### 2023–2024 Academic Year

**When: **Thursday, April 18^{th} from 4:30 to 5:30 p.m.

**Where:** LeConte 103

**Speaker:** Michael Loss (Georgia Institute of Technology)

**When:** Thursday, April 11^{th} from 4:30 to 5:30 p.m.

**Where:** LeConte 103

**Speaker:** Tonghai Yang (University of Wisconsin–Madison)

**When:** Thursday, April 4^{th} 2024 from 4:30 to 5:30 p.m.

**Where: **LeConte 103

**Speaker:** Eric Chitamber (University of Illinois at Urbana-Champaign)

**When: **Friday, February 2^{nd} 2024 at 3:30 p.m.

**Where:** LeConte 103

**Speaker: **Talitha Washington (Clark Atlanta University)

**Abstract: **While data science is in high demand both in research and in the workplace, what is
needed to prepare students for a data-driven workforce remains unclear. The workforce,
including academia, industry, and government, continues to make innovative advancements
via data-driven approaches. The priorities of the workforce may differ by sector,
yet demand assurance of global competitiveness in an evolving landscape. This talk
will explore how mathematics academic programs can meet these priorities by re-envisioning
how to equip students to lead data-driven innovations in academia, industry, and government.

**When: **Thursday November 2nd, 2023 from 4:30 - 5:30pm

**Where: **LeConte 444

**Speaker:** Eitan Tadmor (University of Maryland)

** Abstract: **We introduce a new Swarm-Based Descent (SBD) method for non-convex optimization. The
swarm consists of agents, each identified with position, x, and mass, m. There are
three key aspects to the SBD dynamics: (i) persistent transition of mass from high
to lower ground; (ii) a random choice of descent marching direction, which is aligned
with the orientation of the steepest gradient descent; and (iii) a time stepping protocol,
h(x,m), which decreases with m. The interplay between positions and masses leads to
dynamic distinction between "leaders" and "explorers". Heavier agents lead the swarm
near local minima with small time steps. Lighter agents explore the landscape in random
directions with large time steps, and lead to improved position, i.e., reduce the
"loss" for the swarm. Convergence analysis and numerical simulations demonstrate the
effectiveness of SBD method as a global optimizer.

**When: **Thursday October 26th, 2023 from 4:30 - 5:30pm

**Where: **LeConte 444

**Speaker:** Yuan Gao (Purdue University)

**Abstract: **We focus on the transition path problem and mean field games for Markov jump processes
on graph and in general on any Borel space. We first formulate the transition path
problem for Markov jump processes as a stochastic optimal control problem in an infinite
time horizon. Using the Girsanov transformation for pure jump processes, we choose
the certain relative-entropy type running cost and a terminal cost for the stochastic
optimal control problem with a stopping time. Unbounded terminal cost serves as a
hard constraint, which guarantees the almost sure transition between metastable sets
and can be taken care using Gamma -convergence. We prove a closed formula solution
for optimal control computed via the discrete committor function. In the deterministic
finite time horizon, both transition path and mean field game problem can be formulated
as convex optimization for measures. Moreover, disintegration formula puts both finite
time and infinite time (stochastic) optimal control into one framework, which are
convex optimization problem for path measures.

**When: **Thursday October 12th, 2023 from 4:30 - 5:30pm

**Where: **LeConte 444

**Speaker:** Xingxing Yu (Georgia Institute of Technology)

**Abstract: **The planar Turan number of a graph *H*, *ex*_{P}* *(*n*,*H*), is the maximum number of edges in an *n*-vertex planar graph without *H* as a subgraph. We discuss recent work on *ex*_{P}* *(*n*,*H*), in particular when *H=C _{k}*, the cycle of length

*k*, including our work on

*ex*(

_{P}*n*,

*C*). We prove an upper bound on

_{7}*ex*(

_{P}*n*,

*C*) for

_{k}*k, n ≥*4, establishing a conjecture of Cranston, Lidicky, Liu, and Shantanam. The discharging method and previous work on circumference of planar graphs will be used. This is based on joint work with Ruilin Shi and Zach Walsh.

