Proposed topics

E. Calabi, born in Milano in 1923, conjectured the existence of a special sorts of manifolds. These manifolds turn out to enjoy pretty interesting properties and have many applications to theoretical physics. Nowadays, the study of CY surfaces (especially K3) and threefolds becomes a large body in mathematics.

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

Proposed by: Ziqi Liu

There's the normal procedure in many modern areas of pure mathematics: our work would be based on results of other people, and it is becoming increasingly the case that we don't know the proofs of these results. There is more and more evidence that some of results in the work are either not completely proved, or are completely proved but the proofs are not in the literature and may never be in the literature.

Nowdays we are now entering an age where the computer will not just help us to compute, but help us to reason. Even more, it helps us to do some mathematics at post-undergraduate level:

• 2004: Gonthier formally verified the Four-Color Theorem using "Coq"

• 2005: Hales verified the Jordan Curve Theorem using "Isabelle/HOL"

• 2021: A team lead by Johan Commelin formalised the proof of a fundamental theorem of liquid vector spaces (which is a very new mathematics developed by Peter Scholze and Dustin Clausen) using "Lean"

So formalising mathematics in a computer proof checker would change mathematics for the better! It can be also used to teach undergraduates mathematics in the new way (that is what Kevin Buzzard is doing with his Xena Project).

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

Proposed by: Kaixing

A Kähler manifold is a smooth manifold of even dimension endowed with a Riemannian metric and a complex structure which is parallel with respect to the Levi-Civita connection induced by the metric. Alternatively, one can define a Kähler manifold by considering its holonomy group, which must be contained in the unitary Lie group. Kähler geometry has proven to be a wide research area, both from an algebraic and a geometric viewpoint, due to the richness given by the existence of many intertwined structures on a Kähler manifold.

In 1954, Calabi conjectured the existence of a unique solution for a "prescribed curvature" problem on Kähler manifolds: more precisely, he stated that, given a differential 2-form representing the first Chern class, there exists a unique Kähler metric in each Kähler cohomology class such that this form is in fact the Ricci form of the manifold, representing then the Ricci curvature of this Kähler metric. In particular, the validity of the conjecture implies that, given a Kähler manifold whose first Chern class vanishes, each Kähler class contains exactly one Ricci-flat metric.

The proof of the uniqueness of the solution is due to Calabi, while the proof of the existence is due to Yau (1977-1978): their resolution is innovative, since they managed to transform a geometric problem into an analytical one, transporting the cohomological setting into the PDE world and using many different tools of Geometric Analysis. 

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

Proposed by: DD

From Wikipedia: "Tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition". It provides a way to transform difficult algebro-geometric problems in much easier ones. Plus, you get nice pictures. 

IT HAS REAL WORD APPLICATIONS, also from a computational point of view.

The idea for this seminar is to introduce the topic and then freely explore it, according to the speakers interests.

Topic areas: MAT/03 Geometry (+ other fields as applications)

Proposed by: Federico

Classically, a Riemann surface is a connected one-dimensional complex manifold. There are rich theories from various perspectives on it, which provide a road to higher dimensional study. So different people learn/introduce it in different ways: from function theory to complex analysis and differential equations, from analytic functions to number theory or analytic geometry, from vector bundles to algebraic geometry, etc.

In the seminar, we would like to understand Riemann surfaces from as many aspects as possible (from the ways mentioned above, but also from others like homotopy type, representation theory) to satisfy the tastes of the most of us and help others learn something new (like somebody who knows little about algebraic/analytic geometry can see how vector bundles work in the geometry).

In addition, the key role in the very recent work on geometrization of local Langlands correspondence, found by Fargues and Scholze, is a "Riemann Surface" defined by Fargues and Fontaine. So if people are curious about the link between "Riemann surfaces" and local Langlands correspondence, we can also discuss Fargues's idea in the seminar, at least in terms of simple cases.

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

Proposed by: Kaixing

A seminar after the works of Dustin Clausen and Peter Scholze. They provide a way towards "analytic geometry", an algebraic and topological perspective on analytic concepts. The main topics involved here are topology, complex geometry, mathematical analysis. 

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

Proposed by: Federico

Explore the works on packing the n-dimensional space with spheres: what is the optimal way to dispose (n-1)-dimensional equal spheres in R^n if we want to occupy the most volume? What if we allow different sizes?
This problem has fascinated mathematicians for centuries, and is still of great interest nowadays. A 2016 paper on this problem resulted in the award of the Fields medal to the mathematician Maryna Viazovska. Over the years, it has been tackled with the most different techniques. Nowadays, along with topology and mathematical analysis, numerical methods play a significant role.

Topic areas: MAT/02 Algebra, MAT/03 Geometry, MAT/05 Mathematical analysis, MAT/08 Numerical analysis 

Proposed by: Federico

In 1967, Quillen introduced model categories as a new framework for doing homotopy theory. As the homotopy theory is frequently applied in K-theory and algebraic geometry, this language has been used among more mathematicians. So it would be interesting to understand the motivation and the applications of the theory of model categories.

