- Why is a multi-stage selection needed for candidates to join our group?
- Stage 1: applications to Materials Science, your skills and expectations.
- Stage 2: tuition fees and potential funds for your living costs at Liverpool.
- Stages 3-4: University PhD requirements about past degrees and English.
- Stages 5-6: your CV and video introduction of motivations and skills.
- Stages 7-8: problem solving and a 30-min informal Skype discussion.
- Stages 9-10: an official online application and a formal interview.
- Traditional exercises to engage with all readers of this blog.

Every week I receive several requests for supervision, mainly from PhD candidates, and also from intern students and postdoctoral fellows. From January 2019 I am supervising

- 1 postdoc as the research mentor,
- 5 PhD students as the first supervisor,
- 5 PhD students as the second supervisor,
- 1 PhD student as the third supervisor.

The multi-stage selection is justified by the following reasons.

- An objective and transparent process described below is better than ad-hoc decisions.
- PhD admissions at the University of Liverpool follow rules from higher authorities.
- Since I’m already overloaded, there is a physical limit of potential supervision.
- I seriously consider any supervision, hence a selection process is also serious.
- Mentees seem to be much more motivated after they were properly selected.

Though a few current students complete their PhDs in different areas, my main research is for the Materials Innovation Factory. Hence all future PhD projects will be focused on applications of geometry and topology to Materials Science, in particular solid crystalline materials.

The previous knowledge of Chemistry is not needed, however you should be open-minded enough to learn new research topics. Mathematical crystallographers and computational chemists are welcome, because the group already has strong skills in theoretical areas.

Most relevant subjects are Mathematics (linear algebra, geometry or topology, group theory) and Computer Science (graph algorithms, computational geometry, optimisation). The key requirement is your strong programming experience: C++ (preferable) and/or Python. We use software libraries in C++ and Python APIs (Application Programming Interfaces) of databases.

Due to the high load, new PhD students or postdocs can expect on average 1 hour per week of my time to discuss their research, sometimes with co-supervisors or other colleagues, because I also teach big classes of undergraduates and have important administrative commitments.

Short term project students or interns can expect maximum 1 hour (a half-hour on average) per week of my time. These students may more frequently talk to postdocs or senior PhD students. Collaborative projects are often discussed at the weekly group seminar or in smaller groups.

Everything is possible and highly motivated students, who make regular substantial progress and can present their results in a clear concise form, may naturally get more attention (hence more time for discussions) and recognition for their genuine enthusiasm and hard work.

So Stage 1 is your own evaluation of your skills and motivations. You have passed this stage if

- you are motivated to work with other people and learn new topics;
- you can demonstrate your programming skills in C++ and/or Python;
- you have sufficient expertise in Mathematics and/or Computer Science;
- (for PhD candidates) you agree with the rules of the PhD programme.

This stage is most important for non-European PhD candidates and intern students.

All available PhD positions are funded up to £20K per year covering tuition fees only for EU/UK students (£4260 per year) and a bursary (minimum £14777 a year, the rest is for research expenses), which is tax free.

The tuition fees for international students are much higher (almost £20K per year). A UK embassy might ask for a proof of your funds for living costs when you apply for a visa. The university recommends a minimum £9135 for a single person per year.

The recommended websites for checking prices and accommodation at Liverpool are Liverpool Student Homes, Rightmove, Zoopla. You could say in advance how you plan to cover costs.

The good news is the list of scholarships for international PhD candidates. The financial requirements are outside my control. I get no financial rewards for supervising students or postdocs.

These stages are formal requirements by the university for all PhD candidates.

Stage 3 is to make sure that your past degree is equivalent to at least 2:1 (roughly grade point average 60%) in the UK classification. Please e-mail the PGR office for more details.

If you are a citizen of a country, where English is the main language, or you have completed your degree in such a country, your English is acceptable and you can go to next stage 5.

In all other cases Stage 4 is to think about required IELTS grades (overall 6.5, minimum 6 in each component) or any equivalent. These grades should be obtained not later than 2 years before a PhD start date. You could postpone passing IELTS until after receiving an offer conditional on IELTS grades. Please e-mail the PGR office for more details.

Most candidates start from Stage 5 by sending their CV, which is ok for postdoctoral fellows or local students interested in final year projects or summer dissertations. PhD candidates could follow Stages 1-4 above. Intern candidates may think about finances at Stage 2.

Most efficient e-mails are short (as this phrase). All details can be in your CV (needed) and informal cover letter (optional, only if you would like to express your motivations and highlight relevant experience). File names could include your name, e.g. Last_name.First_name.CV.pdf.

You could include a brief description of (or give links to) your past programming projects, e.g. what software libraries have you used and what challenges have you overcome?

The most important expertise to work in our diverse group is your communication skills. You are expected to communicate well with colleagues from different research areas and industry.

