What is... Seminar

What is a simplicial set?

Maru Sarazola

Abstract: Simplicial sets are a type of mathematical structure that we encounter *all the time* when doing algebraic topology. In this talk, I will explain what they are, how topologists like to think about them, and illustrate their usefulness through some examples.

What are some analytical questions about "elements" in a number ring?

Yashi Jain

Abstract: An obvious generalisation of the Riemann zeta function for integers is a version of Riemann zeta function for number rings stated in terms of the ideals of the number ring. Mathematicians have talked about various generalisations of analytic questions for integers for number rings as well. A few examples are: the restatement of prime number theorem for ideals, or the generalisation of Dirichlet’s theorem on primes in arithmetic progression (called the Chebotarev density theorem). But all these analytic questions are stated in terms of ideals. What if we want to ask analytic questions about the elements in the number ring? This talk will deal with exactly this question.

What is the space of modular forms?

Rui Chen

Abstract: Modular forms are a special class of holomorphic functions on the upper half plane, they have symmetry with respect to the modular group SL2(Z), and their fourier coefficients encode rich arithmetic information. In this talk, we will introduce the definition of the space of modular forms and discuss the dimension formula for modular forms.

What is a minimal surface?

Letian Chen

Abstract: Minimal surface is the mathematical model of idealized soap film. The study of minimal surfaces dates back to Lagrange, and they have since become a central object of study in geometric analysis. In this talk I will give various definitions of minimal surfaces originating from different aspects of mathematics. I will then talk about two major problems in the study of minimal surfaces, the Plateau problem and the Bernstein problem.

What is birational geometry?

Daniil Serebrennikov

Abstract: The classification of algebraic varieties up to isomorphism is very complicated. Meanwhile, it seems possible to classify algebraic varieties up to birational modification. In other words, it is an isomorphism outside lower-dimensional subsets. In this talk, I will explain why such classification is a natural method in algebraic geometry, introduce some basic examples, and formulate the minimal model program.

What is cohomology?

Luqiao Xu

Abstract: What is cohomology? From chain cohomology to singular cohomology, various fields of mathematics have their own cohomology theories, such as De Rham cohomology, sheaf cohomology etc. What unites them as one? Without using derived functors or resolutions, we’ll use spectrum to define and derive important properties of cohomology.

What is forcing?

Benjamin Dees

Abstract: In the 1960s, Paul Cohen solved the continuum hypothesis, by showing that there is a model of the ZFC axioms of set theory which does not satisfy this hypothesis. The crucial new technique Cohen developed is called "forcing." The key facts about forcing are that it allows one to construct new models of set theory out of old ones, and it relates facts in these new models to facts in the old. While we will not explore the details of this technique, this talk will aim to provide context and a high-level explanation of the key ideas for forcing.

What is constructive mathematics?

Sean O'Connor

Abstract: Constructive mathematics has been growing in popularity recently, but what is it? It is not merely the study of classical mathematics with one hand needlessly tied behind your back. On the contrary, in certain situations you can only work with constructive mathematics, but far from being a hindrance, this can open up entirely new perspectives on existing mathematical subjects, such as in the case of synthetic differential geometry.

What is a tqft (for a mathematician)?

Tomas Mejia Gomez

Abstract: Topological quantum field theories (TQFTs) are theoretical frameworks originating from quantum physics which compute topological invariants of geometric objects such as manifolds. In this talk, we will explore the Atiyah-Segal functorial formalism for TQFTs and the algebraic structures that it entails in low dimensions. This will allow us to approach interesting examples in 2D, but some heuristics of theories in higher dimensions will be provided.

What is an affine scheme?

Sean Owen

Abstract: Schemes are one of the main objects of study in modern algebraic geometry. The field of algebraic geometry starts with the study of solution sets of systems of polynomial equations over fields, and seeks to understand them by examining the corresponding polynomial rings. Applying these techniques to arbitrary commutative rings then yields schemes. Passing to this broader class of geometric objects means accepting more abstraction and technical machinery, but in return, not only do we gain a richer set of tools for studying polynomial systems, we also gain access to novel objects that can be used to study other problems. A key class of examples are arithmetic schemes, which are associated to rings of algebraic integers and their relatives and have applications in number theory.

What is topological K-theory?

Alvaro Belmonte

Abstract: Topological K-theory is a (generalized) cohomology theory. In this talk we will introduce the main objects used to define the algebraic invariant (up to homotopy), define the functor K^n(-) and show that these form a spectrum.

What is a model category?

Fan Huang

Abstract: Homotopy theory, roughly speaking, is the study of maps up to homotopy. A model category is a framework for “doing homotopy theory” in a general setting. We will explain the structure by reviewing homotopy on topological spaces, as well as several examples of other model categories. Finally, we will show some applications of model category in the theory of infinity categories.