This seminar may consist of the following aspects:

• Motivation, definitions and easy examples

• Nice properties of model categories

• Model categories and dg-categories

• Model categories and (∞,1)-categories

• Model categories and motives

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

Proposed by: ZL

Moduli spaces are spaces describing a certain sort of objects up to a certain kind of equivalence. The study of the various moduli spaces is very old but also very new. In this seminar we will discuss your favourite moduli space such as:

• moduli space of certain varieties

• moduli space of certain sheaves

• moduli space of complex structures

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

Proposed by: ZL

The concept of phase transition (p.t.) is ubiquitous in physics and mathematics as well . Examples go from magnetization to percolation theory, and many others. Despite the variety of environments where they appear, p.t.  always carry on some key features (i.e. breaking of analyticity, polynomial correlation length, universality of critical exponents) that allow to consider p.t. as a natural phenomenon fully described by mathematics. The proposal is to dive into some examples and then extrapolate the general picture.

Topic areas: MAT/01 Mathematical logic, MAT/03 Geometry, MAT/05 Mathematical analysis, MAT/06 Probability and mathematical statistics, MAT/07 Mathematical physics, MAT/08 Numerical analysis

Proposed by: zamon

J. Lurie is well know to be a great mathematician, a genius actually, and a great writer: he has  written three gigantic books.  

The first two of them aim to define the mathematical tools for the the last one, which is the masterpiece (if you are a fan of J. R. R. Tolkien this is exactly the same relationship between Silmarillion and The Lord of the Rings) where, for some better mathematicians than me, he wants to refound Grothendieck's work "EGA".

The program of this seminar includes the study of the first chapter of the last book, entitled Fundamentals of Spectral Algebraic Geometry.

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

Proposed by: Teox

As a matter of fact, nowadays it is important for many different mathematicians to have knowledge of Higher Category Theory. 

In this seminar we will give an introduction of this theory in the most model-independent way known today, that is, using the very recent ∞-cosmoi Theory.

The program of this seminar involves studying the note of a course that E. Pavia (former doctoral student in Milan and dear friend) gave last years at the International School for Advanced Studies of SISSA, entitled: "A very concise introduction to higher category theory and homotopical algebra", available here.

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

Proposed by: Teox

Traditionally, computations in mathematics were done solely by hand. When computers first came about mathematicians were eager to use them to speed up calculations. Think back to the days when the likes of Euler had to find the digits of π by hand! The development of computers has been tremendous and one can now carry out calculations on a personal laptop that would take years to complete by hand in just a matter of seconds. This clearly was a huge advantage for contemporary mathematicians and physicists. Imagine how much more Euler would have achieved if he had a symbolic package like Mathematica to help with the mundane tasks of obtaining integrals and series.

And this brings us to the heart of the matter. We often forget, when reading the final and polished journal papers, that theoretical research - even in pure mathematics - begins with experimentation. Gauss would plot the number N(x) of primes less than a given real number x and notice that it is approximated by x/log(x), long before complex analysis was invented to offer a proof. Einstein would add a Λ constant to his field equations just to toy with a cosmological model long before the theories of inflation or evolution. This is what mathematicians do; we experiment and then we formalize.

(above paragraphs are almost directly copied from arXiv:2303.12626)

In this case, we might be interested in applying machine learning method to study math. Though I do not know how to, but it seems that they are helpful. Also, it might trigger meaningful discussions among people doing different areas. Here are some possible topics:

• ML and number theory (e.g. arXiv:1911.02008, arXiv:2012.04084)

• ML and complex geometry (e.g. arXiv:2209.10157)

• ML and pure analysis (such as ML and PDE)

• ML and operative research (many many I believe)

Of course, we also need to learn about machine learning which is closely related to statistics and numerical analysis.

Topic areas: MAT/02 Algebra, MAT/03 Geometry, MAT/05 Mathematical analysis, MAT/06 Probability and mathematical statistics, MAT/07 Mathematical physics, MAT/08 Numerical analysis, MAT/09 Operative research

Proposed by: ZL

In plane geometry, the einstein problem asks about the existence of a single tile that can tessellate space but only in a nonperiodic way. Such a shape is called an einstein, a wordplay on ein Stein, German for "one stone".

Several variants of the problem, depending on the particular definitions of nonperiodicity and the specifications of what sets may qualify as tiles and what types of matching rules are permitted, were solved beginning in the 1990s. The strictest version of the problem was solved in 2023 (the initial discovery was in November 2022, with a preprint published in March 2023), pending peer review. 

Our goals will be:

• Delve into the various formulations of the tessellation problem, making explicit the connections between the many areas of mathematics involved;

• Try to present the results obtained over the years, to understand the path that led to the recent candidate solution;

• Investigate some possible real-world applications of this quite simple to formulate problem;

• Of course, provide plenty of colored pictures and drawings!

