Departement Mathematik

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Zurich graduate colloquium

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Frühjahrssemester 2024

Datum / Zeit Referent:in Titel Ort
5. März 2024
15:30-16:30 		Laura Marino
IMJ Paris
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Zurich Graduate Colloquium

Titel What is... a rational tangle?
Referent:in, Affiliation Laura Marino, IMJ Paris
Datum, Zeit 5. März 2024, 15:30-16:30
Ort KO2 F 150
Abstract ''Knot theory is the study of knots, that is embedded circles into R^3 (or S^3) considered up to isotopy. A generic intersection of a knot with a 3-ball is called a tangle. In this talk, I will focus on tangles of a particular kind, called rational tangles. These tangles are in 1:1-correspondence with rational numbers and share many of their properties. After introducing the relevant ingredients and results, I will present the proper rational unknotting number, an invariant of knots defined in terms of rational tangles.
What is... a rational tangle?read_more
KO2 F 150
12. März 2024
16:30-17:30 		Segev Gonen Cohen
ETHZ
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Zurich Graduate Colloquium

Titel What is... Hilbert's 17th problem?
Referent:in, Affiliation Segev Gonen Cohen, ETHZ
Datum, Zeit 12. März 2024, 16:30-17:30
Ort KO2 F 150
Abstract ''Suppose that we are given a polynomial which can be expressed as a sum of squares of polynomials with real coefficients, for example f(X,Y,Z) = (X-4Y)^2 + (17Z^3 - 4XYZ)^2. Then it is clear that no matter what (real) inputs we put in, we will get something positive. Minkowski, in his PhD defence, asked about the converse - suppose we have a polynomial that takes only positive values, must this be because it is a sum of squares? Hilbert, who was sat in the audience, realised that the answer is no; but modified the question slightly, and famously included it as the 17th in a list of open problems presented at the 1900 ICM in Paris. The main part of this talk will be devoted to the solution to this problem, relying on some algebraic foundations of Artin, and a nice model-theoretic proof due to Robinson. Time permitting we will look at modern day generalisations of this question, partial solutions, and (depending on audience interest) their applications to the Connes Embedding Problem, Kazhdan's Property (T), norm computability, and more. No knowledge of any of these topics will be assumed.
What is... Hilbert's 17th problem?read_more
KO2 F 150
19. März 2024
16:30-17:30 		Ana Marija Vego
ETHZ
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Zurich Graduate Colloquium

Titel What is... an Iwasawa algebra?
Referent:in, Affiliation Ana Marija Vego, ETHZ
Datum, Zeit 19. März 2024, 16:30-17:30
Ort KO2 F 150
Abstract ''The Iwasawa algebra is a key object in the study of p-adic L-functions, which are a central topic in number theory. The Iwasawa algebra arises naturally in this context as a tool for understanding the behavior of certain arithmetic invariants, such as Selmer groups and class groups, in towers of number fields. It provides a framework for studying these invariants in a unified way over all the levels of the tower. This allows us to investigate the arithmetic properties of number fields and their associated objects, particularly in the context of p-adic L-functions and Galois representations. It has applications in various areas of number theory, including the study of special values of L-functions, the Birch and Swinnerton-Dyer conjecture, and the structure of class groups of number fields. In this talk we will introduce Iwasawa algebras and give some basic properties. We'll then explore how these algebras are used in constructing Euler systems and obtaining p-adic L-functions. If time allows, we'll also touch on the main conjecture of Iwasawa theory.
What is... an Iwasawa algebra?read_more
KO2 F 150
9. April 2024
16:30-17:30 		Thomas Jacob
Universität Zürich
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Zurich Graduate Colloquium

Titel What is... a motive?
Referent:in, Affiliation Thomas Jacob, Universität Zürich
Datum, Zeit 9. April 2024, 16:30-17:30
Ort KO2 F 150
What is... a motive?
KO2 F 150
16. April 2024
16:30-17:30 		Yuriy Tumarkin
Universität Zürich
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Zurich Graduate Colloquium

