# Department of Mathematics

## Colloquia and Seminars

The Department of Mathematics is home to a dynamic group of scholars who regularly share their expertise with others. Many of our events and activities are free and open to the public, and we encourage you to get involved.

### Colloquia Fall 2019

When: Thursday, October 3, 2019  - 4:30 p.m. to 5:30 p.m.
Where: LeConte 412 (map)

Speaker: Jinchao Xu, Pennsylvania State University

Abstract:  In this talk, I will first present a recently developed uniform framework, known as Extended Galerkin (XG) method, for derivation and analysis of many different types of Galerkin methods, including conforming, nonconforming, discontinuous, mixed and virtual finite-element methods. I will then discuss the question (with some answers and some open problems) if it is possible to give a universal construction and analysis of convergent finite element methods for elliptic boundary value problems. Finally, I will discuss the function class given by deep neural networks and its relationship with finite element and applications to solution of partial differential equations. [PDF]

When: Thursday, October 24, 2019  - 4:30 p.m. to 5:30 p.m.
Where: LeConte 412 (map)

Speaker: Yen-Hsi Richard Tsai, University of Texas at Austin

Abstract: I will review a general framework that is called the implicit boundary integral methods. It is a general framework that can be applied to solve a variety of problems that involve non-parametrically represented surfaces. The main idea is to formulate appropriate extensions of a given problem defined on a surface to ones in the narrow band of the surface in the embedding space. The extensions are arranged so that the solutions to the extended problems are equivalent, in a strong sense, to the surface problems that we set out to solve.  Such extension approaches allow us to analyze the well-posedness of the resulting system, develop systematically and in a unified fashion numerical schemes for treating a wide range of problems that involve both differential and integral operators, and deal with similar problems in which only point clouds sampling the surfaces are given. We will apply this framework to solve some surface PDE problems, boundary integral equations, and optimal control problems.

When: Thursday, November 21, 2019  - 4:30 p.m. to 5:30 p.m.
Where: LeConte 412 (map)

Speaker: Patricia Hersh, North Carolina State University

Abstract: Sergey Fomin and Michael Shapiro proved that the totally nonnegative real part of the unipotent radical of a Borel in a semisimple, simply connected algebraic group has a cell decomposition with Bruhat order as its poset of closure relations, and they conjectured that (after deconing) this was a regular CW complex homeomorphic to a closed ball. Much of the interest in these spaces comes from their interpretation as images of maps related to Lusztig's theory of canonical bases. I will briefly discuss my proof of this conjecture, then turn to new joint work with Jim Davis and Ezra Miller regarding the structure of the fibers of these maps. This will include telling much of the back-story leading up to this work as well as providing motivation and background in this area along the way. [PDF]

When: Thursday, December 5, 2019  - 4:30 p.m. to 5:30 p.m.
Where: LeConte 412 (map)

Speaker: Miao-Jung Yvonne Ou, University of Delaware

Abstract: It has been a long quest in mathematical material sciences to study the relation(s) between microstructure and various effective properties of composite materials. The class of methods based on Nevanllina-Herglotz functions was pioneered in physics by David Bergman and further developed mathematically by Grame Milton, Ken Golden, Elena Cherkaev and many others in the context of using this method to find bounds for effective properties for given constituents with constraints on volume fractions or on microstructural symmetries. The key in this class of method is the Integral representation formula (IRF) of a Nevanllina-Herglotz function or its 'cousins'. In this talk, a brief review of the history of the method will be given. A detailed explanation of the recent development on the IRF for the viscodynamic operator of poroelastic media will also be presented. Finally, the implication of this method in handing the memory term in solving the wave equations will be made clear with numerical examples. [PDF]

### Colloquia Spring 2019

When: Tuesday, January 15, 2019  - 4:30 p.m. to 5:30 p.m.
Where: LeConte 412 (map)

Speaker: Leonardo Zepeda-Nunez, Lawrence Berkeley National Laboratory

Abstract: Deep learning has rapidly become a large field with an ever-growing range of applications; however, its intersection with scientific computing remains in its infancy, mainly due to the high accuracy that scientific computing problems require, which depends greatly on the architecture of the neural network.

In this talk we present a novel deep neural network with a multi-scale architecture inspired in H-matrices (and H2-matrices) to efficiently approximate, within 3-4 digits, several challenging non-linear maps arising from the discretization of PDEs, whose evaluation would otherwise require computationally intensive iterative methods.

