Math 595, Special Topics: Interacting Particle Systems

Professor:Kay Kirkpatrick
Office:231 Illini Hall
Contact:kkirkpat(at)illinois.edu
Course site:https://faculty.math.illinois.edu/~kkirkpat/595spring2017.html
Lectures: 2:00-2:30pm MWF in 445 Altgeld Hall, 2 credits for the first 8 weeks, plus 2 credits for the remainder of the semester if you register for the second course.
Office hours: Mondays 1:00-1:50 pm, and Wednesdays 3:00-3:50pm, or by appointment, in my office, 231 Illini Hall. I would be happy to answer your questions in my office anytime as long as I'm not otherwise engaged, and before and after class are good times to catch me either in my office or in the classroom.
Textbook: The main text will be chapters written by Seiringer and Solovej in the book Quantum Many Body Systems, ed. Giuliani, Mastropietro, and Yngvason, Lecture Notes in Mathematics Vol. 2051, Springer (2012) available through our library online at this link.
Grading policy: Homework/Attendance: 50% of the course grade
Midterm/Final Projects: 50%, a short midterm presentation and a full-length final presentation on a paper or section of a book closely related to the course. Some suggestions are below, but are not exhaustive. Connections to machine learning are encouraged.
Classroom policy: I support individuals with diverse backgrounds, experiences, and ideas across a range of social groups including race, ethnicity, gender identity, sexual orientation, abilities, economic class, religion, and their intersections. I expect that we will all treat each other with respect. Please contact me to request disability accommodations.


Homework (due many Fridays in class or to my mailbox in AH 250 by the end of class): You may work together or turn in your homework separately. Late homework will not be graded, so I will drop your lowest homework score.

HW #1 is due the second Friday in class: info sheet handed out on first day, also available here.

Marci Blocher can give you an override to register for the second half of the semester.

HW #2, your project proposal, is due Fri 2/10 via email. Please type this and aim for a length of about one page (your references can go into a second page), addressing each of the following:
1. In the opening, identify the paper you will focus on, your topic, and your main message, also known as a thesis statement or a motivation that your final project will address.
2. Think about your audience: your classmates, not just me; people in STEM, not just in your field. Write a paragraph addressing some or all of the following questions: Why should your audience care? What do you want them to take away from your presentation(s)? How can you clarify the benefits of your project to your audience?
3. Specifics: What kinds of audiovisual aids will you be choosing to use? Your project should have at least one item of visual interest (picture, simulation, etc.), and at least one item of technical interest (theorem, algorithm, etc.). Which two or three definitions or key ideas will you introduce to your audience? What is a good example (think n=2) that illustrates the main point of your project? Can you find a story that's related to your topic?
4. Annotated bibliography: can go beyond the 1 page limit, and should contain at least 3 references. Please annotate each reference by indicating which sections of the references are most relevant to your project and what their main messages are.

Friday Feb 17 is a class field trip to a talk in CSL B02 at 2pm.

HW #3, your midterm title and abstract, due Fri 2/24 by email. Midterm presentations will be March 3, 6, 8, and 10. Everybody, please read "How to give a good 20-minute math talk" by William Ross. If you're giving a computer talk, please test your connection to the A/V system well in advance, and read this downloadable booklet on slide design for scientific talks or "Slides are not all evil"--both written by Jean-luc Doumont.

HW #4, instead of Wed 3/1 lecture, please attend a talk of your choice, and send me a short email (by Fri 3/3 afternoon) reporting on it: the topic, one thing you liked about it, and one thing you'd do differently if you were the speaker.

HW #5, due soon after your midterm talk: Send me an email attaching your slides or notes (either a draft or an edited version is fine), and write a reflection about your talk, discussing what you thought worked well, what you would do better, and what you plan to do differently in the final presentation (if applicable).

