## Spring 2022

### This Semester's Colloquia

During this semester, the colloquia series will be held via Zoom. Note that all times are given in Eastern US time. We switch from EST (UTC – 5h) to EDT (UTC – 4h) on March 13, 2022.

Zoom Meeting ID: 946 6437 7663

Passcode: 643639

**When: **Thursday, February 10, 2022 - 4:30 p.m. to 5:30 p.m.

**Where: **Zoom Meeting (see description above)

**Speaker: **Lin Lin, University of California at Berkeley

**Abstract:** The two "quantum supremacy" experiments (by Google in 2019 and by USTC in 2020, respectively)
have brought quantum computation to the public's attention. In this talk, I will first
cover some backgrounds in quantum computation. I will then introduce efficient quantum
algorithms for finding the smallest eigenvalue of a Hermitian matrix (also called
the ground state energy), which has wide applications in quantum physics, quantum
chemistry, materials science etc. I will discuss two classes of quantum algorithms:
a near-optimal algorithm that almost saturates complexity lower bounds and is suitable
for full-scale fault-tolerant quantum computers, and a simpler algorithm more suitable
for early fault-tolerant quantum computers.[1] L. Lin, Y. Tong, Near-optimal ground state preparation, Quantum, 4, 372, 2020[2] L. Lin, Y. Tong, Heisenberg-limited ground state energy estimation for early fault-tolerant
quantum computers, PRX Quantum. in press

**When: **Thursday, March 3, 2022 - 4:30 p.m. to 5:30 p.m.

**Where: **Zoom Meeting (see description above)

**Speaker: **Professor, Yihong Du, Fellow of the Australia Academy of Science, University of New England, Australia

**Abstract:** In this talk I will discuss some of the mathematical theories on nonlinear parabolic
equations motivated by the desire to better understand the propagation phenomena.
The talk will start with a brief review of classical works of Fisher, Kolmogorov-Petrovsky-Piskunov
and Aronson-Weinberger, and then it will focus on recent results on free boundary
models with local as well as nonlocal diffusion, which are variations of the reaction-diffusion
models in the classical works.

** **

**When: **Thursday, April 21 - 4:30 p.m. to 5:30 p.m.

**Where: **Zoom Meeting (see description above)

**Speaker:** Tryphon Georgiou, University of California, Irvine

**Abstract:** The recent confluence of three subjects, Stochastic Control, Optimal Mass Transport,
and Stochastic Thermodynamics, has allowed deeper understanding of the mechanism by
which physical contraptions (whether engineered or biological) can transform heat
gradients and information into useful work. Our goal in the talk is to overview some
of these developments and highlight the geometric framework that allows quantitive
assessments on the performance that stochastic thermodynamic engines are capable of.
We will then specifically focus on Brownian gyrating engines that consist of over-damped
particles that are fed by sources of stochastic excitation and reside in a controlled
potential.

The talk is based on joint works with Rui Fu (UCI), Olga Movilla (UCI), Amir Taghvaei (UCI) and Yongxin Chen (GaTech). Research funding by NSF and AFOSR is gratefully acknowledged.

** **

### This Semester's Seminars

- Algebra, Geometry, and Number Theory Seminar
- Analysis Seminar
- Applied and Computational Mathematics Seminar
- Applied and Computational Mathematics Student Seminar
- Discrete Math and Combinatorics Seminar
- Graduate Colloquium
- Number Theory Seminar

### Previous Colloquia

**When: **Thursday, September 23, 2021 - 4:30 p.m. to 5:30 p.m.

**Where: **Zoom Meeting (see description above)

**Speaker: **Long Chen, University of California at Irvine

**Abstract: **A Hilbert complex is a sequence of Hilbert spaces connected by a sequence of closed
densely defined linear operators satisfying the property: the composition of two consecutive
maps is zero. The most well-known example is the de Rham complex involving grad, curl,
and div operators. A finite element complex is a discretization of a Hilbert complex
by replacing infinite dimensional Hilbert spaces by finite dimensional subspaces based
on a mesh of the domain. Usually inside each element of the mesh, polynomial spaces
are used and suitable degree of freedoms are proposed to glue them to form a conforming
subspace. The finite element de Rham complexes are well understood and can be derived
from the framework Finite Element Exterior Calculus (FEEC).