In particular, we focus on the notoriously difficult Kohn-Sham map arising from Density Functional Theory (DFT). We show that the proposed multiscale neural network can efficiently learn this map, thus bypassing an expensive self-consistent field iteration. In addition, we show the application of this methodology to ab-initio molecular dynamics, for which we provide examples for 1D problems and small, albeit realistic, 3D systems.

Joint work with Y. Fan, J. Feliu-Faaba, L. Lin, W. Jia, and L. Ying. [PDF]

When: Thursday, January 24, 2019  - 4:30 p.m. to 5:30 p.m.

Where: LeConte 412 (map)

Speaker: Maziar Raissi, Brown University

Abstract: grand challenge with great opportunities is to develop a coherent framework that enables blending conservation laws, physical principles, and/or phenomenological behaviours expressed by differential equations with the vast data sets available in many fields of engineering, science, and technology. At the intersection of probabilistic machine learning, deep learning, and scientific computations, this work is pursuing the overall vision to establish promising new directions for harnessing the long-standing developments of classical methods in applied mathematics and mathematical physics to design learning machines with the ability to operate in complex domains without requiring large quantities of data. To materialize this vision, this work is exploring two complementary directions: (1) designing data-efficient learning machines capable of leveraging the underlying laws of physics, expressed by time dependent and non-linear differential equations, to extract patterns from high-dimensional data generated from experiments, and (2) designing novel numerical algorithms that can seamlessly blend equations and noisy multi-fidelity data, infer latent quantities of interest (e.g., the solution to a differential equation), and naturally quantify uncertainty in computations. The latter is aligned in spirit with the emerging field of probabilistic numerics. [PDF]

When: Thursday, January 31, 2019  - 4:30 p.m. to 5:30 p.m.

Where: LeConte 412 (map)

Speaker: Simone Brugiapaglia, Simon Fraser University

Abstract: Compressive sensing (CS) is a general paradigm that enables us to measure objects (such as images, signals, or functions) by using a number of linear measurements proportional to their sparsity, i.e. to the minimal amount of information needed to represent them with respect to a suitable system. The vast popularity of CS is due to its impact in many practical applications of data science and signal processing, such as magnetic resonance imaging, X-ray computed tomography, or seismic imaging.

In this talk, after presenting the main theoretical ingredients that made the success of CS possible and discussing recovery guarantees in the noise-blind scenario, we will show the impact of CS in computational mathematics. In particular, we will consider the problem of computing sparse polynomial approximations of functions defined over high-dimensional domains from pointwise samples, highly relevant for the uncertainty quantification of PDEs with random inputs. In this context, CS-based approaches are able to substantially lessen the curse of dimensionality, thus enabling the effective approximation of high-dimensional functions from small datasets. We will illustrate a rigorous noise-blind recovery error analysis for these methods and show their effectiveness through numerical experiments. Finally, we will present some challenging open problems for CS-based techniques in computational mathematics. [PDF]

When: Tuesday, February 5, 2019  - 4:30 p.m. to 5:30 p.m.

Where: LeConte 412 (map)

Speaker: Oren Mangoubi, Ecole Polytechnique Federale de Lausanne

Abstract: Sampling from a probability distribution is a fundamental algorithmic problem. We discuss applications of sampling to several areas including machine learning, Bayesian statistics and optimization. In many situations, for instance when the dimension is large, such sampling problems become computationally difficult.

Markov chain Monte Carlo (MCMC) algorithms are among the most effective methods used to solve difficult sampling problems. However, most of the existing guarantees for MCMC algorithms only handle Markov chains that take very small steps and hence can oftentimes be very slow. Hamiltonian Monte Carlo (HMC) algorithms – which are inspired from Hamiltonian dynamics in physics – are capable of taking longer steps. Unfortunately, these long steps make HMC difficult to analyze. As a result, non-asymptotic bounds on the convergence rate of HMC have remained elusive.

In this talk, we obtain rapid mixing bounds for HMC in an important class of strongly log-concave target distributions encountered in statistical and Machine learning applications. Our bounds show that HMC is faster than its main competitor algorithms, including the Langevin and random walk Metropolis algorithms, for this class of distributions.

Finally, we consider future directions in sampling and optimization. Specifically, we discuss how one might design adaptive online sampling algorithms for applications to problems in reinforcement learning and Bayesian parameter inference in partial differential equations. We also discuss how Markov chain algorithms can be used to solve difficult non-convex sampling and optimization problems, and how one might be able to obtain theoretical guarantees for the MCMC algorithms that can solve these problems. [PDF]

When: Thursday, February 21, 2019  - 4:30 p.m. to 5:30 p.m.