If I have replied to your initial expression of interest, Stage 6 is to e-mail me your short 1-2 min video presentation to introduce yourself. Though your English will be formally checked and some candidates include photos in their CVs, your video will quickly show how you talk.

For your self-presentation, I may ask to highlight any relevant skills depending on your CV. At any stage if I haven’t replied within a week, you could e-mail me again once, please not more.

Congratulations if you have completed previous Stage 6! Depending on your level, at Stage 7 you will solve mathematical problems and programming exercises. You could have 1-2 weeks for preparing your solutions. We can discuss a suitable period if you are currently busy.

If I am relatively free, Stage 8 is to arrange an informal chat by Skype, say up to 30 min. Similarly to a video presentation, a real time discussion will help to establish our future relationships.

For intern students or postdoctoral fellows applying for external grants, this Stage 8 can be the last one. After that I usually talk to the head of department to arrange an invitation letter.

Stage 9 for PhD candidates is to submit a formal application at the university website. If you have passed Stages 1-8, you could mention me as a potential supervisor, possibly a project title with a short description (if already agreed).

Postdoctoral candidates will have another link to the application from a job advert. The advice for postdocs is not to start from this Stage 9, but e-mail me your expression of interest.

Stage 10 is a formal interview for PhD candidates and postdoctoral fellows. All PhD students at Liverpool should have at least two supervisors (the standard split is 80/20). Hence a co-supervisor is involved in an interview, often by Skype for PhD candidates outside the UK.

Postdoctoral candidates will face an interview panel of at least 3 people including a representative from HR. We try to invite European candidates for on-site interviews, though Skype interviews are also possible. If anything seems unclear, feel free to e-mail me your questions. If you contact me for the first time, then I would be grateful if you say how you have found me.

**Q1**. How many stages will PhD candidates pass to be selected for the TDA group?**Q2**. How many people am I supervising from January 2019?

You could write a brief answer or feedback in your comment: reply.

]]>- What is an Atmospheric River?
- Why are Atmospheric Rivers important?
- How are Atmospheric Rivers detected?

An *Atmospheric River* (AR) is a narrow filament of concentrated water vapour in the atmosphere, usually up to several thousand kilometers long and a few hundred kilometers wide.

These filaments were called Atmospheric Rivers in the paper “Atmospheric rivers and bombs” (pdf) in 1994, because a single filament can carry more water than the Amazon River. Hence an Atmospheric River can be informally considered as a “river” flowing in the atmosphere.

The picture above shows the integrated water vapour (IWV) measured in grams over a squared millimetre, formally the mass of water in the vertical column over a square 1×1 mm. Higher values of IWV correspond to the red colour, lower values are shown by the blue colour.

The red box from the picture above is zoomed in the picture below showing how an Atmospheric River hits the California coast in the US.

At any given moment there are 3-5 Atmospheric Rivers on the planet and all of them contribute over 90% to the global north-south water vapour transport. When an Atmospheric River hits a coast, this “river” flows down to the land as heavy rain, which causes severe floods.

These extreme weather events regularly happen along the West Coast of North America, Western Europe and the west coast of North Africa, e.g. read “Rivers in the Sky Are Flooding The World With Tropical Waters” (pdf).

The paper “Winter floods in Britain are connected to atmospheric rivers” (pdf) justifies that all winter floods in the UK in 2000-2010 were caused by Atmospheric Rivers including the 19th November 2009 severe flood on the River Eden in Cumbria (UK).

The input for a detection is a scalar field of the Integrated Water Vapour (IWV) over a regular grid whose lines are usually parallel to meridians and longitudes. The input can be visualised as a matrix of IWV values that are obtained from weather observations or computer simulations. So every node in the regular grid has an associated value of the Integrated Water Vapour and is connected to the four neighbours (north, west, south, east) in the grid.

High moisture regions that bring water vapour from mid-latitudes in the ocean up to the land in the north are called *Atmospheric Rivers *to distinguish them from other high moisture regions that don’t cause floods. A detection algorithm should identify only Atmospheric Rivers.

The picture above shows a big hole in the yellow-red region that doesn’t form an elongated filament. The picture below contains the yellow high moisture region without holes, but this filament doesn’t reach the coast. Hence there are no Atmospheric Rivers in both cases.

The traditional approach to detect an Atmospheric River is to fix a threshold of the Integrated Water Vapour, say 20 g/mm^{2}, and consider all nodes with values above this threshold. If these nodes form a connected component in the regular grid that has expected geometric parameters (length and width) and also joins the mid-latitude region (the bottom line of the chosen box) with the California coast, the latest detection algorithm in the TECA software (Toolkit for Extreme Climate Analysis, pdf) says that an Atmospheric River is detected.