In this talk, we will construct more finite element complexes: the Hessian complex, the elasticity complex, and the divdiv complex. We first give polynomial complexes and Koszul type complexes, which leads to decompositions of polynomial spaces. We then characterize trace operators using Green’s identity as the traces on face and edges plays an important role on the design of degree freedoms. We construct conforming finite elements for tensor functions with extra requirement: symmetric or traceless. We also show the constructed finite element spaces form a complex.

The constructed finite element complexes will have application in the numerical simulation of the biharmonic equation, the linear elasticity, and the general relativity etc.

This is a joint work with Xuehai Huang from Shanghai University of Finance and Economics.

**When: **Thursday, October 14, 2021 - 4:30 p.m. to 5:30 p.m.

**Where: **Zoom Meeting (see description above)

**Speaker: **Leo Goldmakher

**Abstract:** A remarkable theorem due to Khovanskii asserts that for any finite subset A of an
abelian semigroup, the cardinality of the h-fold sumset hA grows like a polynomial
for all sufficiently large h. However, neither the polynomial nor what sufficiently
large means are understood in general. In joint work with Michael Curran (Oxford),
we obtain an effective version of Khovanskii's theorem for any subset of \(\mathbb{Z}^d\)
whose convex hull is a simplex; prior to our work such results were only available
for d=1. Our approach also gives information about the structure of hA, answering
a question posed by Granville and Shakan.

**When: **Thursday, October 21, 2021 - 4:30 p.m. to 5:30 p.m.

**Where: **Zoom Meeting (see description above)

**Speaker: **Robin NeuMayer

**Abstract:** Among all subsets of Euclidean space with a fixed volume, balls have the smallest
perimeter. Furthermore, any set with nearly minimal perimeter is geometrically close,
in a quantitative sense, to a ball. This latter statement reflects the quantitative
stability of balls with respect to the perimeter functional. We will discuss recent
advances in quantitative stability and applications in various contexts. The talk
includes joint work with several collaborators and will be accessible to a broad research
audience.** **

** **

**When: **Thursday, November 4, 2021 - 4:30 p.m. to 5:30 p.m.

**Where: **Zoom Meeting (see description above)

**Speaker: **Kevin Buzzard

**Abstract:** We all know that computers can be used to calculate. What is less well-known (at least
in mathematics departments) is that nowadays they can be used to reason, that is,
to state and prove mathematical theorems, and to check that proofs are valid. I will
talk about how people around the world are using the Lean theorem prover to teach
mathematics in new ways, and to engage with modern research mathematics in new ways.
Rest assured — computers will not be automatically proving the Riemann Hypothesis
any time soon. However, it is not unreasonable to expect that as this technology develops
(and it's developing fast) it will have an impact on how humans do mathematics, just
as digitising music had an impact on how humans stored and consumed it. I will give
an introduction to, and overview of, the area of computer theorem proving. I'll assume
some general mathematical knowledge, but no background in computers will be assumed
at all.

**When: **Thursday, November 11, 2021 - 4:30 p.m. to 5:30 p.m.

**Where: **Zoom Meeting (see description above)

**Speaker: **Felix Leditzky

**Abstract:** Symmetries are a powerful tool in mathematical physics, as they typically simplify
the description of physical processes. In quantum information theory, we study the
information-processing capabilities of quantum systems, for which two particularly
relevant symmetries are unitary and permutation symmetry. In particular, we are interested
in the natural representations of these symmetry groups on tensor products of a fixed
Hilbert space modeling the (multipartite) quantum system of interest. In this situation,
Schur-Weyl duality gives us a powerful framework to study quantum information-theoretic
resources (such as entangled states or quantum channels) that are invariant under
both group actions. I will first give an overview of this technique using the well-known
example of estimating the spectrum of a quantum state. Then, I will focus on a variant
of quantum teleportation called "port-based teleportation", where these representation-theoretic
methods allow us to describe the structure and asymptotic behavior of optimal protocols.