HW #6, due Wednesday, April 12 by 5pm (extensions until Thursday fine):
1. Read or re-read the Doumont papers: downloadable booklet on slide design for scientific talks and (now "and" instead of "or" :) "Slides are not all evil".
2. Revise three of your midterm slides according to the principles that you've learned, drawing or texing up your changes. This may include finding or drawing a picture to illustrate the main point of the slide, or making the wording more efficient to reduce word-wrap. The homework that you turn in should look something like this example, with six slides: 3 originals and 3 improved versions.
3. Update your project proposal with any changes you've made since the first submission.
4. Include your plain text title and abstract (if revised; or if the same, just say so) in an email.



HW #7, in preparation for your final presentation: Read "How to Talk Mathematics" by Paul Halmos or Academic talk advice from a Berkeley CS prof.

HW #8, due 2 or 3 days after your final presentation: Send me an email with a short reflection, discussing what you thought worked well, and what you would do better, and comparing your two presentations. Please attach slides or notes.

The final presentations will run April 19 to May 3. Schedule:
Wed, April 19: Xiao Li
Fri, April 21: Jaime Thissen
Mon, April 24: Abiodun Oki
Wed, April 26: Daniel Inafuku
Fri, April 28: Destry Newton
Mon, May 1: Haojian Li
Wed, May 3: Jon Drobny



Potential papers/topics for your project (others are possible with my approval):

Chen Pavlovic 3-body boson interactions: https://arxiv.org/abs/0812.2740
Kirkpatrick Schlein Staffilani https://arxiv.org/pdf/0808.0505v2.pdf
Ben Arous Kirkpatrick Schlein https://arxiv.org/abs/1111.6999
Kirkpatrick Lenzmann Staffilani https://arxiv.org/abs/1108.6136
Kirkpatrick Zhang http://www.sciencedirect.com/science/article/pii/S0167278915301469
Frohlich Knowles Schlein Sohinger arxiv:1605.07095v2
Lewin Nam Rougerie arxiv:1410.0335v3
Rougerie survey 1507.01440
Zilber arxiv:1604.07745
Shor https://arxiv.org/abs/quant-ph/9508027
quantum typicality: https://www.dias.ie/wp-content/uploads/2012/06/lecture-4-wilde.pdf
quantum entanglement:v http://math.rutgers.edu/~oldstein/papers/GAPtyp150210.pdf
renormalization group (first chapter of textbook) and connections to deep learning: https://calculatedcontent.com/2015/04/01/why-deep-learning-works-ii-the-renormalization-group/ and https://arxiv.org/abs/1301.3124



Midterm talk schedule:
Friday, March 3:
Haojian Li
Tayyab Nawaz
Xiao Li
Mon, Mar 6:
Derek Kielty
Hadrian Quan
Katie Bolan
Wed, Mar 8:
Xiao Zhang
Jaime Thissen
Abiodun Oki
Fri, Mar 10:
Joshua Leveillee
Matthew Quiroz
Destry Newton
Mon, Mar 13:
Daniel Inafuku
Jon Drobny



Names, Titles, and Abstracts for Midterm Talks:

Haojian Li: The final presentation will be about the upper bound on the free energy of dilute Bose gases at positive temperature. Main results are from two papers, Free energy of a dilute Bose gas: Lower bound and Free energies of Dilute Bose Gas: Upper bound. The upper bound of the free energy per volume and its proof will be introduced in details. The basic idea is using variational principle to simplify the problem. Therefore, it suffices to prove an upper bound of a trial state (which would be defined during the presentation.) Since the time is limited, some parts of proof would be omitted. References: 1. Free Energies of Dilute Bose Gas: Upper Bound. Jun Yin. 2010. 2. Free Energy of a Dilute Bose Gas: Lower Bound. R.Seiringer. 2008. 3.The Mathematics of the Bose Gas and its Condensation. E.H.Lieb. R.Seiringer. 2005.