**When: **Thursday October 5th, 2023 from 4:30 - 5:30pm

**Where: **LeConte 444

**Speaker:** Chi-Wang Shu (Brown University)

**Abstract:** In scientific and engineering computing, we encounter time-dependent partial differential
equations (PDEs) frequently. When designing high order schemes for solving these
time-dependent PDEs, we often first develop semi-discrete schemes paying attention
only to spatial discretizations and leaving time *t* continuous. It is then important to have a high order time discretization to main
the stability properties of the semi-discrete schemes. In this talk we discuss several
classes of high order time discretization, including the strong stability preserving
(SSP) time discretization, which preserves strong stability from a stable spatial
discretization with Euler forward, the implicit-explicit (IMEX) Runge-Kutta or multi-step
time marching, which treats the more stiff term (e.g. diffusion term in a convection-diffusion
equation) implicitly and the less stiff term (e.g. the convection term in such an
equation) explicitly, for which strong stability can be proved under the condition
that the time step is upper-bounded by a constant under suitable conditions, the explicit-implicit-null
(EIN) time marching, which adds a linear highest derivative term to both sides of
the PDE and then uses IMEX time marching, and is particularly suitable for high order
PDEs with leading nonlinear terms, and the explicit Runge-Kutta methods, for which
strong stability can be proved in many cases for semi-negative linear semi-discrete
schemes. Numerical examples will be given to demonstrate the performance of these
schemes.

### Previous Colloquia

During this semester, some colloquia series talks will be in-person, while some 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, January 19 - 4:30 p.m. to 5:30 p.m.

Where: LC 444

Speaker: Andreas Veeser , University of Milan

Abstract: Finite elements methods are an important and well-established technique
for the solution of partial differential equations (PDE). They rely on the weak formulation
of the PDE and replace its trial and test spaces by finite-dimensional, discrete counterparts.
If these discrete spaces are contained in the original ones, the method is called
conforming, otherwise nonconforming. A finite element solution is a near best approximation
whenever its error is bounded by the best error in the discrete trial space. Near
best approximation in conforming methods was established quite early and is nowadays
part of an introductory course. Perhaps therefore surprisingly, near best approximation
results for nonconforming methods appeared only recently.

This talk wants to provide an introduction to near best approximation in nonconforming
methods. It bases upon an approach developed in collaboration with Pietro Zanotti
(University of Pavia, Italy).

Andreas will be at USC the entire week; January 16-20, 2023.

When: Monday, March 13 - 11:50 a.m. to 12:50 p.m.

Where: Petigru 213

Speaker: Jozsef Balogh, University of Illinois Urbana-Champaign

Abstract: We prove that 1-o(1) fraction of all k-SAT functions on n Boolean variables are unate (i.e., monotone after first negating some variables), for any fixed positive integer k and as n tends to infinity. This resolves a conjecture by Bollobás, Brightwell, and Leader. The proof uses among others the container method and the method of (computer-free) flag algebras.

The lecture is summarizing results of a paper of Dingding Dong, Nitya Mani, and Yufei Zhao, and a follow-up paper with additional authors Bernard Lidický and the speaker.

The talk is aimed for a general audience, including computer scientists.

When: Tuesday, March 28 - 4:30 p.m. to 5:30 p.m.

Where: LC 444

Speaker: Robert Lemke-Oliver, Tufts University

Abstract: From an algebraic perspective, the square root of 2 can be represented quite simply by saying that it's a root of the polynomial x^2-2. This is also essentially the simplest way to represent this number. Similarly, the polynomial x^3-5 is the simplest way to represent the irrational number that's a cube root of 5. But what about the polynomial x^5 - 7810*x^3 - 121055*x^2 + 2116510*x + 18532349? It can't be factored and its roots are all irrational numbers, but is this complicated polynomial really the "simplest" way to encode those roots? It shouldn't be obvious either way! In this talk, I'll tell you how with a little bit of extra information about the polynomial (its Galois group) it's often possible to encode the roots much more efficiently. Pulling back the curtain, this talk is really about studying number fields of bounded discriminant, and is based on joint work with Frank Thorne and others.

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

Where: LC 444

Speaker: Kirsten Wickelgren, Duke University

Abstract: In a celebrated paper from 1948, André Weil proposed a beautiful connection between algebraic topology and the number of solutions to equations over finite fields: the zeta function of a variety over a finite field is simultaneously a generating function for the number of solutions to its defining equations and a product of characteristic polynomials of endomorphisms of cohomology groups. The ranks of these cohomology groups are the number of holes of each dimension of the associated complex manifold. This talk will describe the Weil conjectures, some A1-homotopy theory, and then combine them to enrich the zeta function to have coefficients in a group of bilinear forms. The enrichment provides a connection between the solutions over finite fields and the associated real manifold. No knowledge of A1-homotopy theory is necessary. The new work in this talk (https://arxiv.org/abs/2210.03035) is joint with Margaret Bilu, Wei Ho, Padma Srinivasan, and Isabel Vogt.

When: Thursday, April 6 - 4:30 p.m. to 5:30 p.m.