** **

**When: **Thursday, March 11, 2021 - 4:30 p.m. to 5:30 p.m.

**Where: **Zoom Meeting (see description above)

**Speaker: **Robert Lipton, Louisiana State University

**Abstract**: A hallmark of fracture modeling using non-local models in computation is the emergence
of cracks simultaneously with elastic deformation. Here, interactions between nearby
points result in global consequences like the emergence of fracture surfaces. Emergent
phenomena can be modeled non-locally and examples include motion of flocks of birds
modeled through the Cuker Smale model. We provide a mathematically well posed non-local
model for calculating dynamic fracture. The force interaction is derived from a double
well strain energy density function. The fracture set emerges from the model and is
part of the dynamics. The material properties change in response to evolving internal
forces eliminating the need for a separate phase field to model the fracture set.
In the limit of zero nonlocal interaction, it is seen that the model reduces to a
sharp crack evolution characterized by the classic Griffith free energy of brittle
fracture with elastic deformation satisfying the linear elastic wave equation off
the crack. The non-local model is seen to encode the well-known kinetic relation between
crack driving force and crack tip velocity. A rigorous connection between the nonlocal
fracture theory and the wave equation posed on cracking domains given in Dal Maso
and Toader is found.

**When: **Thursday, April 15, 2021 - 4:30 p.m. to 5:30 p.m.

**Where: **Zoom Meeting (see description above)

**Speaker: **Ronald DeVore, Texas A&M University

**Abstract:** Deep Learning is much publicized and has had great empirical success on challenging problems in learning. Yet there is no quantifiable proof of performance
and certified guarantees for these methods. This talk will give an overview of Deep
Learning from the viewpoint of mathematics and numerical computation.

**When: **Thursday, April 15, 2021 - 4:30 p.m. to 5:30 p.m

**Where: **Zoom Meeting (see description above)

**Speaker: **Alicia Dickenstein, University of Buenos Aires

**Abstract: **I will try to show in my lecture that the question in the title has a positive answer,
summarizing recent mathematical results about signaling networks in cells obtained
with algebro-geometric tools.

** **

**When: **Thursday, April 22, 2021 - 4:30 p.m. to 5:30 p.m.

**Where: **Zoom Meeting (see description above)

**Speaker: **Ed Barnes, Virginia Tech

**Abstract:** Future technologies such as quantum computing, sensing and communication demand the
ability to control microscopic quantum systems with unprecedented accuracy. This task
is particularly daunting due to unwanted and unavoidable interactions with noisy environments
that destroy quantum information through decoherence. I will present a new theoretical
framework for deriving control waveforms that dynamically combat decoherence by driving
qubits in such a way that noise effects destructively interfere and cancel out. This
theory exploits a rich geometrical structure hidden within the time-dependent Schrödinger
equation in which quantum evolution is mapped to geometric space curves. Control waveforms
that suppress noise can be obtained by drawing closed curves and computing their curvatures.
I will show how this can be done for single- and multi-qubit systems.

** **

**When: **Thursday, September 24, 2020 - 4:30 p.m. to 5:30 p.m.

**Where: **Zoom Meeting (see description above)

**Speaker: **Stanley Osher, University of California - Los Angeles

**Abstract**: Mean field games play essential roles in AI, 5G communications, unmanned aerial vehicle
path planning, social norms, and controlling natural disasters, such as COVID 19.
In this talk, we present several results by our MURI team in the year 2019-2020. We
designed fast and reliable numerical algorithms with connections to AI and machine
learning, and formulated models in mean-field inverse problems, velocity control for
massive rotary-wing UAV’s, controlling COVID 2019 pandemic spreading, etc. Several
numerical examples and engineering experiments will be presented. Future directions
will be discussed. This is based on a joint work with many people at UCLA, University
of South Carolina (Wuchen Li who just moved to UofSC), University of Houston, and
Princeton University