Mean Field Limit of Quantum Mechanics Derived by Different Approaches
Abstract: The many body quantum body system in mean field would be learned by using the Hartree equations and Fock space representations. Since my audience are from different academic backgoudns, the mid-term presentation will be sperated into four parts in order to make the materials more accessible. The first part is about density matrix and reduced density matrix. To help my audience understand the reduced density matrix, a simple example will be shown. The second part would focus on the introduction of basic notations and definitons, such as the definition Hamiltonian, the Hartree equations, the mean field regime, and Fock space. As for the third part, the law of large number and Theorem 1.1 will be displayed. The goal of Part 3 is to convince my audience the evolution of a wave function conseves the factorized wave functions approximately in a proper sense. The final part is the main result of [1], saying that O_t, as a random variable, converges to a Gaussian random variable.

Daniel Inafuku: Using Physics to Understand Deep Learning
Abstract: As access to big data increases, new computational methods are becoming increasingly necessary to evaluate large data sets. Machine learning is a branch of computer science that aims to provide an automated way for computer programs to extract meaningful features from data. A special type of machine learning, known as deep learning, has proven to be exceptionally successful in high-level pattern recognition tasks such as speech recognition and object labeling. However, the theoretical basis for deep learning is poorly understood. Recent attention has turned to theoretical physics to explain some of the aspects of deep learning. In particular, an important scheme often used in high-energy and condensed matter physics, called the Renormalization Group, has gained significant attention due to its ability to model the stacked architecture of deep learning schemes. In this talk, I will present recent advances in the formalization of deep learning that elucidate the connection between the Renormalization Group and deep learning. In addition, I will supplement this discussion with physical models that utilize the Renormalization Group to give an intuitive picture of this connection.

Joshua Leveillee: Machine learning on exchange-correlation functional optimization for many fermion systems in real materials
Abstract: Density functional theory calculates the interaction energy between fermions in an ionic background by integrals over the charge density of all electrons. This is a mean-field theory approach, and the explicit electron-electron interaction effects of exchange and correlation (XC) are estimated in an XC functional. XC functionals are the major source of error in DFT prediction of real material phenomena. In the work by Wellendorff et al., the GGA exchange correlation function is polynomials expanded, with the coefficients comprising the optimization parameters found by machine learning. The Tikhonov regularization method is used to minimize the cost function, while reducing over-fitting error, given a particular training set and optimization parameter space (the XC coefficients). Multiple training sets are assessed for physical accuracy. Through this method, the BEEF-vDW XC semi- empirical XC functional is developed. Assessment of the error associated with BEEF- vDW is determined by a Bayesian statistics method to compare to known-error XC functionals. The BEEF-vDW XC functional is then used in DFT to examine its predictive power for assessing reaction barriers, non-covalent forces, and surface adsorption. It is observed that BEEF-vDW is particular good at predicting the energetics of carbon monoxide chemisorption on platinum and rubidium surfaces. After the discussion of paper results, I will briefly discuss alternative methods used to increase exchange-correlation accuracy, including perturbation theory.

Xiao Zhang: Quantum Support Vector Machine: Theory and Experimental Realization
Abstract: Support vector machine (SVM) is an important method in supervised machine learning. This particular method takes into two different groups of data in a certain space and constructs a hyperplane separating them. SVM allows one to classify a given query point into one of the two groups. SVM has a solution that is polynomial in the product of the dimension of the space N and the number of training samples M. This makes it difficult for large scale simulations where both number could grow large. Recent studies have found that with quantum algorithm for solving linear system, the cost of SVM can be reduced to the logarithm of MN. In this presentation, first the concept of a support vector machine will be introduced. After that, the basic theory of using quantum computing to solve for SVM problem will be introduced, along with the reason of the reduction in cost. Finally, an example of using quantum SVM to classify hand written character "6'' and "9'' will be presented to show the validity of the method.

