Courses: past

Registration for this course is not currently available.

Discrete optimization focuses on developing efficient methods to determine the maximum or minimum value of a function over a finite (discrete) domain. This course will cover a wide range of topics in discrete optimization which may include linear programming, semi-definite programming, dynamic programming, matroids, combinatorial algorithms, duality, hardness reductions, among others. We will also see many interesting applications of tools from Discrete Optimization to problems in combinatorics and other areas of mathematics and computer science.

Course Webpage

This course will have an accompanying webpage

• https://extremalcombinatorics.com/optimization/

Materials related to the course, links and other updates will be posted to the course webpage as the course proceeds.

Class Schedule

• Monday, Thursday 1:00-2:20pm (PT)

Remote Access

Remote access for this course will be provided via zoom. This course will be taught from the UVic Multiaccess classroom HHB 110. The room is equipped with multiple cameras in the ceiling which can capture two blackboard areas and TV screens that can be used to show the Zoom gallery. A demonstration of this system can be seen in the instructor’s existing Extremal Combinatorics Network Wide Course playlist. Notes and other course related material will be made available on the instructor’s website (see e.g. notes for Extreemal Combinatorics).

Lectures will also be live-streamed on the instructors YouTube channel and also be available to view there asynchronously.

Availability

This course may be open to students from universities outside of the PIMS network, and those coming from industry/government.

Registration for this course is not currently available.

This course is an introduction to the theory of elliptic curves and modular forms at the graduate level. Elliptic curves will be introduced through both their classical analytic construction over the real and complex numbers and their algebraic realizations via normal forms over arbitrary fields. Moduli and monodromy considerations lead us to study the special role of the elliptic modular group SL(2,Z) and the crucial notions of modular functions and forms. Studying torsion points and level structure then motivates the extension to finite index subgroups and the theory of modular curves. Throughout the course, there will be an emphasis on hands-on explicit computations. Directed by the instructor, each student will complete a final project, presentation, and paper. Possible topics could include post-quantum elliptic curve cryptography, applications in string theory, geometry of elliptic modular surfaces, features of periods and Picard-Fuchs operators, etc.

Class Schedule

• Tuesday and Thursday, 16:00 - 17:20 MST

Remote Access

The course will be taught over Zoom using a tablet and shared screen. Lecture notes will be written out live on a tablet. There will also be pre-prepared slides on certain topics. The in-class lecture notes will be saved and distributed as .pdf files. There will be a course webpage to host all of these plus additional course materials and readings.

The format will be Zoom based, with videos on. Breakout rooms will be used periodically for small group work. Students will be encouraged to “raise hands” with questions at any time.

Availability

This course may be open to students from universities outside of the PIMS network, and those coming from industry/government.

Registration for this course is not currently available.

This graduate course is designed to provide graduate students with key concepts and practical applications in Biophysics, with an emphasis on the quantitative tools as they are used in current research. Biophysics is a highly interdisciplinary field—the researchers who attend the annual Biophysical Society meeting, for example, come from departments spanning all of the STEM disciplines. Nevertheless, they share a common interest to establish a quantitative understanding of living matter. Despite growing interest however, a gap remains in graduate training to prepare students to contribute effectively to this broad and rapidly evolving field. This course aims to address this gap by covering both foundational and advanced concepts and applications that are commonly used by practicing biophysicists today. The structure of the course will follow selected advanced material from Physical Biology of the Cell by Rob Phillips, Jane Kondev, Julie Theriot, and Hernan G. Garcia. Each topic will be introduced conceptually, developed mathematically, and explored through real biological case studies using both textbook material and current literature. Given student interest, the course may include interviews with leading biophysicists on their recent published work. Topics will include:

• Diffusion problems in biology
• Enzymatic reactions including ODEs, diffusion-limited reactions, and Michaelis-Menton reactions
• Statistical mechanics as it applies to Biology, including Gibbs free energy of biochemical reactions
• Liquid-liquid phase separation, and its role in the cell and in transcription
• Polymer physics; DNA looping, persistence length, polymer entropy
• Heterogeneous mixtures and osmotic pressure
• Quantitative analysis of genetic networks
• Expression distributions of transcription and translation
• Phase portrait analysis and stability/metastability of cellular states
• Genetics of enhancers – from a biophysical perspective
• Pattern formation including Turing patterns, symmetry breaking in an embryo
• Quantitative genomics (time permitting)