Where: LeConte 412 (map)

Speaker: Alexander Kiselev, Duke University

Abstract: The Euler equation describing motion of ideal fluid goes back to 1755. The analysis of the equation is challenging since it is nonlinear and nonlocal. Its solutions are often unstable and spontaneously generate small scales. The fundamental question of global regularity vs finite time singularity formation remains open for the Euler equation in three spatial dimensions. I will review the history of this question and its connection with the arguably greatest unsolved problem of classical physics, turbulence. Recent results on small scale and singularity formation in two dimensions and for a number of related models will also be presented. [PDF]

Host: Changhui Tan

When: Friday, March 1, 2019  - 4:30 p.m. to 5:30 p.m.

Where: LeConte 412 (map)

Speaker: Robert Calderbank, Duke University

Abstract: Quantum error-correcting codes can be used to protect qubits involved in quantum computation. This requires that logical operators acting on protected qubits be translated to physical operators (circuits) acting on physical quantum states. I will describe a mathematical framework for synthesizing physical circuits that implements logical Clifford operators for stabilizer codes. Circuit synthesis is enabled by representing the desired physical Clifford operator as a partial 2m × 2m binary symplectic matrix, where N = 2m.  I will show that for an [[m, m − k]] stabilizer code every logical Clifford operator has 2k(k+1)/2 symplectic solutions, and I will describe how to obtain the desired physical circuits by decomposing each solution as a product of elementary symplectic matrices, each corresponding to an elementary circuit. Assembling all possible physical realizations enables optimization over the ensemble with respect to any suitable metric.

Explore https://github.com/nrenga/symplectic-arxiv18a for programs implementing these algorithms, including routines to solve for binary symplectic solutions of general linear systems and the overall circuit synthesis algorithm.

This is joint work with Swanand Kadhe, Narayanan Rengaswamy, and Henry Pfister. [PDF]

Host: George Androulakis

When: Thursday, March 7, 2019  - 4:30 p.m. to 5:30 p.m.

Where: LeConte 412 (map)

Speaker: Mikhail Ostrovskii, St. John's University

Abstract: Embeddings of a discrete metric space into a Hilbert spaces or a "good" Banach space have found many significant applications. At the beginning of the talk I plan to give a brief description of such applications. After that I plan to present three of my results: (1) On L1-embeddability of graphs with large girth; (2) Embeddability of infinite locally finite metric spaces into Banach spaces is finitely determined; (3) New metric characterizations of superreflexivity. [PDF]

Host: Stephen Dilworth

### Colloquia 2018

When: Thursday, January 25, 2018  - 4:30 p.m. to 5:30 p.m.
Where: LeConte 412 (map)

Speaker: Zhen-Qing Chen, University of Washington

Abstract: Anomalous diffusion phenomenon has been observed in many natural systems, from the signalling of biological cells, to the foraging behaviour of animals, to the travel times of contaminants in groundwater. In this talk, I will first discuss the interplay between anomalous diffusions and differential equations of fractional order. I will then present some recent results in the study of these two topics, including the counterpart of DeGiorgi-Nash-Moser-Aronson theory for non-local operators of fractional order. No prior knowledge in these two subjects is assumed. [PDF]

Host: Hong Wang

When: Thursday, February 8, 2018 - 4:30 p.m. to 5:30 p.m.
Where: LeConte 412 (map)

Speaker: Changhui Tan, Rice University

Abstract: Self-organized behaviors are commonly observed in nature and human societies, such as bird flocks, fish swarms and human crowds. In this talk, I will present some celebrated mathematical models, with simple small-scale interactions that lead to the emergence of global behaviors: aggregation and flocking. The models can be constructed through a multiscale framework: from microscopic agent-based dynamics, to macroscopic fluid systems. I will discuss some recent analytical and numerical results on the derivation of the systems in different scales, global well-posedness theory, large time behaviors, as well as interesting connections to some classical equations in fluid mechanics. [PDF]

When: Thursday, February 15, 2018 - 4:30 p.m. to 5:30 p.m.
Where: LeConte 412 (map)

Speaker: Andrei Tarfulea, University of Chicago

Abstract: Understanding the behavior of solutions to physically motivated evolution equations is one of the most important areas of applied analysis. Developing strong bounds and asymptotics are crucial for anticipating the behavior of simulations, simplifying the methods needed to model the physical phenomena. The focus will be on recent results in three physical models: homogenization and asymptotics for nonlocal reaction-diffusion equations, a priori bounds for hydrodynamic equations with thermal effects, and the local well-posedness for the Landau equation (with initial data that is large, away from Maxwellian, and containing vacuum regions). Each problem presents unique challenges arising from the nonlinearity and/or nonlocality of the equation, and the emphasis will be on the different methods and techniques used to treat those difficulties in each case. The talk will touch on novelties in viscosity theory and precision in nonlocal front propagation for reaction-diffusion equations, as well as the emergence of "dynamic" self-regularization in the thermal hydrodynamic and Landau equations. [PDF]

