Osmosis and the Oracle
Pure Mathematics Student Seminar
This seminar, run by the School of Mathematics and Statistics at the University of Melbourne, is for maths students of all walks and stages of life: undergraduates, graduates, provers, poets, learners, teachers, thinkers and inspirers.
The Pure Mathematics Student Seminar is held as a joint event with the Pure Mathematics Seminar. Everyone is welcome (and encouraged) to attend both.
2023 Semester 2
Time and place
Fridays 4:30 - 5:30pm
Peter Hall Building, Room 162
In Semester 2 2023, talks will be guided by two principles: Osmosis and the Oracle.
Osmosis
It is a policy of maths education that the most useful and needed topics not be covered formally in coursework: things like tensor products, the Koszul complex, K-theory, etc. This seminar, aimed at undergraduates and beyond, attempts to identify and provide a venue for filling in some of these topics. Your requests for topics to be included will be greatly appreciated!
Ask the Oracle
When working on research an important part of the process is to formulate questions for the Oracle. If you had a direct line to the Oracle, what would you ask? What is it you really need to know? Sometimes you’ve been looking for the definition of blonkdupzis for weeks and can’t find it, and that is holding up your understanding (and your PhD thesis). In this seminar the speakers will give talks about their work and formulate their 'ask the Oracle' questions. The organisers will then do their best to make the phone call for you and follow up on the answers.
Organisers
Mailing list
To subscribe to our mailing list, contact Davood Nejaty, or visit the link below (University of Melbourne login required):
https://lists.unimelb.edu.au/info/pure-maths-student-seminar
Upcoming talks
Past talks
13 October
Kurt Stoekl (University of Melbourne)
The Three Little Graces and the Big Bad Basis
In this talk, we will discuss algebraic operads and a general method for proving they are Koszul. First, we will introduce operads and three key examples, known as the graces. These are the operads whose algebras/representations are associative, commutative, and Lie algebras respectively. After discussing what it means for an operad to be Koszul, we will show that this property is implied by the existence of a conceptually simpler, confluent terminating rewrite system, i.e. a Groebner basis. Finally, we will work through examples showing that the three graces are Koszul, and discuss further applications of this technique to other algebraic structures.
6 October
Yau Wing Li
Soergel bimodules
The theory of Soergel Bimodules plays an important role in representation theory, combinatorics, and topology. Soergel associated every Coxeter group with a set of graded bimodules over a polynomial ring. In the case of a Weyl group, they correspond to the (equivariant) intersection cohomology of Schubert varieties. In this talk, I will provide a brief introduction to Soergel Bimodules.
22 September
Zhihang Yu
Kazhdan-Lusztig theory
Let $\mathfrak{g}$ be a semisimple Lie algebra. In the representation theory of semisimple Lie algebras, an important question is to determine the formal characters of the irreducible modules of $\mathfrak{g}$. In the 70's, Kazhdan and Lusztig defined what is now known as the Kazhdan-Lusztig polynomial, and conjectured a formula which allows one to calculate formal characters from this polynomial (the KL conjecture). The conjecture was proven independently by Beilinson-Bernstein and Brylinski-Kashiwara.
In this talk I will explain how to define the KL polynomial from a certain Hecke algebra associated to $\mathfrak{g}$, and some basics about a certain subcategory of $\operatorname{Rep}(\mathfrak{g})$, named BGG category O. We will also discuss some results of KL,BB and BK.
15 September
Ethan Fursman (University of Melbourne)
W-algebras
W-algebras are a type of vertex algebra with rich mathematical structure. However what exactly is a W-algebra? In this talk I will give a very basic overview of conformal invariance and vertex algebras. This will be followed by an accessible introduction to the two flavours of W-algebras: finite-type and affine-type. Finally I will briefly mention how the types are related, and how W-algebras are useful in mathematics and physics.
8 September
Ali Khalili Samani (University of Melbourne)
Categorical Hopf map
I introduce the categorical Hopf map, higher analog of the Hopf map. I explain its relation to the Hopf map. I show that we can write the categorical Hopf map as a composition of the Hopf map and the basic bundle gerbe over a three-dimensional sphere. If time permits, I will talk about some motivations for the categorical Hopf map. As I review most of the background, your undergraduate knowledge is enough.
18 August
Fei Peng (University of Melbourne)
Reconstruction problems in geometry
Reconstruction problems form another class of problems of great interest in geometry. In this talk, I will provide a gentle introduction to the problem of reconstruction in the context of algebraic geometry, and list a few examples and results of reconstruction in algebraic geometry.
4 August
Arun Ram (University of Melbourne)
The secrets of a successful career around mathematics
28 July
Oliver Li (University of Melbourne)
A gentle introduction to moduli spaces in geometry
Moduli spaces are everywhere in geometry and mathematical physics, but it is perhaps in algebraic geometry that they are the most studied. In this talk, I will give a very accessible introduction to the types of questions one thinks about when studying moduli. The only prerequisite is some abstract algebra.