Class Schedule

• Wed, Fri 10am-11:30am (classes will begin Sept. 10)

Remote Access

Remote participation will be via zoom. Lectures will also be recorded and shared via UBC’s media capture system Panopto. Annotated notes on pre-distributed PDF slides are made during class using an iPad, recorded in real time, and uploaded to UBC’s Canvas server after class, along with links to the lecture recording.

Availability

This course may be open to students at universities outside of the PIMS network.

Grading

There will be no Final exam, instead there will be a final project. I’ve tentatively planned for an in-class midterm, but as a directed studies class, this may become a “take-home” midterm. For percentages, here is a tentative breakdown:

Homework (45%)

Homework due each Sunday at 11:59 pm.

Homework is assigned a week prior to the due date. I will allow late homework, however a 10% penalty per day after the due date is applied to late HW. There will be ~11 homework assignments. Assignments are roughly the same length; The points for each assignment will be weighted proportional to the number of problems in it.

Midterm (20%)

The midterm will take place on evening of Nov 13 or Nov 14 at 6 pm.

The content cut-off for the material will be up and including the Friday before midterm break (Nov 7) and the format will be discussed in class in advance. Email me by Friday Sept 12 if you have a conflict (steve@phas.ubc.ca). The in-class midterm may be replaced with a take-home midterm. |

Final Project (20%)

Date and time the final project is due will be announced, but it will be some time during the Dec exam period. We will discuss how to best design these projects to be most beneficial to everyone.

Guest speaker interviews and Lab Tutorial (15%)

I am thinking of including interviews of postdocs or professors, as I have done in the past. If we do this, we will read an assigned publication by a guest scientist or their lead first-author student/PDF, and we will then interview that scientist/student on the paper. Both general and specific questions may be asked during each interview. However, technical questions specific to the paper are essential for credit. The number of interviews we have is TBD.

One of your planned assignments will also be to complete a lab tutorial involving some more open-ended questions about experimental observations, which will be described further in Module 7 of the course.

Textbook

The textbook for Physics 305 is Physical Biology of the Cell (2nd edition), by Rob Phillips, Jane Kondev, Julie Theriot, and Hernan Garcia. It is available from multiple resources including Amazon as either a paperback or eTextbook. Downloadable resources for the text are here. We will also use material from select journal sources.

Registration for this course is not currently available.

This course is a bridge into the machine learning literature for graduate students in mathematics. Compared to existing course offerings in our neighbouring departments (mainly https://www.cs.ubc.ca/~dsuth/532D/23w1 (https://www.cs.ubc.ca/~dsuth/532D/23w1)) we will assume that you know somewhat more analysis, but prior coding experience will not be required. Briefly, the learning objectives are:

• understand the different “learning paradigms” considered in ML (supervised learning, unsupervised learning, reinforcement learning, etc.) and their relation with existing statistical theory
• be comfortable with mathematical tools (eg. kernel methods) which appear commonly in the ML literature but are not well known among pure mathematicians
• see some natural connections between ML theory and: optimization/calculus of variations, measure theory, PDE, etc
• gain fluency reading ML papers (which can be less trustworthy than pure math papers)
• start to think about how to bring your area of mathematical expertise to bear on ML problems.

Outline:

• Unit 0: (~1 week) What is machine learning?
• Unit 1: (~4 weeks) Supervised learning: The statistical learning theory framework. Inference in high dimension. Falsibiability of models and measures of model complexity. Regression and classification. Kernel methods. Learning with neural networks. Double-descent and failure of Ockham’s razor.
• Unit 2: (~3 weeks) Unsupervised learning: Clustering and dimensionality reduction. Manifold hypothesis. Geometric graph methods. Inferring probability distributions: density estimation, sampling, generative models.
• Unit 3: (~4 weeks) Reinforcement learning: Exploration-exploitation tradeoff. Sequential decision problems. Markov decision processes and connections with control theory. Efficient exploration for bandit problems and small-scale games. Complexity notions and learnability for large scale games.