**YouTube**: UofSC Math Colloquium talk on Sep 24

**When: **Thursday, October 1, 2020 - 4:30 p.m. to 5:30 p.m.

**Where: **Zoom Meeting (see description above)

**Speaker: **Jeremy Rouse, Wake Forest University

**Abstract**: To classify mathematical objects, mathematicians create invariants: functions f
defined on the objects that one seeks to classify, so that if A and B are isomorphic
objects, then f(A) = f(B). I will give examples of several situations where these
invariants fail to classify number theoretic objects, as well as give a discussion
of two reasons why these invariants do not suffice. Most of the talk will consist
of examples, including non-isometric lattices with the same theta function, non-isomorphic
number fields with the same Dedekind zeta function, non-equivalent trinomials defining
the same number field, and further examples involving elliptic curves and modular
curves.

**YouTube:** UofSC Math Colloquium talk on Oct 1

**When: Monday**, October 12, 2020 - 4:30 p.m. to 5:30 p.m.

**Where: **Zoom Meeting (see description above)

**Speaker:** Ricardo Nochetto, University of Maryland

**Abstract**: We analyze the Oliker-Prussner method and a two-scale method for the Monge-Ampere
equation with Dirichlet boundary condition, and explore connections with a Bellman
formulation. We also study a two-scale method for a fully nonlinear obstacle problem
associated with convex envelopes. We derive pointwise error estimates that rely on
the discrete Alexandroff maximum principle and the geometric structure of these PDEs
for both classical and non-classical solutions.

**YouTube**: UofSC Math Colloquium talk on Oct 12

**When: Monday**, October 19, 2020 - 4:30 p.m. to 5:30 p.m.

**Where: **Zoom Meeting (see description above)

**Speaker: **Xiaolin Li, Stony Brook University

**Abstract**: Front tracking is a Lagrangian method to model the fluid interface problems, This
method has been used to study the fluid interface instabilities and phase transition
problems. Recently, we have also applied it to the fluid-structure interaction problem.
In this talk, I will introduce a mesoscale dual-stress spring-mass model based on
Rayleigh-Ritz analysis to mimic the fabric surface as an elastic membrane using the
front tracking data structure and functions. Our model is coupled with both incompressible
and compressible fluid solvers through the immersed boundary and impulse method. We
apply this method to the simulation of parachute inflation. I will discuss both the
numerical and physical aspects of this project, including numerical stability, verification
and validation study, porosity modeling, and coupling with turbulence model in the
simulation.

**YouTube**: UofSC Math Colloquium talk on Oct 19

**When: **Thursday, November 5, 2020 - 4:30 p.m to 5:30 p.m.

**Where: **Zoom Meeting (see description above)

**Speaker:** Elisenda Grigsby

**Abstract**: One can regard a neural network as a particular type of function \( F: \mathbb{R}^n
\rightarrow \mathbb{R}^m \) , where \( \mathbb{R}^n \) is a (typically high-dimensional)
Euclidean space parameterizing some data set, and the value, \( F( \mathbf{x} ) \)
, of the function on a data point \( {\bf x} \) is used to predict the answer to a
question of interest. For example, when the question of interest is a binary classification
task (e.g., "Is this e-mail spam?"), the neural network output is 1-dimensional, and
the neural network partitions the domain into decision regions labeled "yes" or "no"
depending on whether they are in the super-level or sub-level set of a chosen threshold,
\( t \) .

It is a classical result in the subject that a sufficiently complex neural network can approximate any function on a compact set. In 2017, J. Johnson and B. Hanin-M. Sellke independently proved that universality results of this kind depend on the architecture of the neural network (the number and dimensions of its hidden layers). Their argument(s) were novel in that they provided explicit topological obstructions to representability of a function by a neural network, subject to certain simple constraints on its architecture. I will begin by telling you just enough about neural networks to understand and appreciate their result. Then I will describe a joint on-going project with K. Lindsey aimed at developing a general framework for understanding how the architecture of a neural network constrains the topological features of its decision regions.