Topic areas: MAT/02 Algebra, MAT/03 Geometry, MAT/06 Probability and mathematical statistics, MAT/07 Mathematical physics

Proposed by: Garibaldi

Have you ever 'looked' at your jean pant? Would you believe if someone told you that your jean pant could be used for some mathematical classifications? Now, take a look…What do you see?

Apart from discovering that it hasn’t been washed for a while, we also evidently notice some holes. With a topological lens, we can see that there are actually 3 holes; two on the bottom (on the legs) and one on the top (along the waist). This simple model forms a basis for the classification problem of manifolds and has opened up an exciting area of research by vast generalization of the jean pant model. It is called the (CO-)BORDISM theory.

In Layman's language, "a cobordism is an equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold". Two manifolds of the same dimension are 'cobordant' if their disjoint union is the boundary of a compact manifold with one dimension higher. 

So, coming back to our pant: if we denote the upper circle (waist) by A, and the lower circles (legs) by B, C, we see that A, B, and C together make up the boundary of the pant P. Thus, the pant P is a cobordism between the circles A, B, and C. 

OK, but what's the deal here? 

Unlike diffeomorphism or homeomorphism, cobordism is a much coarser equivalence relation on manifolds; And so, it is significantly easier to study and compute. For e.g., it is not possible to classify manifolds up to diffeomorphism or homeomorphism in dimensions ≥ 4, but it is possible to classify manifolds up to cobordism! 

Moreover, cobordism has several exciting features and is closely interconnected between other theories such as Morse theory or topological quantum field theory, that find applications in both pure and applied mathematics.

Possible topics of seminar:

1. Cobordism from the categorical perspective and several MINDBLOWING examples and survey of the literature; Ref: Chapter 1 and 4 of [IV].

2. Surgery classification of manifolds and effectiveness of cobordism among other equivalence relations; Ref: Chapter 1 of [III]

3. Basics of Morse theory and h-cobordism leading to the handle decomposition theorem; Ref: Chapter 2.3, 2.4 of [III]

4. Definition of the (un-)oriented and framed cobordism group, basics elements of Complex cobordism = Universal oriented cohomology theory and some applications; Ref: [II], 6.2 and 6.3 of [III], this, this

5. Leray-Hirsch theorem and Thom isomorphism, Thom-Pontryagin construction; Ref: Chapter 2, 3 of [II], [VI], this

6. Basic overview of algebraic cobordism (analogous theory for varieties and schemes) and fundamental properties, relations with complex cobordism; Ref: [VII], this

7. Basic ideas of TQFT and introduction to the Cobordism hypothesis; Ref: [VIII], [IX], [X], this 

Possible references:

[I] Basic ideas from Wikipedia: link
[II] A lecture course on Cobordism Theory - Johannes Ebert: link
[III] Algebraic and Geometric surgery - Andrew Ranicki: link
[IV] Notes on cobordism theories – Robert E. Stong: link
[V]   Topology from the differential viewpoint - John W. Milnor: link
[VI]  Characteristic Classes and Cobordism - Alberto San Miguel Malaney: link
[VII] Algebraic cobordism – Marc Levine, Fabien Morel: link
[VIII] Introductory lectures on topological quantum field theory - Nils Carqueville, Ingo Runkel: link
[IX] On the Classification of Topological Field Theories - Jacob Lurie: link
[X] Introduction to the Cobordism hypothesis after Hopkins-Lurie - Araminta Amabel: link

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

Proposed by: Krishna

Mean field games (MFGs) have been an active theme of research for almost two decades, started in the mid 2000’s from the seminal works of Lasry and Lions and of Huang, Malhamé and Caines.

Roughly speaking, MFG theory is the study of strategic decision making by small interacting agents in very large populations. It lies at the intersection of game theory with stochastic analysis and control theory. In other words, each agent acts according to his minimization or maximization problem taking into account other agents’ decisions and because their population is large we can assume the number of agents goes to infinity and a representative agent exists.

There are plenty of applications, from economics to population dynamic, from financial markets to epidemic models.

Since theory is cool and easy, but in the end the real world needs numerics, our aim will be:

•  give a quick introduction to the MFGs theory and models, presenting significant examples and applications, as well as the main approaches to them (analytical and probabilistic);

• present some of the classical numerical methods for MFG;

• try to understand the more recent techniques, that involve for instance deep-learning and reinforced-learning;

• show many coloured graphs and funny examples;

• LEARN HOW TO BECOME A TRADER AND GAIN MILLIONS CLICK HERE FOR DETAILS.

The idea is to follows more ore less some recent lectures by Mathieu Laurière. 

Topic areas: MAT/05 Mathematical analysis, MAT/06 Probability and mathematical statistics, MAT/07 Mathematical physics, MAT/08 Numerical analysis, MAT/09 Operative research

Proposed by: Baldigari