Main references: for textbook references we will use a couple chapters from each.

• Unit 0: Vapnik, “The nature of statistical learning theory”.
• Unit 1: Wainwright, “High-dimensional statistics”. Bach, “Learning theory from first principles”.
• Unit 2: There is no good textbook for unsupervised learning that I am aware of. I have course notes. We will also look at some classic research papers, for example for geometric graph methods we will read “Laplacian eigenmaps for dimensionality reduction and data representation” by Belkin and Niyogi.
• Unit 3: Foster and Rakhlin, RL theory notes: https://arxiv.org/abs/2312.16730

Class Schedule

• TBA

Remote Access

Remote access to this course will be via zoom. The delivery mechanism will be either blackboard or via tablet depending on available rooms. A PDF textbook and/or research article readings will be distributed in advance of each class.

Availability

This course may be open to students from universities outside of the PIMS network.

Registration for this course is not currently available.

In this course we are learning to build and analyse nonlinear partial differential equation models. The focus of the course will be models of ecological systems, but the techniques learned apply broadly across application areas. We learn a wide variety of analytic, graphic, and simplification techniques which elucidate the behaviour of these mathematical models, whether or not a closed-form solution is available. By the end of the class, the students will be able to competently read and follow a research paper presenting and analysing a differential equation model from a wide variety of application areas. Broadly, the topics that we cover are applications of ecological applications of travelling waves, disease models, and pattern formation in reaction-diffusion and reaction-diffusion-chemotaxis models.

Class Schedule

TBA

Remote Access

Lectures will be livestreamed via zoom. The lecturer will be writing on a whiteboard interspersed with pdf presentations. Lecture notes will be posted on Canvas.

Availability

This course may be open to students from universities outside of the PIMS network.

Registration for this course is not currently available.

In the past 25 years or so, the subject of “Arithmetic Statistics”, beginning with the work of Bhargava’s success in enumerating rings and fields of low degree and rank, and Bhargava and Shankar’s proof of the boundedness of algebraic rank of elliptic curves, is an enormously exciting subject. We will give an introduction to the subject centred on the work of Bhargava and his coworkers.

Remote Access

Lectures will be conducted via zoom, using electronic slides. Slides, assignments and exams will be distributed electronically. Lectures will be recorded and made available to registered students.

Registration for this course is not currently available.

The purpose of this graduate course is to equip graduate students with cutting-edge techniques in data-driven mathematical and computational modelling, analysis and simulations of semi-linear parabolic partial differential equations (PDEs) of reaction-diffusion type. It will cover diverse areas in data-driven modelling using PDEs in biology. I will cover approaches on formulating models from data using first principles, mathematical analysis of reaction-diffusion systems such as linear stability analysis, basic concepts on bifurcation analysis and numerical bifurcation analysis. The second part will focus on numerical methods for PDEs including finite difference methods, and finite elements. This part will also deal with time-stepping schemes and nonlinear solvers for nonlinear PDEs. If time allows, we will look at applications of reaction diffusion theory to cell motility and pattern formation. To support theoretical modelling and numerical analysis, numerical algorithms will be developed and implemented in MATLAB as well as in open finite element source software packages such as FeNiCs, deal.ii and others. Students will be allowed to use packages of their choice as appropriate. Expertise and skills sets to be acquired through this course

1. Acquire data-driven modelling skills and techniques in PDEs and their applications to biology
2. Acquire techniques and knowledge in mathematical analysis of reaction-diffusion systems
3. Acquire expertise and skills in bifurcation analysis, numerical bifurcation, and sensitivity analysis
4. Acquire numerical analysis techniques and skills to compute approximate numerical solutions
5. Acquire expertise and knowledge in finite difference methods for semi-linear parabolic PDEs
6. Acquire expertise and knowledge in finite element methods for semi-linear parabolic PDEs
7. Gain some knowledge in bulk-surface PDEs, and their analysis (might be covered if time allows) Key