**When: **Thursday, November 12, 2020 - 4:30 p.m. to 5:30 p.m.

**Where: **Zoom Meeting (see description above)

**Speaker: **Matthew P. A. Fisher, University of California - Santa Barbara

**Abstract**: The inexorable growth of non-local quantum entanglement is the key feature that
distinguishes quantum from classical systems. Monitoring an open system (by making
projective measurements) can compete against entanglement growth, leading to a many-body
quantum Zeno effect. A hybrid quantum circuit model consisting of both unitary gates
and projective measurements exhibits a quantum dynamical phase transition between
a weak measurement phase and a quantum Zeno phase. Detailed properties of the weak
measurement phase - including relations to quantum error correcting codes - and of
the critical properties of this novel quantum entanglement transition will be described.

**When: **Thursday, February 20, 2020 - 4:30 p.m. to 5:30 p.m.**Where: **LeConte 412 **(map)**

**Speaker: **David Galvin, University of Notre Dame

**Abstract**: The Stirling numbers of the second kind, introduced in 1730, arise in many contexts
— combinatorial, analytic, algebraic, probabilist, …. I’ll introduce these versatile
numbers, and describe some of their interpretations and applications.

The standard combinatorial interpretation of the Stirling numbers involves set partitions,
and this interpretation has a natural generalization to graphs. I’ll discuss an application
of this generalization to a problem coming from the Weyl algebra (the algebra on alphabet
\( \{x, D\} \) with the single relation \( Dx=xD+1 \) ). This is joint work with Hilyard
and Engbers. [PDF]

**When: CANCELLED. Postponed until Fall 2020. Details TBA.****Where: **LeConte 412 **(map)**

**Speaker: **Xiaolin Li, Stony Brook University

**Abstract**: In this talk, I will introduce a mesoscale dual-stress spring-mass model based on
Rayleigh-Ritz analysis to mimic the fabric surface as an elastic membrane using the
front tracking data and function structures. Our model is coupled with both incompressible
and compressible fluid solvers through the immersed boundary and impulse method. We
apply this method to the simulation of parachute inflation. I will discuss both the
numerical and physical aspects of this project, including numerical stability, verification
and validation study, porosity modeling, and coupling with turbulent flow in the simulation.
[PDF]

**When: ****CANCELLED. Postponed until Fall 2020. Details TBA.**

**Where: **LeConte 412 **(map)**

**Speaker: **Jeremy Rouse, Wake Forest University

**Abstract**: TBA

**When: CANCELLED. Postponed until Fall 2020. Details TBA.****Where: **LeConte 412 **(map)**

**Speaker:** Annalisa Quaini, University of Houston

**Abstract**: Membrane fusion is a potentially efficient strategy for the delivery of macromolecular
therapeutics into the cell cytoplasm. However, existing nano-carriers formulated to
induce membrane fusion suffer from a key limitation: the high concentrations of fusogenic
lipids needed to cross cellular membrane barriers lead to toxicity in vivo. To overcome
this limitation, we are developing complimentary in silico and in vitro models that
will explore the use of membrane phase separation to achieve efficient membrane fusion
with minimal concentrations of fusion-inducing lipids and therefore reduced toxicity.
The in silico research will be based on a novel multiphysics model formulated in terms
of partial differential equations posed on evolving surfaces. [PDF]

**When: **Thursday, April 23, 2020 Time TBA**Where:** TBA

**Speaker:** Dr. Talitha Washington of Howard University and the NSF

**Abstract:** TBA

**When: ****CANCELLED. Postponed until Fall 2020. Details TBA.**

**Where: **LeConte 412 **(map)**

**Speaker: **Anthony Várilly-Alvarado, Rice University

**Abstract:** TBA

**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:** Monday, December 2, 2019 - 4:00 p.m. to 5:00 p.m.**Where:** LeConte 405 **(map)**

**Speaker:** Wei Zhu, Duke University

**Abstract:** With the explosive production of digital data and information, data-driven methods,
deep neural networks (DNNs) in particular, have revolutionized machine learning and
scientific computing by gradually outperforming traditional hand-craft model-based
algorithms. While DNNs have proved very successful when large training sets are available,
they typically have two shortcomings: First, when the training data are scarce, DNNs
tend to suffer from overfitting. Second, the generalization ability of overparameterized
DNNs still remains a mystery despite many recent efforts.