The state-of-the-art algorithms work only for carefully chosen parameter values. Many Climate Scientists propose different values. That is why we are now working on a parameterless approach combining ideas of Topological Data Analysis with Machine Learning.

Which of the pictures below show Atmospheric Rivers in your opinion and why?

You could write a brief answer or feedback in your comment: reply.

]]>This post motivates the new research area of *Topological Computer Vision*.

- Why don’t we have self-driving cars yet?
- Stability under noise is still a big problem
- First key results of
*Topological Computer Vision*

Here is the response by Dr. Andreas Wendel (Google) from his invited talk *Self-Driving Cars* at the CVPR 2015 workshop Computer Vision in Vehicle Technology: “We can’t predict all possible road accidents. In the weirdest case our car stopped and waited for an old lady in a wheelchair chasing a duck with a broomstick … in the middle of a road!”

A Google car makes about 200 decisions per second. If any of these 200 decisions is wrong, there could be a fatal accident. Self-driving cars will appear on the market when the error rate is less than 0.01%. My collaborator Andrew Fitzgibbon from Microsoft Research Cambridge has predicted that we might wait for another 10 years.

The current flagship method in speech and image recognition is *Deep Learning*. Briefly, an algorithm is trained (often for weeks) to predict correct outputs from big labelled data. For instance, the ImageNet database has more than 14M images split into over 21K categories like cars, frogs etc. These images were manually labelled by humans, which required about 25K Amazon Mechanical Turks.

During the training, the algorithm finds features that best split all labelled images into required categories. During validation, the algorithm chooses the category whose features are closest to those of a new given image. The overall error rate, when the algorithm mis-classifies images, is about 6.7%, see page 20 in ImageNet Large Scale Visual Recognition Challenge (arXiv/1409.0575, 8M). However, exercises below how this approach fails in the presence of little noise.

*Topological Computer Vision* was introduced as a *new research area* within Topological Data Analysis (TDA) in the invited talk at the scoping workshop of the Alan Turing Institute at Oxford on 10th September 2015.

The first key concept is a *Homologically Persistent Skeleton* (HoPeS) depending only on a point cloud C without extra input parameters. HoPeS(C) is the first structure that provides a closed geometric approximation to an unknown graph given only by a noisy sample C. Details are in

- the slides Topological Computer Vision (pdf, 11M) and in the paper
- A one-dimensional Homologically Persistent Skeleton of an unstructured point cloud in any metric space (extended pdf, 14 pages, 3.2M), Computer Graphics Forum, v. 34, no. 5 (2015), p. 253-262.

The big aim is to combine the stable-under-noise persistence from TDA with the state-of-the-art tools of Deep Learning that currently suffer from noise.

**Q1**. The middle image below is the difference (multiplied by 10) between 2 dog images. One image is correctly recognized by the state-of-the-art Deep Learning Net as a dog. However, another image with little added noise is misclassified as an … ostritch. Where is the noisy image: on the left or on the right?**Q2**. This is an example from CVPR 2015, the top conference in Computer Vision and Pattern Recognition). Can you guess how the image below is mis-classified?

(Hint for possible answers: kid’s drawing, pedestrian, school bus, trademark).

You could write a brief answer or feedback in your comment: reply.

]]>- What does the word
*data*mean? - What are the practical aims?
- What is a typical problem?

The usual data input in topological data analysis is a noisy sample of points in a Euclidean or in a more general metric space. For example, a black-and-white image can be given as a finite sample of black points in the plane.

More generally, data can be sampled from any topological shape (or a space). Examples of shapes below are a graph, a figure-eight shape in the plane, a 2-dimensional torus.

The ultimate goal is to understand the meaning of data. The practical aims are the following:

- Represent data in an easy way for further processing. Why?

We need to find and easily encode extra structures to work later

with structured mathematical objects rather than with raw data. - Quantify or measure given data by topological invariants. Why?

Extra structures allow us to find topological invariants that

depend only on the shape of data, not on extra structures. - Make robust statistical predictions about topology of data. Why?

If data are huge, any algorithm inevitably works on subsamples and

we should be able to take (say) the average of all topological outputs.

We give links to more details about each of the 3 practical aims above:

- our post on representing data by simplicial complexes
- the wonderful post Measuring shape by Prof Gunnar Carlsson
- the paper Stochastic Convergence of Persistence Landscapes and Silhouettes by F.Chazal, B.Fasy, F.Lecci, A.Rinaldo, L.Wasserman.

Our trained human eye can recognize a familiar heart shape in the cloud of red points below. The red heart shape is easy enough and we could connect each point with its two nearest neighbors to get a reasonable contour.

However, a robust contour detection in noisy images is still a hot problem in computer vision. Topological data analysis looks for methods beyond the simplest nearest neighbor search. The key idea is not to fix a scale parameter when searching for neighbors, but analyze a summary of data over all scales so that this summary is stable under noise.