Where: LC 444

Speaker: Iman Marvian, Duke University

Abstract: According to a fundamental result in quantum computing, any unitary transformation on a composite system can be generated using so-called 2-local unitaries that act only on two subsystems. Beyond its importance in quantum computing, this result can also be regarded as a statement about the dynamics of systems with local Hamiltonians: although locality puts various constraints on the short-term dynamics, it does not restrict the possible unitary evolutions that a composite system with a general local Hamiltonian can experience after a sufficiently long time. In this talk, I show that this universality does not hold in the presence of conservation laws and global continuous symmetries: generic symmetric unitaries on a composite system cannot be implemented, even approximately, using local symmetric unitaries on the subsystems. Based on this no-go theorem, I propose a method for experimentally probing the locality of interactions in nature. I also argue that in some cases this no-go theorem can be circumvented using ancilla qubits. For instance, any rotationally-invariant unitary on qubits can be realized using the Heisenberg exchange interaction, which is 2-local and rotationally invariant, provided that the qubits in the system interact with a pair of ancilla qubits. Furthermore, I briefly discuss qudit systems with SU(d) symmetry, and show that there is a surprising distinction between the case of d=2 and d>2. Finally, I present a general characterization of the group generated by k-local symmetric unitaries, in the case of Abelian symmetries.

### This Semester's Seminars

When: Thursday, September 15 - 4:30 p.m. to 5:30 p.m.

Where: Zoom Meeting (see description above)

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 nanocarriers 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 in silico 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 models we consider are formulated in terms of partial differential equations posed on evolving surfaces. For the numerical solution, we use a fully Eulerian hybrid (finite difference in time and trace finite element in space) discretization method. The method avoids any triangulation of the surface and uses a surface-independent background mesh to discretize the problem. Thus, our method is capable of handling problems posed on implicitly defined surfaces and surfaces undergoing strong deformations and topological transitions.

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

Where: LC 444

Speaker: Wendy Smith, University of Nebraska, sponsored by CTE

Abstract: Although education research has many implications for improving student outcomes in mathematics, more work is needed to successfully and sustainably translate research findings into practice.

This interactive colloquium will focus on what we know as a field about effective and efficient ways to improve student outcomes in mathematics, and what that could or should mean for mathematics departments. Improving student outcomes sustainably involves not just making individual, isolated changes, but involves whole departments investing in improving complex systems.

This session will seek to inspire attendees with next steps to address equitable teaching and learning mathematics in South Carolina.

Presentation PDFs:

- Lunch and Learn (11:30am): Equitable Teaching Practices: Responding to Microaggressions
- Math Department Colloquium (4:30pm): Improving Student Outcomes in Mathematics: What Do We Know? What Can We (Reasonably) Do?

When: Thursday, October 20 - 4:30 p.m. to 5:30 p.m.

Where: LC 444

Speaker: Caroline Moosmueller, University of North Carolina at Chapel Hill

Abstract: Detecting differences and building classifiers between distributions, given only finite samples, are important tasks in a number of scientific fields. Optimal transport (OT) has evolved as the most natural concept to measure the distance between distributions and has gained significant importance in machine learning in recent years. There are some drawbacks to OT: computing OT can be slow, and it often fails to exploit reduced complexity in case the family of distributions is generated by simple group actions.

If we make no assumptions on the family of distributions, these drawbacks are difficult to overcome. However, in the case that the measures are generated by push-forwards by elementary transformations, forming a low-dimensional submanifold of the Wasserstein manifold, we can deal with both of these issues on a theoretical and on a computational level. In this talk, we’ll show how to embed the space of distributions into a Hilbert space via linearized optimal transport (LOT), and how linear techniques can be used to classify different families of distributions generated by elementary transformations and perturbations. The proposed framework significantly reduces both the computational effort and the required training data in supervised learning settings. We demonstrate the algorithms in pattern recognition tasks in imaging and provide some medical applications.

This is joint work with Alex Cloninger, Keaton Hamm, Harish Kannan, Varun Khurana, and Jinjie Zhang.

When: Friday, November 11 - 4:30 p.m. to 5:30 p.m.

Where: LC 444

Speaker: Ken Ono, University of Virginia

Abstract: The theory of modular forms enjoys some of the most significant recent advances in number theory. This includes the discovery of ideas that underly the proof of Fermat’s Last Theorem, provide the framework for much of the Langlands Program, and gives glimpses of deep connections between geometry, number theory and physics. Despite these deep advances, the innocent (in appearance) Conjecture of D. H. Lehmer on the non-vanishing of Ramanujan's tau-function remains open. In this talk the speaker will tell the story of this important function, and describe recent results on variants of Lehmer’s Conjecture, where much progress has been made in the last few years.

Additional Information: Ken Ono's visit is part of the Phi Beta Kappa Visiting Scholar program.

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.

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]

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.

Host: Michael Filaseta