1. The art of mathematical modelling
   1. An introduction to the art of mathematical modelling
   2. The physical origins of partial differential equations and their applications
      1. Derivation of the heat equation: Heat Transfer (A taster of what to come)
      2. General classification of PDEs
   3. Mathematical Notations and Definitions
   4. Physical laws
   5. Exercises
2. Reaction-diffusion systems on stationary domains: modelling, analysis and simulations
   1. Introduction
   2. Derivation of reaction-diffusion systems on stationary domains
   3. Classical nonlinear reaction kinetics
      1. Activator-depleted reaction kinetics
      2. Gierer-Meinhard reaction kinetics
      3. Thomas reaction kinetics
   4. Non-dimensionalisation – unit free
      1. Reaction-diffusion system with activator-depleted reaction kinetics
      2. Reaction-diffusion system with Gierer–Meinhardt reaction kinetics
      3. Reaction-diffusion system with Thomas reaction kinetics
3. Stability analysis of reaction-diffusion systems on stationary domains and the generation of parameter spaces
   1. Introduction
      1. Preliminaries
   2. Linear stability analysis of reaction-diffusion systems on stationary domains
      1. Linear stability in the absence of spatial variations
      2. Linear stability in the presence of spatial variations
   3. Eigenfunctions in one dimension and on special domains in two dimensions
      1. Eigenfunctions in one dimension
      2. Eigenfunctions of a rectangle
4. Numerical Methods for Reaction-Diffusion Systems on Stationary Domains
   1. Finite Difference Methods for Reaction-Diffusion Systems on Stationary Domains
      1. Finite Difference Stencils in 2- and 3-Dimensional Domains
      2. Forward Euler Method
      3. Backward Euler Method
      4. Crank-Nicholson Method
      5. Fractional-Step 𝜃 method
      6. Implicit and explicit (IMEX) time-stepping schemes for reaction-diffusion systems on stationary domains
   2. Finite Element Methods for Reaction-Diffusion Systems on Stationary Domains
      1. Sobolev Spaces
      2. Weak Variational Form
      3. Space discretisation
      4. Mesh Generation
      5. Time discretisation
   3. Fully implicit time-stepping schemes and non-linear solvers for systems of reaction-diffusion equations
   4. Algorithm development and implementation using finite element open source software pages
      1. Introduction to PDE computing with FeNiCs
      2. Algorithm development and testing in FeNiCs
5. Introduction to reaction-diffusion systems on evolving domains and surfaces
   1. Reaction-diffusion systems on deforming domains and surfaces . . . . . .
   2. Finite element methods for reaction-diffusion systems on deforming domains and surfaces
6. Summary of the course taught.

Remote Access

We will use zoom for each lecture. Course notes will be distributed in advance and lecture notes will be distributed after each lecture.

Registration for this course is not currently available.

The course is a first semester of algebraic topology. Broadly speaking, algebraic topology studies spaces and shapes by assigning algebraic invariants to them. Topics will include the fundamental group, covering spaces, CW complexes, homology (simplicial, singular, cellular), cohomology, and some applications.

Lecture Schedule

Lectures will take place Monday, Wednesday and Friday 12:30 - 1:20 PM Regina time.

Remote Access

The class will be in a hybrid format hosted in a classroom equipped with hyflex technology.

Lecture notes will be projected on the screen, shared simultaneously on Zoom, and posted afterwards on the course website.

Registration for this course is not currently available.

This course provides an introduction into analytic number theoretic methods with applications to arithmetic geometry. We will study Dirichlet series with applications to distributions of prime numbers and as examples of L-series. We will also look at modular forms and their applications to the arithmetic of elliptic curves and their moduli spaces. We will also consider results in diophantine approximation, such as lower bounds on linear combinations of logs of algebraic numbers, with as application Siegel’s theorem on finiteness of integral points on elliptic curves.

Lecture Schedule

• Wednesday/Friday 2:30-4:20pm Pacific Time See the SFU Calendar for more details.

Remote Access

The class will be held in a room equipped with controllable cameras. The instructor will write on whiteboards in this room and the camera controls used to provide clear views of the boards. Zoom links will be available on the course webpage (via Canvas, which will be available to enrolled students).