In this talk, I will discuss two recent works to “inject” the “modeling” flavor back
into deep learning to improve the generalization performance and interpretability
of DNNs. This is accomplished by deep learning regularization through applied differential
geometry and harmonic analysis. In the first part of the talk, I will explain how
to improve the regularity of the DNN representation by imposing a “smoothness” inductive
bias over the DNN model. This is achieved by solving a variational problem with a
low-dimensionality constraint on the data-feature concatenation manifold. In the second
part, I will discuss how to impose scale-equivariance in network representation by
conducting joint convolutions across the space and the scaling group. The stability
of the equivariant representation to nuisance input deformation is also proved under
mild assumptions on the Fourier-Bessel norm of filter expansion coefficients.

**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]

**When:** Friday, December 6, 2019 - 2:30 p.m. to 3:30 p.m.**Where:** LeConte 412 **(map)**

**Speaker:** Wuchen Li, UCLA

**Abstract:** Nowadays, optimal transport, i.e., Wasserstein metrics, play essential roles in data
science. In this talk, we briefly review its development and applications in machine
learning. In particular, we will focus its induced optimal control problems in density
space and differential structures. We introduce the Wasserstein natural gradient in
parametric models.

The Wasserstein metric tensor in probability density space is pulled back to the one
on parameter space. We derive the Wasserstein gradient flows and proximal operators
in parameter space. We demonstrate that the Wasserstein natural gradient works efficiently
in learning, with examples in Boltzmann machines, generative adversary networks (GANs),
image classifications, and adversary robustness etc.

**When:** Monday, December 9, 2019 - 4:00 p.m. to 5:00 p.m.**Where:** LeConte 405 **(map)**

**Speaker:** Lise-Marie Imbert-Gerard, University of Maryland

Abstract: Trefftz methods rely, in broad terms, on the idea of approximating solutions to PDEs using basis functions which are exact solutions of the Partial Differential Equation (PDE), making explicit use of information about the ambient medium. But wave propagation problems in inhomogeneous media are modeled by PDEs with variable coefficients, and in general no exact solutions are available. Generalized Plane Waves (GPWs) are functions that have been introduced, in the case of the Helmholtz equation with variable coefficients, to address this problem: they are not exact solutions to the PDE but are instead constructed locally as high order approximate solutions. We will discuss the origin, the construction, and the properties of GPWs. The construction process introduces a consistency error, requiring a specific analysis.

**When:** Friday, December 13, 2019 - 4:00 p.m. to 5:00 p.m.**Where:** LeConte 405 **(map)**

**Speaker:** Bao Wang, UCLA

Abstract:

Deep learning achieves tremendous success in image and speech recognition and machine translation. However, deep learning is not trustworthy.

- How to improve the robustness of deep neural networks? Deep neural networks are well known to be vulnerable to adversarial attacks. For instance, malicious attacks can fool the Tesla's self-driving system by making a tiny change on the scene acquired by the intelligence system.
- How to compress the high-capacity deep neural networks efficiently without loss of accuracy? It is notorious that the computational cost of inference by the deep neural network is one of the major bottlenecks for applying them to mobile devices.
- How to protect the private information that is used to train the deep neural network? Deep learning-based artificial intelligence systems may leak the private training data. Fredrikson et al. recently showed that a simple model-inversion attack can recover the portraits of the victims whose face images are used to train the face recognition system.

In this talk, I will present some recent work on developing PDE principled robust neural architecture and optimization algorithms for robust, accurate, private, and efficient deep learning. I will also present some potential applications of the data-driven approach for bio-molecule simulation.

**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

**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