**In conclusion**, we highlight answers to the questions posed at the beginning of this post:

- the input data are finite clouds of points without much structure
- the practical aims are to represent and measure data in easy ways
- a typical problem is to reconstruct contours from a noisy sample.

**Q1**. What topological shape can one reconstruct from the cloud of points given

at the beginning of the post?**Hint**. You have seen a partial yellow shape above.**Q2**. What can you see in the black-and-white image below?**Hint**. Find an animal.

You could write a brief solution or feedback in your comment: reply.

]]>**what**are simplicial complexes and what are they not?**how**can a shape be represented by a simplicial complex?**why**are simplicial complexes easy for representing shapes?

A *simplicial complex* is a high-dimensional generalization of a graph. That is why simplicial complexes are sometimes called *hypergraphs*. The building blocks of a simplicial complex are vertices, edges, triangles and higher-dimensional *simplices* like tetrahedra in \(\mathbb{R}^3\).

The standard k-dimensional simplex is the subset of points in the \(k\)-dimensional space:

\(\Delta^k=\{(x_1,\dots,x_k)\in\mathbb{R}^k \; :\; 0\leq x_i\leq 1,\; \sum\limits_{i=1}^k x_i=1\}\).

We may say that the simplex \(\Delta^k\) is *spanned by* its \(k+1\) vertices marked by (say) \(0,1,\dots,k\). Any subset of \(l\) vertices spans an \(l\)-dimensional *face *(or a subsimplex) of \(\Delta^k\).

Many geometrically different shapes are homeomorphic (topologically equivalent) to the same simplex \(\Delta^k\subset\mathbb{R}^k\). For instance, the 2-dimensional disk \(\{(x,y)\in\mathbb{R}^2\; :\; x^2+y^2\leq 1\}\) is homeomorphic to the standard triangle \(\Delta^2\). Then the combinatorial structure of \(\Delta^2\) induces a similar structure on the disk, namely 3 vertices, 3 edges and one 2-dimensional face.

A *triangulation* of (or the structure of a *simplicial complex* on) a shape is

- a splitting into finitely many pieces homeomorphic to simplices in such a way that
- the intersection of any two simplices in the triangulation is their common face.

A shape is called *triangulable* if it has a triangulation satisfying the above conditions. A simplicial complex may contain simplices of different dimensions as shown in the sun glasses below. The dimension of a simplicial complex is the maximum dimension of its simplices.

Representing a shape as a simplicial complex allows us to introduce later topological invariants of shapes. For instance, the Euler characteristic of a shape is expressed via the number of \(l\)-dimensional simplices in a trianglation for different values of \(l\).

The condition on intersection of simplices in dimension 1 means that any two edges can meet only at a single common vertex. Hence a 1-dimensional simplicial complex can not have loops or double edges with the same endpoints. This condition seems too restrictive for graphs, but is essential for higher dimensions as explained below.

Any simplicial complex with vertices marked by (say) \(0,1,\dots,n\) can be encoded by a list of maximum simplices that are not contained in larger simplices. Any maximum simplex spanned by \((k+1)\) vertices \(i_0,\dots,i_k\) is encoded by the unordered \((k+1)\)-tuple \((i_0,\dots,i_k)\). The sun glasses with 8 vertices above are encoded by the list (012),(345),(05),(14),(26),(37).

To encode a shape, first we should triangulate it. Let us split a round ring into 4 triangles with vertices 0,1,2,3 below. The corresponding list of triples is (012),(013),(023),(123).We may notice that the same list of triples encodes the standard tetrahedron \(\Delta^3\) with four 2-dimensional faces. To avoid this confusion, the definition of a simplicial complex requires that any two simplices should meet along their common face. So the splitting into 4 curved triangles above is not a simplicial complex.

If our code contains triples (123) and (013), we should know that the corresponding triangles share the common edge (13) as in the tetrahedron \(\Delta^3\), not only the endpoints 1 and 3 as in the round ring. A minimum triangulation of a round ring is at the beginning of this post.

**In conclusion**, we highlight answers to the questions posed at the beginning of the post:

- a simplicial complexes is glued from standard simplices along their common faces,
- to represent a shape, we triangulate it into pieces homeomorphic to simplices,
- simplicial complexes can be easily encoded in a computer memory, hence

they provide a convenient representation of a shape for further analysis.

**Q1**. Encode the figure-eight shape below by a simplicial complex.

**Hint***.*Each of 3 boundary contours should have at least 3 vertices.**Q2**. Find a minimum triangulation of the 2-dimensional torus below.

**Hint**A minimum triangulation of the torus has at least 7 vertices.*.*

You could write a brief solution or feedback in your comment: reply.

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