### 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:** A 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 L_{1}-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