Proposed topics for the first edition

The harmonic measure of a subset of the boundary of a bounded domain in R^n can be defined as the probability that a standard Brownian motion, starting inside the domain, first hits that subset of the boundary. I.e. the harmonic measure of a sphere, for a Brownian motion starting in its center, is the standard Lebesgue measure. My question is: what can we say about the harmonic measure of a square? Are we able to describe the density associated? The problem can be approach both from a dynamical point of view or from the point of view of potential theory and harmonic analysis in general. 

Topic areas: MAT/05 Mathematical analysis, MAT/06 Probability and mathematical statistics, MAT/07 Mathematical physics

Proposed by: Zamon

Almost every widespread programming language pertain to the so-called imperative paradigm. In contrast, there exist some purely non-imperative languages: it is the so-called functional paradigm. In particular, they do not admit side-effects in execution. The essential logical structure in functional programming, that enables to avoid side-effects, is the purely categorical concept of monad. My question is double: what is a monad, why overcome the absence of side-effects?

Topic areas: MAT/01 Mathematical logic, MAT/02 Algebra, MAT/03 Geometry

Proposed by: Zamon

Disclaimer: I think Applied Category Theory is, using a metaphorical image, a rock trying to be a hedgehog.

According to Forbes, quantum computing is "a new generation of computer technology on the horizon," and some mathematicians are looking for a language that can help study this new technology and Quantum Categorical Theory is one attempt to satisfying this necessity.

It may be interesting to study this theory now that it is in its primordial state and it does not cost so much effort because, if AND I SAY IF Forbes's right, it could be a fortune.

Also, if I understand correctly, this theory uses symmetric monoidal categories and (co)monad theory; tools that, being present in many areas of mathematics, are useful to know about. 

Topic areas: MAT/02 Algebra, MAT/07 Mathematical physics, MAT/09 Operative research

Proposed by: Teo

Topos theory is a deep mathematical theory with ideas coming from different mathematical backgrounds. In the book Sketches of an Elephant, a.k.a the Bible of Topoi, Johnstones  writes thirteen different definitions in the introduction and, and if my mind does not deceive me, he says that there exists six different approaches to this theory.

I am interested in the geometric (a.k.a sheaves theory), logical and algebraic approaches (definitions: i-iv-viii from the cited list). In particular I would like to understand the following sentence of the Bible : "A Grothendieck Topos is the same thing as a Morita equivalence class of geometric theories".

I have any clue how, but topos theory is used  to do differential geometry; indeed definition (xii) is :" A Topos is a setting for synthetic differential geometry"

Topic areas: MAT/01 Mathematical logic, MAT/02 Algebra, MAT/03 Geometry

Proposed by: Teo

In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor.

Topic areas: MAT/05 Mathematical analysis, MAT/07 Mathematical physics

Proposed by: LadyS.

Introduction to the concepts of metric, change of metrics and so on.

Topic areas: MAT/03 Geometry, MAT/05 Mathematical analysis, MAT/07 Mathematical physics

Proposed by: LadyS.

Today, 2-category theory is widespread in mathematics: the definition of adjunction (one of the most important in category theory) is a 2-categorical definition; in algebraic geometry the Deligne-Mumford stack 2-category plays an important role; in the theory of infinity categories (consequently in homotopy theory) Joyal, Quillen, Hovey, Riehl and Dominic have shown that almost all results of interest can be recast in this theory using the homotopy category of an ∞-cosmos; the 2-category of Hilbert spaces (see the proposed topic in SeMIVA: categorization of 2-Hilbert spaces) is another example.

Given its recurrence, it may be useful to learn about this intricate theory full of pitfalls.

In particular, our aim is to study the scary lax and pseudo structures with the goal of fully understanding Grothendieck's construction, namely the following result:

Theorem (E.Riehl- Two-sided Discrete fibrations in 2-categories and bicategories.) There is a 2-equivalence of 2-categories:

Fib(B) ≃[Bop, Cat]ps,

where the latter is the 2-category of pseudo-functors, pseudo-natural transformations and modifications.

