Louis de Thanhoffer de Volcsey, Ph.D

Currently, I am in my hometown of Antwerp, Belgium looking for my next challenge in data science. I'm also affiliated with Insight in Toronto, Canada. A fellowship that aims to turn academics into well-rounded data scientists

Before that, I was a postdoc in pure mathematics at the University of Toronto where I worked on a wide range of interests. First, I conducted research on a project originally conceived of together with the late Ragnar Buchweitz which lies at the cusp of algebraic geometry, homological methods and noncommutative algebra.
Additionally, I spent my days either teaching undergrads at UTSC , running our department's seminar on new algebraic techniques , or playing around with neural networks and digging into the foundations of machine learning .

Before coming to U of T in 2015, I completed my Ph.D in noncommutative algebraic geometry under the supervision of Michel Van den Bergh in Hasselt, BE. In my thesis , I constructed and classified some fascinating new spaces which have the exact kind of symmetry necessary to allow the strings of string theory to move with the freedom they need.

If you're interested in the geometric research I do, or want to take a look at some of my code, wish to know a little bit more about the math behind machine learning or took one of my courses, just click on the appropriate link above. For anything else, just send me a message

Machine Learning

One would be hard-pressed to find a better place than Toronto to experience the current excitement around machine learning. Here's some of the stuff I've been working on:


  1. Mathematical Foundations:

    Machine learning is rapidly expanding into a vast and intricate web of diverse mathematical techniques. As a result it can sometimes be tricky to maintain a clear overview of the field.
    To contribute to this problem, I recently began the math vs machine project. Its goal is to construct a formal theory of machine learners and consequently explain how the major tools in use today (regression, neural networks, clustering etc.) fit inside this theory.
    A nice byproduct of this project is that is was necessary to describe those techniques in the most abstract context possible, which lead to a much cleaner exposition of classical topics traditionally found in the literature.
    Feel free to have a look at the accompanying blog (still under construction) which offers a relatively informal roadmap of the project.








  2. Fun Projects:

    In recent years, the emergence of miraculous frameworks such as Tensorflow or Keras allows us to build sophisticated neural nets very intuitively. If you're reasonably trained in Python as well, applying your coding skills to create tools in order to solve some fun real-world problems becomes a
    In my spare time, I like to take on projects and build some such algorithms. To name the two most recent ones: I recently fit a neural net that detects whether a banknote is forged with 97% accuracy (as the graduating project of the Udacity ML engineer program ). Currently I'm part of team that's working on a project that uses natural language processing techniques together with a certain implementation of recurrent neural nets in order to recognize how and why youtube comments exhibit some form of toxicity.







  3. Machine Learning today and tomorrow:

    I am particularly excited about how machine learning is adopting beautiful mathematical concepts from very diverse fields in such very clever ways. As abstract concepts such as the Wasserstein metric, persistent homology or even hyperbolic geometry get used in order to build new models, it is becoming clear just how exciting the interplay between machine learning and current mathematics will become in the future. As part of the ML group at U of T, I am actively working towards a gaining a deeper collective understanding of those techniques which will in turn allow us to make the architectures of tomorrow more and more powerful.

    <


Geometry

My research lies at the intersection of algebra, geometry and physics. Specifically, I am interested in the math that describe the geometry used by string theorist to describe the vibrations of strings. This type of geometry is called 3CY geometry (after Eugenio Calabi and Shin-Tung Yau.) In recent years mathematicians discovered that this 3CY property can be viewed as very general symmetric behaviour which can also be percieved in the world of noncommutative algebras and more even more generally in category theory. Concurrently, as algebraic geometry kept refining its viewpoint, it kept including more general structures as well...

Starting with the connection between varieties and coordinate rings discovered by Hilbert , the torch was passed on to Grothendieck who introduced schemes as the geometric counterpart to all commutative rings, and developped homological algebra as a way to investigate them. His students then embraced the idea that a space should first and foremost be studied through its homological properties alone and found a way to encode all homogical information eautifully in a mathematical concept called the derived category of coherent sheaves of the space. In the last few decades this idea was take yet further when such people as Kontsevich argued that any category similar to a derived category should itself in fact be considered the space. This idea is the core philosophy of noncommutative geometry (the word noncommutative refers to the idea that the cateogry may be built by starting with a space, or come from a different part of math altogether. Crucially, only the structure of the category matters in this theory, so that we are able to do geometry without knowing the space, or even without knowing of a space actually exists!)