When: Tuesday, February 20, 2018 - 4:30 p.m. to 5:30 p.m.
Where: LeConte 412 (map)

Speaker: Xiu Yang, PNNL

Abstract: Realistic analysis and design of complex engineering systems require not only a fine understanding of the underlying physics, but also a significant recognition of uncertainties and their influences on the quantities of interest. Intrinsic variabilities and lack of knowledge about system parameters or governing physical models often considerably affect quantities of interest and decision-making processes. For complex systems, the available data for quantifying uncertainties or analyzing sensitivities are usually limited because the cost of conducting a large number of experiments or running many large-scale simulations can be prohibitive. Efficient approaches of representing uncertainties using limited data are critical for such problems. I will talk about two approaches for uncertainty quantification by constructing surrogate model of the quantity of interest. The first method is the adaptive functional ANOVA method, which constructs the surrogate model hierarchically by analyzing the sensitivities of individual parameters. The second method is the sparse regression based on identification of low-dimensional structure, which exploits low-dimensional structures in the parameter space and solves an optimization problem to construct the surrogate models. I will demonstrate the efficiency of these methods with PDE with random parameters as well as applications in aerodynamics and computational chemistry. [PDF]

When: Thursday, February 22, 2018 - 4:30 p.m. to 5:30 p.m.
Where: LeConte 412 (map)

Speaker: Daniel Krashen, Rutgers University/University of Georgia

Abstract: Understanding algebraic structures such as Galois extensions, quadratic forms and division algebras, can give important insights into the arithmetic of fields. In this talk, I will discuss recent work showing ways in which the arithmetic of certain fields can be partially described by topological information. I will then describe how these observations lead to arithmetic versions of the Meyer-Vietoris sequences, the Seifert–van Kampen theorem, and examples and counterexamples to local-global principles. [PDF]

Host: Frank Thorne

When: Thursday, March 1, 2018 - 4:30 p.m. to 5:30 p.m.
Where: LeConte 412 (map)

Speaker: Lars Christensen, Texas Tech University

Abstract: Let K be a field, for example that of complex numbers, and let R be a quotient of the polynomial algebra $$Q = K [x,y,z]$$. The minimal free resolution of R as a module over Q is a sequence of linear maps between free Q-modules. One may think of such free resolutions as the result of a linearization process that unwinds the structure of R in a
series of maps. This point of view, which goes back to Hilbert, already yields a wealth of information about R, but there is more to the picture: The resolution carries a multiplicative structure; it is itself a ring! For algebraists this is  Gefundenes Fressen, and in the talk I will discuss what kind of questions this structure has helped answer and what new questions it raises. [PDF]

Host: Andrew Kustin

When:  Thursday, April 5, 2018  - 4:30 p.m. to 5:30 p.m.
Where: LeConte 412 (map)

Speaker: Anthony Bonato, Ryerson University

Abstract: The intersection of graph searching and probabilistic methods is a new topic within graph theory, with applications to graph searching problems such as the game of Cops and Robbers and its many variants, Firefighting, graph burning, and acquaintance time. Graph searching games may be played on random structures such as binomial random graphs, random regular graphs or random geometric graphs. Probabilistic methods may also be used to understand the properties of games played on deterministic structures. A third and new approach is where randomness figures into the rules of the game, such as in the game of Zombies and Survivors. We give a broad survey of graph searching and probabilistic methods, highlighting the themes and trends in this emerging area. The talk is based on my book (with the same title) co-authored with Pawel Pralat published by CRC Press. [PDF]

Bio: Anthony Bonato’s research is in Graph Theory, with applications to the modelling of real-world, complex networks such as the web graph and on-line social networks. He has authored over 110 papers and three books with 70 co-authors. He has delivered over 30 invited addresses at international conferences in North America, Europe, China, and India. He twice won the Ryerson Faculty Research Award for excellence in research and an inaugural Outstanding Contribution to Graduate Education Award. He is the Chair of the Pure Mathematics Section of the NSERC Discovery Mathematics and Statistics Evaluation Group, Editor-in-Chief of the journal Internet Mathematics, and editor of the journal Contributions to Discrete Mathematics.