Derek Kielty: Exponential Decay of Eigenfunctions in Many Body Quantum Systems
Abstract: The classic exactly solvable single-particle quantum systems (eg. finite potential well and harmonic oscillator) all exhibit exponential decay of their eigenfunctions. The exactly solvable multi-particle systems (eg. hydrogen atom) also exhibits exponential decay of eigenfunctions. This inspires the question of whether this phenomenon can be proven for a large class of potentials. This question has affirmative answers in both the single and multi-particle cases. I will discuss the results and some of their implications for many-body problems.

Abiodun Oki: Field Theoretic Description of Interfacial Polycondensation
Abstract: My presentaton will focus on an approach to develop a field-theoretic based description for reaction-diffusion processes of the type: A + B => C After developing a field-theoretic based model for reaction-diffusion, I will go further to apply the model for rational design of nanofiltration and reverse osmosis membrane synthesized via interfacial polycondensation. It turns out that the aforementioned reaction-diffusion system is a generalization of the process that occurs during interfacial polycondensation although for interfacial polycondensation a high molecular weight thin film polymer is formed. This thin film is then used as a barrier for separation purposes. Interfacial polycondensation is the most commonly used approach to synthesize polymeric membranes materials used for reverse osmosis and nanofiltration. Interfacial polycondensation occurs via a non-trivial reaction-diffusion process and it is not fully understood, it is a reaction between two polymer precursors at the interface of two immiscible liquid phases. A better understanding of interfacial polycondensation will make it possible to rationally design materials for separations. The main technical result will be how to use field-theoretic approaches to describe reaction diffusion processes and hence interfacial polycondensation. I intend to expand on how to map a master equation for a reaction-diffusion process for arbitrary occupation numbers into a Fock-space formalism.Future aspect of this work will include the use of a cox representation approach for solving the inverse problem of learning a stochastic reactionñdiffusion process from data.In this way, we can develop a famework for in silico rational design of separation membranes.Audiovisuals this project will contain include pictures describing the interfacial polycondensation process and the principle of operation of nanofiltration and reverse osmosis membranes.

Tayyab Nawaz: Shor's Algorithm for Quantum Factorization
Abstract: In quantum computing, Shor's algorithm solves the quantum factorization problem in polynomial time. The RSA algorithm in cryptography uses factorization of large numbers for encryption/decryption. I will explain the Shor algorithm using the problem of prime factorization for some simple integer problem and will explain it's importance in cryptography. In this talk, I am planning to give some simple examples to explain the main idea.

Xiao Li: Spin Glasses and Information Processing
Abstract: In this talk I will introduce the concept of spin glasses as examples of interacting particle systems. I hope to get to talk about the replica method and its significance. I will give a quick introduction to Error Correcting codes and basic information theoretical ideas like the Shannon bond. Finally I will compare Error Correcting codes with spin glasses and try to discuss the spin glass representation of Error Correcting codes.

Katie Bolan: GAP Distribution of Conditional Wavefunctions for Entangled Systems
abstract: The Gaussian adjusted projected (GAP) measure on a Hilbert space H is obtained by starting with a Gaussian measure G(p), with mean 0 and covariance matrix p. The adjusted Gaussian measure is then defined by GA(p)(dy) = norm(y)^2 G(p)(dy), for a Gaussian-distributed random vector . This distribution is then projected onto the unit sphere in H . Considering a system in state y in H, suppose H is the product of two entangled subspaces, H1 and H2. A conditional wavefunction y1 for subsystem 1 can be found in the following way. A random element bj of an orthonormal basis of H2 is chosen. Then y1 is the normalized inner product of bj and y, the wavefunction of the entire system. The distribution of conditional wavefunctions, averaged over all orthonormal bases, is the GAP distribution. Therefore, the GAP measure can be used to describe this entangled subsystem.