My research tries to use this noncummutative geometry to shed some light on the 3CY property of varieties,algebras or categories described above. More specificaly, my research focusses on the noncommutative geometry of certain types of surfaces called Del Pezzo . They are interesting for a number of reasons: because one can decompose their derived categories into simple building blocks that have a deep connection to algebra, which in turn allows us to work a little more explicitely. Moreover, one can perform a simple geometric construction on such surfaces to obtain a 3CY geometry.
Recently, we managed to give a complete classification of Del Pezzo surfaces of a certain type, but there is still much more (esciting) work to be done, if we hope to gain a complete picture some day!


Teaching

This semester, I am responsible for two courses: STAB57 (statistics) and MATC32 (graph theory).
I will be teaching both for the first time, and although the material is (mostly) similar to the previous sessions, I’ll be using my own approach and highlighting the aspects that I personally find important. So don’t rely too much on past notes (and disregard past exams in particular).
I intend to guide you through the practical applications of the course -think machine learning, baseball or real estate in STAB57 and finding gps directions, determining weak points in networks, setting up exam timetables or even matching you with a study buddy optimally in MATC32 ). It’s important to know that for both courses, we will be putting mathematical rigour at the forefront. This means that week by week, we'll discuss theorems extensively, taking the time to prove them and put them in context. You will find a brief summary under the appropriate week tab. Make sure you understand everything to ensure you keep up as the material dive more and more into the material.

Finally, experience tells me that the word ‘proving’ petrifies studentsma..to make sure you have everything you need to to succeed, you will be given a comprehensive list of exactly everything you need to know for tests/exams..A few more details you may need to know:
  1. MATC32

    You’ll be tested through a midterm/final and by weekly assignments. These consist of problems, some of which are straightforward, but most of which aren’t. I really encourage you to work with your fellow students and come see me during the office hours..i won’t solve your problem, but I can definitely get you unstuck if I see you’re putting in the work



  2. STAB57

    We’ll be operating through the standard weekly tutorial sessions during which you will have a suggested list of problems as well as a biweekly quiz (usually about 3 standard questions testing your basic understanding of the material), together with a midterm/final. Here also, both myself as well as the ta’s are at your disposal to discuss anything about the course



Above all, make sure that you stay on top of the courses, and feel free to get in touch with either the TA's or myself if you feel you could benefit from our help or advice

Seminar

This semester, in the seminar our central theme will be to study a variety \(X\) through its derived category \(\mathcal{D}^b(X)\). The focal point of which will be to determine how one can break down the category into two simpler components: \(\mathcal{D}^b(X)=\langle \mathcal{A},\mathcal{B}\rangle \) -technically known as a semi-orthogonal decomposition or SOD. The most common way to find such a decomposition is to look for a sequence of objects \(\mathbb{E}=(E_1,\ldots , E_n)\) with very rigid homological properties known as exceptional . The reason being that the category \(\mathcal{A}=\langle \mathbb{E}\rangle \) generated by \(\mathbb{E}\) is always a component of an SOD. Moreover, tilting theory tells us that \(\mathcal{A}\) is equivalent to the derived category of a quiver with relations \(\mathcal{D}^b(Q,w)\). Which is a category one can work with relatively explicitely. In certain situations a miracle happens where \(\mathcal{A}\) is in fact the whole category \(\mathcal{D}^b(X)\), so that \( \mathcal{D}^b(X) \) in fact can be described by the quiver with relations \((Q,w)\)! In such cases, we call \(\mathbb{E}\) full. Once one first goal of the seminar:

Goal 1: Get a good insight into which varieties have exceptional sequences (which may or may not be full)

A lot of results towards in direction have been made. Arguably the first (and most famous) result was established by Beilinson for the case \(X=\mathbb{P}^n\). This paper gives a nice overview of what is currently known.

A second aspect of this question is to see which conclusions can be drawn by understanding the other component \(\mathcal{B}\) of the SOD. The most commonly used approach here is to compute invariants such as such as the Grothendieck group \(K_0(\mathcal{B})\), the Hochschild cohomology \(\text{HH}^\bullet(\mathcal{B})\) etc... One then starts to look for a space \(Y\) such that \(\mathcal{D}^b(Y)\) matches those invariants. Of course it is well known that simply matching invariants does not necessarily imply that \(\mathcal{B}\cong \mathcal{D}^b(Y)\) (spaces with cool-sounding names like phantoms provide counterexamples). However, seminal work by Kuznetsov provides techniques in which in certain cases, indeed allow one to construct a tentative space \(Y\) from \(\mathcal{B}\). Summarizing

Goal 2: Understand how to build a space \(Y\) from \(\mathcal{B}\) (and if \(\mathcal{B}\cong \mathcal{D}^b(X)\))