algorithm, recently introduced which is totally different from the proximal point algorithm or any of its modifications, and yields strong convergence to a zero of a maximal monotone map even in real Banach spaces much more general than real Hilbert spaces. Finally, we present numerical examples to illustrate the strong convergence of the sequence of our algorithm.

Biographical sketch

Charles Ejike Chidume received his PhD degree in Mathematics from The Ohio State University, Columbus, Ohio. He returned immediately to his home country, Nigeria and worked at the University of Nigeria, Nsukka, for six years during which he rose to the rank of full professor of Mathematics. He then moved, on invitation, to join the International Centre for Theoretical Physics (ICTP) of IAEA and UNESCO of the United Nations, in Trieste, Italy, where he worked as the coordinator of the Postgraduate Diploma Program (Mathematics) and Research Mathematician. In 2009, Professor Chidume, on invitation, joined the African University of Science and Technology (AUST), a Nelson Mandela Institution offering only Postgraduate programs, located in Abuja, Nigeria. He is currently at this University serving as the Department Head of Pure and Applied Mathematics, Director of the Mathematics Institute, and Acting President of the University.

Professor Chidume has supervised 19 PhD theses and over 80 MSc and Postgraduate Diploma of the ICTP thesis. He publishes copiously and is an Associate Editor in several top journals.

Faculty host: Geraldo de Souza, Professor Emeritus

**DMS Colloquium: Virginia Vassilevska Williams**

**Sep 27, 2019 04:00 PM**

Speaker: **Virginia Vassilevska Williams** (MIT)

Title: Limitations on All Known (and Some Unknown) Approaches to Matrix Multiplication

Abstract: In this talk we will consider the known techniques for designing Matrix Multiplication algorithms. The two main approaches are the Laser method of Strassen, and the Group theoretic approach of Cohn and Umans. We define generalizations that subsume these two approaches: the Galactic and the Universal method; the latter is the most general method there is. We then design a suite of techniques for proving lower bounds on the value of \(\omega\), the exponent of matrix multiplication, which can be achieved by algorithms using many tensors \(T\) and the Galactic method. Some of our techniques exploit `local' properties of \(T\), like finding a sub-tensor of \(T\) which is so `weak' that \(T\) itself couldn't be used to achieve a good bound on $\omega$, while others exploit `global' properties, like \(T\) being a monomial degeneration of the structural tensor of a group algebra.

The main result is that there is a universal constant \(\ell>2\) such that a large class of tensors generalizing the Coppersmith-Winograd tensor \(CW_q\) cannot be used within the Galactic method to show a bound on \(\omega\) better than \(\ell\), for any \(q\). We give evidence that previous lower-bounding techniques were not strong enough to show this.

The talk is based on joint work with Josh Alman, which appeared in FOCS 2018. More recently, Alman showed how to extend our techniques so that they apply to the Universal method as well. In particular, Alman shows that the Coppersmith-Winograd tensor cannot yield a better bound on \(\omega\) than 2.16805 even using the Universal method.

Faculty host: Luke Oeding

**DMS Colloquium: Dr. Zuofeng Shang**

**Sep 20, 2019 04:00 PM**

Speaker: **Dr. Zuofeng Shang**, New Jersey Institute of Technology

Title: Statistical Optimality of Deep Neural Networks in Regression and Classification

Abstract: In this talk, I will discuss statistical optimality of deep neural network methods in regression and classification. In the first part, I will consider linear regression model with instrumental variables that characterize endogenous errors. Deep neural network provides a flexible way to characterize the relationship between the design variable and instrumental variable. Asymptotic distribution and semiparametric efficiency are established for the proposed estimator. In the second part, I will consider nonparametric classification in which classifiers are characterized by deep neural networks. Tight upper and lower bounds for the classification risk are established. The proposed methods enjoy the so-called intrinsic dimension phenomenon.

Bio sketch:

Dr. Zuofeng Shang is an associate professor in the Department of Mathematics and Statistics at New Jersey Institute of Technology.

His research aim focuses on statistical foundation of modern data science fields, primarily in developing statistical frameworks for data science objects, efficient learning algorithms for scientific tasks and theoretical understanding on the nature of the problems. He has published more than 18 top journal papers. He is also the principle investigator of two NSF grants.