Jon Drobny: Classical, Semi-Classical, and Quantum Simulations of the Overdense One Component Plasma Ground State Energy
Overdense plasma is an extreme state of matter rich with complex physics. Overdense plasmas are predicted to exist in Inertial Confinement Fusion (ICF) devices, Jupiter's core, and the crusts of neutron stars. Overdense plasmas, composed of charge particles that interact via Coulombic forces, exhibit strongly coupled behavior. Strongly coupled systems are not easily analyzed through analytic methods. Instead, they are best analyzed using computational simulation techniques such as Path Integral Monte Carlo, Molecular Dynamics (MD), and variational methods, depending on the density and temperature of the system. At very high densities, overdense plasmas can crystallize into a state known as a Wigner crystal. At low temperatures, quantum effects begin to play a role in the system dynamics, leading to changes in the properties of the crystalline state. The simplest overdense plasma system is the One Component Plasma (OCP), composed of a single charged species in a neutralizing background. In this study, the ground state energy per particle of the low-temperature Wigner crystal for the OCP is found using three techniques: classical MD, semi-classical MD with quantum modifications to the pair interaction potential, and a variational method using a Gaussian trial wavefunction to represent electron lattice sites. Additional results from the MD simulations include a prediction of crystal structure (simple cubic for the classical case, results for the semi-classical case to come) and radial distribution functions.

Jaime Thissen: An Overview of Polynomial-time Algorithms Utilizing Prime-Factoring
Computer science is in certain respects an emerging field with many fundamental challenges. One such challenge is the overall efficiency of computing systems for more effective machine learning. This is especially apparent in the gap between classical and quantum computing. In theory polynomial-time algorithms are necessary for the establishment of quantum computing. Quantum computing could enable substantial advances in machine learning. The hypothesis of this overview further states these algorithms do not have to be conclusively solved in order to operate, namely in terms of assessing error. The asymptotic “process for creating O((log n)2(log log n)(log log log n)) in conjunction with the polynomial (in log n) for post-processing on a classical computer” as described by Shor is examined in further detail. The particular method used here is prime-factoring. Anticipated applications of this overview include advances in computer science (namely machine learning) as well as cryptology. The less steps taken in any algorithm will allow for faster processing time and less possibility of error. Future work may include combining advanced algorithms with advances in materials science.

Destry Newton: Truly Quantum Neural Networks
Due to the difficulty of designing complex automated tasks by hand, it is becoming increasingly necessary to rely on machine learning methods to learn the task instead. There exists a biologically inspired model, called a neural network, used in a wide variety of applications to solve such problems. Neural networks pose their own difficulties though, requiring potentially vast amounts of data storage and a not insignificant amounts of time to learn their task. Quantum computers have been shown to be far better than classical computers at some tasks, massively reducing the amount of time and data necessary for some computations. The field of Quantum Neural Networks has emerged to examine the possibility that neural networks running in quantum systems could alleviate some of issues that neural networks currently posses. In this talk, I will present the basics of both quantum computation and neural networks, followed by an overview of the field of quantum neural networks and some of the current prominent models(and weather they are actually quantum or not).

Hadrian Quan: Bosons, Gibbs Measures, and The Variational Principle
In this talk, we'll discuss recent results of Lewin et. al on how to derive non-linear Gibbs measures from many-body quantum mechanics. The focus of this talk will be the role the variational principle plays in their argument, and how it bridges a gap between mathematics and the natural sciences. Time permitting, we'll discuss the dynamics these Gibbs measures are invariant under: namely those of Non-linear Schrodinger flow.


Syllabus

This is a graduate course on interacting particle systems. This course will provide an introduction to the mathematical theory of quantum many-body systems, including the many-particle Schrodinger equation and its various limits such as the Hartree and Gross-Pitaevskii (cubic nonlinear Schrodinger) equations. Some possible topics, if time permits and if there is interest among the audience, include: mean-field interactions, Coulomb interactions, localizing mean-field-type interactions, quantum probability, and quantum groups. Prerequisites: some linear algebra, probability, and PDE or quantum mechanics--or a willingness to catch up through studying.

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