Important fact: the above result is related to categorical Galois theory (another topic proposed in SeMIVA); indeed, Grothendieck probably finds it to generalize Galois theory.

Topic areas: MAT/02 Algebra, MAT/03 Geometry, MAT/05 Mathematical analysis, MAT/07 Mathematical physics

Proposed by: Teox

In a elegant paper in ‘73, Lawvere showed that the category of metric spaces is R-enriched.

In mathematical analysis one of the fundamental concept is Hilbert Spaces, which are normed metric spaces.

After reading the preceding lines a mathematician, especially if they are a fan of movies like Inception and Tenet, will have immediately wondered: for Hilbert Spaces, is there an analogous result to Lawvere’s?

According J. Baez the answer is yes!

The object of this seminar is to study this analogy and its goal is understand the following sentence in one of Baez’s notes:

“We define a 2-Hilbert space to be an abelian category enriched over Hilb with a ∗-structure, conjugate-linear on the hom-sets, satisfying” some axioms.

Topic areas: MAT/01 Mathematical logic, MAT/02 Algebra, MAT/03 Geometry, MAT/04 Complementary mathematics, MAT/05 Mathematical analysis, MAT/07 Mathematical physics

Proposed by: Teox

Borceaux and Janelodze begin their book on Galois theory with: "E. Galois (1811-1832) would certainly be surprised to see how often his name is mentioned in the mathematical books and articles of the twentieth century, in topics which are so far from his original work.”

And as he could not be! His idea, thank the work of important mathematicians as Artin, Poincare, Grothendieck etc., turned out to be only the tip of an iceberg.

Nowadays we use this theory in logics, homotopy theory, cohomology theory, sheaves, covering spaces, field extension, topos theory, Hodge theory and so on.

I personally  find useful for my research that that this theory allows us to see fibrations over a category B as functors to the category of groups on B. If this last statement upsets you no problem,  pretend you haven’t read it and replace it with:  connected covering spaces of a (nice) space B is classyfing by subgorups of the fundamental group π(B).

One of the version of categorical Galois theory is the Grothendieck construction (another proposed topic in SeMIVA).

Topic areas: MAT/01 Mathematical logic, MAT/02 Algebra, MAT/03 Geometry, MAT/04 Complementary mathematics

Proposed by: Teox-PoCax

Here are some reasons why elliptic curves have become fundamental objects during the last 150 years of Mathematics:

• Several classic problems in Number Theory turn to be equivalent to finding integer or rational solutions for the equation defining an elliptic curve

• As complex varieties, elliptic curves are biholomorphic to the complex tori

• Elliptic curves can be defined over any algebraic field: in particular, elliptic curves over finite fields are crucial tools in modern Cryptography 

• They are the simplest instance of abelian variety, i.e. they are projective varieties such that they possess a group structure which naturally accords with the geometric structure

• They hold many properties not satisfied by a generic curve: they are isomorphic to their Jacobian, morphisms between elliptic curves are unramified, and so on

• They are deeply linked to complex variable functions denoted as modular forms. Studying their connection resulted, among the other things, in a proof for Fermat's Last Theorem

• Elliptic curves are related to one of the Millenium Problems: the Birch-Swinnerton Dyer conjecture.

Topic areas: MAT/01 Mathematical logic, MAT/02 Algebra, MAT/03 Geometry, MAT/04 Complementary mathematics, MAT/05 Mathematical analysis, MAT/08 Numerical analysis, MAT/09 Operative research

Proposed by: BatFra

In mathematics, the geometric Langlands correspondence is a reformulation of the Langlands correspondence obtained by replacing the number fields appearing in the original number theoretic version with function fields and applying techniques from algebraic geometry. The geometric Langlands correspondence relates algebraic geometry and representation theory.