Faculty host: Guanqun Cao

**DMS Colloquium: Beatrice Riviere**

**Sep 06, 2019 04:00 PM**

Speaker: **Béatrice Rivière**, Rice University

Title: High order discontinuous Galerkin methods for solving the miscible displacement problem in porous media

Abstract: The accurate prediction of flow and transport in porous media is essential in optimizing the clean-up of contaminated groundwater or the production of hydrocarbons from oil reservoirs. In the miscible displacement problem, a solvent is injected and mixes with the resident fluid (contaminant or oil). The fluid mixture then propagates through the set of connected pores. At the Darcy scale, the mathematical model is a system of partial differential equations coupling flow and transport. This talk presents high order interior penalty discontinuous Galerkin (IPDG) methods and hybridizable discontinuous Galerkin (HDG) methods for solving the nonlinear system of equations. HDG methods retain the positive features of IPDG, but the number of globally coupled degrees of freedom for high order HDG is significantly smaller. The proposed numerical methods are shown to be accurate on coarse meshes when the polynomial degree increases. The numerical approximations of the solvent concentration exhibit sharp fronts even in highly heterogeneous media. Finally, the discontinuous Galerkin methods in space can accurately model viscous fingering. Viscous fingering in porous media may occur when a fluid with low viscosity is used to displace a fluid with high viscosity. For this type of flow instability, a tiny perturbation can be amplified exponentially, which triggers a finger-like pattern in the fluid concentration profile during the fluid displacement. Simulations in two and in three dimensions show the growth and propagation of fingers for large mobility ratios and large Peclet numbers. Results are compared with those obtained by using a generic cell-centered finite volume method.

**Short Bio-sketch**

Dr. Rivière has worked extensively of the development and analysis of numerical methods applied to problems in porous media and in fluid mechanics. Her current research deals with the development of high-order methods in time and in space for multiphase multicomponent flows (in rigid and deformable media); the modeling of pore scale flows for immiscible and miscible components; the numerical model of oxygen transport in a network of blood vessels; the analysis of PDE-based neural networks for image segmentation and the design of iterative solvers.

Dr. Rivière is an associate editor for the SIAM Journal on Numerical Analysis, for the SIAM Journal on Scientific Computing, for Results in Applied Mathematics, and a member of the editorial board for Advances in Water Resources. She has graduated a total of fourteen Ph.D. students, with eight working in academia and five in industry.

Faculty host: Thi-Thao-Phuong Hoang

**DMS Colloquium: Wei Cai**

**Aug 30, 2019 04:00 PM**

Speaker: **Wei Cai**, Southern Methodist University

Title: Algorithms for wave scattering of random media: Fast multipole method in layered media and a phase shift deep neural network for wideband learning

Abstract: In this talk, we will present two algorithms and numerical results for solving electromagnetic wave scattering of random meta-materials. Firstly, a fast multipole method for 3-D Helmholtz equation for layered media will be presented based on new multipole expansion (ME) and multipole to local translation (M2L) operators for layered media Green's functions. Secondly, a parallel phase shift deep neural network (PhaseDNN) is proposed for wideband data learning. In order to achieve uniform convergence for low to high frequency content of data, phase shifts are used to convert high frequency learning to low frequency learning. Due to the fast learning of many DNNs in the low frequency range, PhaseDNN is able to learn wideband data uniformly in all frequencies.

**Short Bio-sketch**

Prof. Wei Cai obtained his B.S. and M.S. in Mathematics from the University of Science and Technology of China in 1982 and 1985, respectively, and his Ph.D. in Applied Mathematics at Brown University in 1989. Currently, he is the Clements Chair professor of applied mathematics at SMU. Before he joined SMU, he was an assistant and then associate professor at the University of California at Santa Barbara during 1995-96, and a full Professor at the University of North Carolina after 1999. He also taught and conducted research at Peking University, Fudan University, Shanghai Jiaotong University. His research interest focuses on the development of deterministic and stochastic numerical methods for studying electromagnetic and quantum phenomena with applications in meta-materials, nano-photonics, nano-electronics, biological systems, and quantum systems. He has published over 110 refereed articles in top international journals and is the author of the book "Computational Methods for Electromagnetic Phenomena: Electrostatics in Solvation, Scattering, and Electron Transport" published by Cambridge University Press, 2013.

Faculty hosts: Junshan Lin and Yanzhao Cao

**DMS Colloquium: Matthias Heikenschloss**

**Apr 26, 2019 04:00 PM**

Speaker: **Matthias Heikenschloss**, Rice University

Title: Risk averse PDE constrained optimization

Last Updated: 09/11/2015