Other Information

Please see the SFU Calendar for more details about this course.

Registration for this course is not currently available.

This course covers classical problems and modern techniques in extremal combinatorics. The first part of the course is on extremal properties of families of sets: e.g.

• What is the largest size of a collection of k-element subsets of a set of size n in which any two sets in the collection intersect?
• What is the largest size of a collection of subsets of a set of size n in which no set is properly contained within another?

Other topics may include VC dimension, Kneser’s Conjecture, the Kruskal-Katona Theorem and the Littlewood Offord Problem. The rest of the course is on extremal graph theory: e.g.

• What is the maximum number of edges in a triangle-free graph on n vertices?
• What is the minimum number of 6-cycles in a graph with n vertices and m edges?
• What is the minimum size of an independent set in a triangle-free graph?

Other topics may include the Szemerédi Regularity Lemma, Shannon Capacity, the Entropy Method, the Container Method and Stability. The course webpage, which includes a link to a preliminary version of the course notes, can be found here.

Lecture Schedule

This course will run Sept. 4th-Dec. 4th, 2024. Lectures will take place every Tuesday, Wednesday and Friday from 10:30am-11:20am (Pacific Time). See the UVic course catalog entry for more details.

Remote Access

Lectures will be livestreamed via Zoom. The lecturer will write on chalkboards which will be shared via Zoom. Recordings of the lectures will be available for asynchronous viewing. Preliminary lecture notes are available on the course website and assignments will be distributed electronically.

Registration for this course is not currently available.

The course will be an introduction to the behaviour of fluids (liquids and gases) from an applied math perspective, starting with an introduction to the Navier-Stokes equations and other PDEs used to model fluids. The emphasis will be on physically relevant properties of solutions that can be deduced mathematically. The course will have more mathematics than a typical physics or engineering fluids course, including basic functional analysis and variational methods, and it will have more physics than a pure PDE analysis course. Through detailed study of several fundamental model systems, we will see PDE examples of topics that may be more familiar in the context of ODE dynamical systems, such as linear stability, nonlinear stability, bifurcations and chaos. Undergraduate knowledge of PDEs and real analysis are assumed.

There will be no exams, only assignments access and submitted online via Crowdmark.

Class Schedule

• M/Th 11:30 AM - 12:50 PM PST

Remote Access

The lecturer will use zoom for each lecture. Typed lecture notes will be distributed electronically.

Registration for this course is not currently available.

The most basic example of an arithmetic group is $\Gamma=SL_n(Z),$ and understanding the cohomology of this group (and its close relatives) will be the basic theme of this course. The cohomology we are interested in can also be identified with that of the locally symmetric space $\Gamma \setminus X$ where, in this case, $X= SL_n(R)/ SO(n)$ is a generalization of the (complex) upper half plane. As such, a diverse set of techniques, stemming from geometry, topology, harmonic analysis, and number theory can be used to analyze the situation. After carefully developing the basics of the subject, we will present some of the major developments in this area (mostly from the 1960s-1970s), and then end with an overview of modern directions.

Lecture Times

• Dates: Sep. 3 - Dec. 9
• Class Time: Tuesday/Thursday, 16-17:20 (Mountain Time)

Remote Access

The lecturer will use a tablet connected to zoom/camera to live stream lectures and notes. Hand written (from table) and typed lecture notes will be distributed.

Registration for this course is not currently available.

This course presents classical mechanics to a mixed audience of mathematics and physics undergraduate and graduate students. It is complementary to regular phsyics courses in that while the physics background will be developed the emphasis will be on the resulting mathematical analysis. Physics topics may include Newtonian mechanics and Galilean symmetry, Lagrangian mechanics, conservation laws and Noether’s Theorem, rigid body motion, Hamiltonian mechanics. Mathematical topics may include existence and uniqueness of solutions to ODE, calculus of variations, convexity and Legendre transformations, manifolds, tangent and cotangent vectors, rotations and the orthogonal group.

Course Website

Full information about this course is available on the course website.