Host: Linyuan Lu

When:  Friday, April 27, 2018  - 3:30 p.m. to 4:30 p.m.
Where: LeConte 412 (map)

Speaker: Richard Anstee, The University of British Columbia

This is a special Colloquium and reception in honor of Jerry Griggs' retirement

Abstract: Extremal Combinatorics asks how many sets (or other objects) can you have while satisfying some property (often the property of avoiding some structure). We encode a family of n subsets of elements {1,2,..,m} using an element-subset (0,1)-incidence matrix. A matrix is simple if it has no repeated columns. Given a p × q (0,1)-matrix F, we say a (0,1)-matrix A has F as a configurationconfiguration if there is submatrix of A which is a row and column permutation of F. We then defi ne our extremal function forb(m,F) as the maximum number of columns of any m-rowed simple (0,1)-matrix which does have F as a configurationconfiguration. Jerry was involved in some of the initial work on this problem and the construction that led to an attractive conjecture. Two recent results are discussed. One (with Salazar) concerns extending a p × q configurationconfiguration F to a family of all possible p × q configurationconfigurations G with F less than or equal to G (i.e. only the 1's matter). The conjecture does not extend to this setting but there are interesting connections to other extremal problems. The second (with Dawson, Lu and Sali) considers extending the extremal problem to (0,1,2)-matrices. We consider a family of (0,1,2)-matrices which appears to have behaviour analogous to (0,1)-matrices. Ramsey type theorems are used and obtained. [PDF]

Host:  Linyuan Lu

When: Thursday, November 8, 2018  - 4:30 p.m. to 5:30 p.m.
Where: LeConte 412 (map)

Speaker: Paul S. Aspinwall, Duke University

Abstract: Superstring theory is hoped to provide a theory of all fundamental physics including an understanding of quantum gravity. While theoretical physicists like to describe spacetime in terms of differential geometry, we will show how stringy geometry is better explained in terms of representation theory of certain algebras and this can be more easily described in terms of algebraic geometry. We will discuss how mirror symmetry arises and how the derived category of coherent sheaves is useful in this context. [PDF]

Host:  Matthew Ballard

When: Thursday, November 15, 2018  - 4:30 p.m. to 5:30 p.m.
Where: LeConte 412 (map)

Speaker: Claudio Canuto, Politecnico di Torino

Abstract: Discrete Fracture Network (DFN) models are widely used in the simulation of subsurface flows; they describe a geological reservoir as a system of many intersecting planar polygons representing the underground network of fractures. The mathematical description is based on Darcy’s law, supplemented by appropriate interface conditions at each intersection between two fractures. Efficient numerical discretizations, based on the reformulation of the equations as a PDE-constrained optimization problem, allow for a totally independent meshing of each fracture.
We consider stochastic versions of DFN, in which certain relevant parameters of the models are assumed to be random variables with given probability distribution. The dependence of the quantity of interest upon these variables may be smooth (e.g., analytic) or non-smooth (e.g., discontinuous). We perform a non-intrusive uncertainty quantification analysis which, according to the different situations, uses such tools as stochastic collocation, multilevel Monte Carlo, or multifidelity strategies. [PDF]

Host:  Wolfgang Dahmen

When: Friday, December 7, 2018  - 4:30 p.m. to 5:30 p.m.
Where: LeConte 412 (map)

Speaker: Bruce C. Berndt, University of Illinois at Champaign-Urbana

Abstract: Beginning in May, 1977, the speaker began to devote all of his research efforts to proving the approximately 3300 claims made by Ramanujan without proofs in his notebooks. While completing this task a little over 20 years later, with the help, principally, of his graduate students, he began to work with George Andrews on proving Ramanujan's claims from his "lost notebook.” After another 20 years, with the help of several mathematicians, including my doctoral students, Andrews and I think all the claims in the lost notebook have now been proved. One entry from the lost notebook connected with the famous Dirichlet Divisor Problem remained painfully difficult to prove. Borrowing from Sherlock Holmes, G.N. Watson's retiring address to the London Mathematical Society in November, 1935 was on the "final problem," arising from Ramanujan's last letter to Hardy. Accordingly, we have called this entry the "final problem," because it was the last entry from the lost notebook to be proved. Early this summer, a proof was finally given by Junxian Li, who just completed her doctorate at the University of Illinois, Alexandru Zaharescu (her advisor), and myself. Since I will tell you how I became interested in Ramanujan and his notebooks, part of my lecture will be historical. [PDF]

Host:  Michael Filaseta