In mathematics, the classical Langlands correspondence is a collection of results and conjectures relating number theory and representation theory. Formulated by Robert Langlands in the late 1960s, the Langlands correspondence is related to important conjectures in number theory such as the Taniyama–Shimura conjecture, which includes Fermat's Last Theorem as a special case. Establishing the Langlands correspondence in the number theoretic context has proven extremely difficult. As a result, some mathematicians have posed the geometric Langlands correspondence.

I am interested in understanding the geometric picture, in particular the relation between representations of complex algebraic groups and D-modules.

Topic areas: MAT/02 Algebra, MAT/03 Geometry

Proposed by: Filippo M

The Enriques–Kodaira classification of compact complex surfaces states that every nonsingular minimal compact complex surface is of exactly one of the 10 types listed on this page; in other words, it is one of the rational, ruled (genus > 0), type VII, K3, Enriques, Kodaira, toric, hyperelliptic, properly quasi-elliptic, or general type surfaces.

Topic areas: MAT/02 Algebra, MAT/03 Geometry

Proposed by: DD

Whenever we have to compute some (co)homology, we are used to thinking at some Mayer-Vietoris approach at first. However, when spaces get more complicated, it might not be enough to solve our computations. This is when spectral sequences enter the stage: an algebraically more refined means to decompose and gradually approximate (co)homology of a nicely behaved space. (Spoiler alert: some work for homotopy as well!)

The idea for this seminar is to follow [McCleary, J. (2000). A User's Guide to Spectral Sequences]: we will give an algebraic introduction to the topic (chapters 1-3) and we will have a look at the Leray-Serre spectral sequence (chapters 5-6). Time permitting, we may introduce some more exotic object, like the the Adams (chapter 9) or the Bockstein (chapter 10) spectral sequences.

Topic areas: MAT/02 Algebra, MAT/03 Geometry

Proposed by: TeoX and BigBro

Nowadays many mathematicians are fascinated by the world of higher categories, as they are proving to be the real categorical context where to study homotopy (and homology consequently).

Many references will guide us: from the "classical" [Jacob Lurie, Higher Topos Theory] and [Jacob Lurie, Higher Algebra] to more modern expositions, like [Markus Land, Introduction to Infinity-Categories], [Denis-Charles Cisinski, Higher Categories and Homotopical Algebra] or [Emily Riehl and Dominic Verity, Elements of ∞-Category Theory]. We will begin with a general introduction to the subject. Then, according to the interests of the participants, we could stick to a purely categorical approach or see some applications.

Topic areas: MAT/02 Algebra, MAT/03 Geometry

Proposed by: BigBro

The study of varieties over general fields from the classical perspective (i.e. as zero-loci) is unsatisfactory, because assembling points to form subvarieties is far from being trivial. This is the reason why people (Grothendieck mainly) introduced schemes and invented modern algebraic topology.

But as for standard topological spaces, it is unreasonable to classify schemes using strict criteria (for instance, up to isomorphism). People then decided to combine schemes and homotopy theory to create a perfect perspective on algebraic topology. None has succeeded so far (27 March 2023), but many interesting theories arose.

The idea for this seminar is to take a glimpse into those. There are many references and many different perspectives we could choose. A good starting point could be [Lecture Notes on Motivic Cohomology. Carlo Mazza, Vladimir Voevodsky, and Charles Weibel] or my favorite [Motivic Homotopy Theory. Bjørn Ian Dundas, Marc Levine, Paul Arne Østvær, Oliver Röndigs, Vladimir Voevodsky].

Topic areas: MAT/02 Algebra, MAT/03 Geometry

Proposed by: BigBro

$$$$$$$$$$$$$$$$$ MONEY ALERT $$$$$$$$$$$$$$$$$$$

(from one of the references)

"Signature-based techniques give mathematical insight into the interactions between complex streams of evolving data. These insights can be quite naturally translated into numerical approaches to understanding streamed data, and perhaps because of their mathematical precision, have proved useful in analysing streamed data in situations where the data is irregular, and not stationary, and the dimension of the data and the sample sizes are both moderate."