Lecture Schedule

• Tuesday/Thursday 11:00am-12:30pm Pacific Time

Remote Access

Lectures will be held in-person on the UBC campus and on Zoom. Lectures will be recorded and the videos posted to an unlisted but openly accessible YouTube playlist. There will be Zoom office hours and a Piazza discussion board.

Registration for this course is not currently available.

In this course we are learning to build and analyse nonlinear differential equation models. The focus of the course will be models of ecological systems, but the techniques learned apply broadly across application areas. We learn a wide variety of analytic, graphic, and simplification techniques which elucidate the behaviour of these mathematical models, whether or not a closed-form solution is avalable. By the end of the class, the students will be able to competently read and follow a research paper presenting and analysing a differential equation model from a wide variety of application areas.

Class Schedule

The class will meet on Monday, Wednesday and Friday during term from 12pm-1pm (Pacific Time).

Textbook

• Mathematical Biology, 3rd Edition, volumes I and II, by James Murray

Remote Access

Lectures will be livestreamed via zoom. The lecturer will be writing on a whiteboard.

Registration for this course is not currently available.

Spectral methods are numerical methods for solving PDEs. When the solution is analytic, the convergence rate is exponential. The first part of this course gives an introduction to spectral methods. The emphasis is on the analysis of these methods including truncation and interpolation error estimates, and convergence and condition number estimates. The second part of the course focuses on fast algorithms for orthogonal polynomials. These algorithms leverage data-sparsities that are present in many of the problems when solved by orthogonal polynomial expansions.

Class Schedule

This class will meet Mondays and Wednesdays from 11am-12:15pm (CDT)

Remote Access

Lectures will be delivered via Zoom using iPad with GoodNotes.

Registration for this course is not currently available.

This is a one semester three credit hour course. We shall first briefly introduce some basic concepts and results on stochastic processes, in particular the Brownian motions. Then we will discuss stochastic integrals, Ito formula, the existence and uniqueness of stochastic differential equations, some fundamental properties of the solution. We will concern with the Markov property, Kolmogorov backward and forward equations, Feynman-Kac formula, Girsanov formula. We will also concern with the ergodic theory and other stability problems. We may also mention some results on numerical simulations, Malliavin calculus and so on.

Remote Access

We will use zoom for each lecture. The eclass website will be used to post lecture slides, homework collections, monitor midterm and final examinations

Registration for this course is not currently available.

Advances in imaging techniques have enabled the access to 3D shapes present in a variety of biological structures: organs, cells, organelles, and proteins. Since biological shapes are related to physiological functions, biological studies are poised to leverage such data, asking a common statistical question: how can we build mathematical and statistical descriptions of biological morphologies and their variations? In this course, we will review recent attempts to use advanced mathematical concepts to formalize and study shape heterogeneity, covering a wide range of imaging methods and applications. The main mathematical focus will be on basics of image processing (segmentation, skeletonization, meshing), Diffeomorphisms and metrics over shape space, optimal transport theory with application for image analysis, manifold learning, with some other concepts covered in specific applications (e.g. quasiconformal mapping theory for shape representation, 3D reconstruction in Fourier space…). Students will be encourage to work in groups to present research papers and do a small project to pass the course. This course will also build on the recent BIRS workshop, Joint Mathematics Meetings, and the upcoming SIAM workshops (LSI 2024, SIMODS 2024) on this topic, with some participants to these events invited to contribute to this course and present their research.

Lecture Schedule

Remote Access

Remote access will be via zoom. A combination of prepared slides and hand written notes will be used. The hand written notes will be on a blackboard or tablet depending on room availability. The lecturer will distribute lecture notes online.

Registration for this course is not currently available.

Translation surfaces and their moduli spaces have been the objects of extensive recent study and interest, with connections to widely varied fields including (but not limited to) geometry and topology; Teichmüller theory; low-dimensional dynamical systems; homogeneous dynamics and Diophantine approximation; and algebraic and complex geometry. This course will serve as an introduction to some of the big ideas in the field, centered on the ergodic properties of translation flows and counting problems for saddle connections, and associated renormalization techniques, without attempting to reach the full state of the art (an aim that is in any case impossible given the speed at which the field is evolving).