Titel What is... a translation surface?
Referent:in, Affiliation Yuriy Tumarkin, Universität Zürich
Datum, Zeit 16. April 2024, 16:30-17:30
Ort KO2 F 150
Abstract ''If you take a square and glue the opposite sides together, you get a flat torus. What happens if you start with a different polygon instead, say a regular octagon? The result is a translation surface, a central object in the field of Teichmüller Dynamics. Of particular interest is the straight-line flow on a translation surface, a simple to define dynamical system that arises from the study of billiards in polygons. In this talk I will give a friendly introduction to a few of the key concepts of the field, such as the moduli space of translation surfaces, and the idea of using renormalisation to study the dynamics on a translation surface.
What is... a translation surface?read_more
KO2 F 150
30. April 2024
16:30-17:30 		Konstantin Andritsch
ETHZ
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Zurich Graduate Colloquium

Titel What is... an adelic torus orbit?
Referent:in, Affiliation Konstantin Andritsch, ETHZ
Datum, Zeit 30. April 2024, 16:30-17:30
Ort KO2 F 150
Abstract ''As the term suggests - Adelic torus orbits - are nothing but the orbit of an algebraic torus over the ring of Adeles. They provide a powerful tool to collectively study the behavior of collections of geometric data given by arithmetic data. In this talk we will motivate the use of adelic torus orbits by looking at a concrete example: > Already in the 19th century Gauss studied integral binary quadratic forms. He observed that there are essentially only finitely many different integral binary quadratic forms with fixed discriminant. In more modern terms, these different forms arise through a natural action of the ideal class group of a quadratic number field. To study the properties of different forms at the same time it is convenient to consider the Adelic extension of the modular curve. We will see that forms who are not equivalent over the integers might be equivalent over the Adeles. After introducing the necessary concepts and motivating the idea behind Adelic torus orbits we will discuss how they can be used to prove equidistribution results on (real) homogeneous spaces.
What is... an adelic torus orbit?read_more
KO2 F 150
7. Mai 2024
16:30-17:30 		Diane Saint Aubin
Universität Zürich
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Zurich Graduate Colloquium

Titel What is... Bose-Einstein condensation?
Referent:in, Affiliation Diane Saint Aubin, Universität Zürich
Datum, Zeit 7. Mai 2024, 16:30-17:30
Ort KO2 F 150
Abstract ''A Bose-Einstein condensate (BEC) is a state of matter that is formed when a low-density gas of bosons is cooled to near absolute zero. Under these conditions, a majority of the particles occupy the same quantum state and quantum effects become apparent. First predicted by Bose and Einstein a century ago, BECs were realised in laboratories over eight decades later and led to the Nobel price in 2001. Since then, many progresses have been made in the rigorous study and understanding of quantum many-body systems from a mathematical point of view. In this talk we will give an introduction to quantum mechanics and to the mathematical description of quantum many-body systems. We will then present a formal definition of BEC.
What is... Bose-Einstein condensation?read_more
KO2 F 150
14. Mai 2024
16:30-17:30 		Alessio Cangini
Uni Basel
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Zurich Graduate Colloquium

Titel What is... the Schinzel-Zassenhaus conjecture?
Referent:in, Affiliation Alessio Cangini, Uni Basel
Datum, Zeit 14. Mai 2024, 16:30-17:30
Ort KO2 F 150
Abstract ''An algebraic integer is a complex number which is a root of a monic irreducible polynomial with integer coefficients. The complete set of zeros of such a polynomial is called a conjugate set of algebraic numbers. Bounding the maximum absolute value of elements in these sets from below has been studied intensively over the years by number theorists. We will call this maximum the house of an algebraic integer. In 1965, Schinzel and Zassenhaus proposed the following conjecture. There exists an absolute positive constant C such that the house of every non-zero algebraic integer which is not a root of unity is at least 1 + C/d. The above conjecture was proved in 2019 by Dimitrov. In this talk we will introduce the relevant notions and go over Dimitrov's proof.
What is... the Schinzel-Zassenhaus conjecture?read_more
KO2 F 150
28. Mai 2024
16:30-17:30 		Dr. Samir Canning
ETHZ
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Zurich Graduate Colloquium

Titel What is... the cohomology of moduli spaces of curves?
Referent:in, Affiliation Dr. Samir Canning, ETHZ
Datum, Zeit 28. Mai 2024, 16:30-17:30
Ort KO2 F 150
Abstract ''The moduli space of curves was first studied by Riemann. I will explain what it is, how to compactify it, and how to attempt to compute the cohomology of its compactifications. Ideas from a broad range of mathematics are necessary, including low dimensional topology, algebraic geometry, and number theory.
What is... the cohomology of moduli spaces of curves?read_more
KO2 F 150

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