Since I don't know the topic, see the literature below. I only know that there are some interesting applications in finance ($$$$$$$) and that rough paths theory is used (so it's a nice way to learn it).

https://arxiv.org/abs/1603.03788

https://arxiv.org/abs/2206.14674

Topic areas: MAT/05 Mathematical analysis, MAT/06 Probability and mathematical statistics, MAT/08 Numerical analysis

Proposed by: MM

Monte Carlo methods are a broad class of algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used when it is difficult or impossible to use other approaches. One of the main applications is to generate samples from a probability distribution.

When the probability distribution is parameterized, mathematicians often use a Markov chain Monte Carlo (MCMC) sampler. The central idea is to design a judicious Markov chain model with a prescribed stationary probability distribution. That is, in the limit, the samples being generated by the MCMC method will be samples from the desired (target) distribution. By the ergodic theorem, the stationary distribution is approximated by the empirical measures of the random states of the MCMC sampler. 

Topic areas: MAT/06 Probability and mathematical statistics, MAT/08 Numerical analysis, MAT/09 Operative research

Proposed by: JB

In the study of artificial neural networks (ANNs), the neural tangent kernel (NTK) is a kernel that describes the evolution of deep artificial neural networks during their training by gradient descent. It allows ANNs to be studied using theoretical tools from kernel methods. 

https://proceedings.neurips.cc/paper_files/paper/2018/file/5a4be1fa34e62bb8a6ec6b91d2462f5a-Paper.pdf

Topic areas: MAT/05 Mathematical analysis, MAT/06 Probability and mathematical statistics, MAT/08 Numerical analysis

Proposed by: JB

In very general terms, twistor theory consists of the association of a complex space (twistor space) to a real space in a natural and useful way. The goal of this lecture is to introduce the twistor theory presenting the main results and techniques. If it possible also to give an idea of the applications of twistor space in physic.

Topic areas: MAT/03 Geometry, MAT/07 Mathematical physics

Proposed by: M∀Я⟂☮

"Give a 10-year old kid a pencil + paper and request them to draw something.

The chances are that its either made up of lines or circles or a combination of these. And the reasoning behind this is absolutely crucial in mathematics. The lines and circles forms a basis for what's called the 'Algebraic Topology', except that we study them in arbitrary dimensions. They behave dually in terms of the topological properties they hold.

For e.g., the line has no hole in any dimension but a circle has a one 1-dim hole. So, in this sense a circle is seen to be more 'complex' than a line.

A natural craving then, is to compare other objects with the circle (via certain maps) and classify them up to certain interesting classes. Objects that imitate lines will be called 'contractible spaces' and those that imitate the circle as 'non-contractibles'. This is a very rough idea of what a homotopy theorists does for a living! Having said this, the life becomes even more delighting (or perhaps not) when algebra shakes hand with geometry. With the intervention of schemes, the modern homotopy theory had taken a new avatar by transferring the classical setup into the realm of algebraic geometry. 

One such successful instance is due to the work of Morel-Voevodsky in the late 90's, by constructing the homotopy theory for schemes (a.k.a Motivic/A1-homotopy theory). Soon after that, several formulations have been simplified by translating back and forth between these fields. This has also established some powerful connections with the arithmetic and complex geometry. But the payback is that lines and circles are no longer the only protagonists here.

Yes, it is wild out here and the only question is... are you up for a safari?

(Here's something for a "pre-safari": https://arxiv.org/abs/1903.07851#)"

Topic areas: MAT/02 Algebra, MAT/03 Geometry

Proposed by: Krish

Derived categories have increasingly become a powerful tool in all of the main avenues of research in algebraic geometry, making connections with other areas of mathematics and physics. Many of the most important and successful applications of derived category techniques have been in conjunction with the introduction of stability conditions on derived categories and the details study of moduli spaces of stable objects. The goal of these lectures is to introduce the subjects (derived categories, stability conditions and moduli spaces) explain which are the relations between them, the main problem and techniques. Some specific examples are required!

Topic areas: MAT/02 Algebra, MAT/03 Geometry

Proposed by: M∀Я⟂☮