We will start by introducing the important motivating example of the flat torus, exploring its geometry, and its associated dynamical and counting problems. The linear flow on the torus and its associated first return map, a rotation of a circle, are amongst the first dynamical systems ever studied. The counting of closed orbits is intricately tied to number theory. We discuss, as motivation, the moduli space of translation surfaces on a torus, a bundle over the well-known modular curve and the action of $GL^+(2,\mathbb R)$ on this space of translation surfaces. Translation surfaces are higher-genus generalizations of flat tori. We will define translation surfaces from three perspectives (Euclidean geometry, complex analysis, and geometric structures), and show how some translation surfaces arise from unfolding billiards in rational polygons. We will give a short introduction to Teichmüller theory and its relation to the study of translation surfaces, and discuss the natural dynamical systems associated to translation surfaces, namely, linear flows and their first return maps, interval exchange transformations. We will explore their ergodicity and mixing properties, and will study an important example of a translation surface flow for which every orbit is dense but not every orbit is equidistributed with respect to Lebesgue measure, a phenomenon that does not occur in the case of linear flows on the torus. We will show how information about the recurrence properties of an orbit of a translation surface under the positive diagonal subgroup of $SL(2, \mathbb R)$ (the Teichmüller geodesic flow) can be used to get information about the ergodic properties of the associated linear flow on an individual translation surface. As another example of the strength of renormalization ideas, we will show how the ergodic properties of the $SL(2, \mathbb R)$-action can be used to obtain counting results for saddle connections and, subsequently. Finally we will discuss examples, characterizations, and properties of surfaces with large affine symmetry groups, known as lattice or Veech surfaces.

Remote Access

The instructor will use a tablet and Zoom. The tablet will be displayed locally in the classroom and via zoom. Lecture notes will be distributed in PDF format.

Class Schedule

This class will meet every Monday, Wednesday and Friday from 1:30-2:50 (Pacific time), starting on March 31st. Remote participation is via zoom

Registration for this course is not currently available.

The course will provide an introduction to algebraic and probabilistic techniques in combinatorics and graph theory. The main topics included will be: Eigenvalues of graphs and their applications, probabilistic methods (first order, second order, Lovasz local lemma), Szemeredi regularity lemma. Recent discoveries like the proof of the Sensitivity conjecture, the use of eigenvalues for equiangular lines, etc., will be part of the course. '

Lecture Times

• Time: Tuesday 10:30-12:20 and Thursday 10:30-12:20
  • First day of classes: January 9
  • Reading break: February 20-25
  • Last day of classes: April 11

Delivery details

Note: This course is also offered through PIMS and WDA (Western Dean’s Agreement) as an online course. A Zoom link will be shared with registered students.

Course outline

Part I

• Introduction (warmup application of graph eigenvalues)
• Eigenvalue basics (including Perron-Frobenius Theorem)
• Eigenvalue interlacing (bounds on the maximum clique and chromatic number)
• Wilf’s Theorem, proof of sensitivity conjecture
• Graph Laplacians (Matrix-tree Theorem, Cheeger inequality)
• Random walks, effective resistance
• Spectral sparsifiers

Part II

• Random graphs, probabilistic method (including Lovasz local lemma)
• Quasirandom graphs
• Eigenvalues of random graphs (Wigner, Tao-Vu)
• Regularity Lemma
• Finding regular partitions
• Random covers and Ramanujan graphs

Grading scheme:

• Homework assignments 30%
• Midterm 30%
• Final exam 40%

Registration for this course is not currently available.

This will be a standard graduate number theory course. Topics will include:

• Number fields, rings of integers, ideals and unique factorization. Finiteness of the class group.
• Valuations and completions; local fields.
• Ramification theory, the different and discriminant.
• Geometry of numbers: Dirichlet’s Unit Theorem. and discriminant bounds.
• Other topics if time permits

The main pre-requisites are basic algebra (rings and fields, rings of polynomials, unique factorization in Euclidean\ndomains), basic number theory (modular arithemtic, factorization into primes) and Galois Theory, but no specific courses are required.

https://personal.math.ubc.ca/~lior/teaching/2324/538_W24/

Other Information