# The mathematics seminar for students

The mathematics seminar is a seminar series for students at the bachelor's programme in mathematics. Other students interested in mathematics are also welcome to participate.

In the autumn seminars, we focus on initial insights into the subject of mathematics. This spring's seminars shed light on ongoing mathematical research.

Contact for the mathematics seminar: Martin Herschend

## Upcoming seminars, spring 2023

### Tuesday 11 april: Lauri Viitasaari

Time: 15:15
Location: Å2002
Title: On integration with respect to functions
Abstract:
Basic integration theory allows to integrate a function f(x) against the variable x, that can be seen as computation of the integral \int f(x)dg(x), where g(x) = x is the identity function. However, in many applications arising e.g. from physics, economics, or AI to simply name a few examples, one needs to consider integrals against more general function g. In particular, in many situations g is itself rather badly behaving function that does not have derivatives, making integration a difficult problem. Moreover, in many applications the function g is not only bad behaving as a function, but has random features in it.

In this talk I will discuss some recent methods to compute such integrals. The focus will be on intuitive ideas behind different concepts, while all the difficult technicalities will be omitted.

### Monday 17 april: Noemi Legout

Time: 15:15
Location: Å4001
Title: Legendrian knots and their algebraic invariants
Abstract:
Legendrian knots are knots satisfying some tangential constraints in the 3-dimensional space equipped with a contact structure. For example, when you skate on a frozen lake, your trajectory follows (the front projection of) a Legendrian. Given two Legendrian knots, it is usually very hard to say if the two are the same or not up to isotopy, i.e. if one can smoothly deform one to the other while keeping the tangential constraints. In order to distinguish and classify Legendrian knots up to Legendrian isotopy, one can associate to them algebraic invariants, which can be very simple to describe (e.g. a number), or more advanced (e.g. a homology theory).

In this talk I will define the mathematical objects introduced above and describe some algebraic invariants of Legendrian knots.

### Monday 24 april: Darius Dramburg

Time: 15:15
Location: Å2002
Title: Om singulariteter, upplösningar och flat-earthers
Abstract:
Vi vet att jorden inte är platt, och det känns självklart att vi alla vet det. Men att jorden har formen av ett klot var en vetenskaplig upptäckt, och det finns fortfarande människor idag som tror att jorden är platt. Hur kommer det sig?

I det här seminariet kommer vi se att det (delvis) beror på ett geometriskt fenomen, nämligen att jorden saknar singulariteter. Vi kommer först se, med jorden som exempel, vad en singularitet är och definiera singulariteter i språket av algebraisk geometri. Efteråt kommer vi försöka att upplösa en singularitet, och förklara varför man skulle vilja göra det. Vi avslutar med att visa hur det leder rätt till moderna forskningsfrågor inom algebraisk geometri.

### Tuesday 22 may: Max Raner

Time: 15:15
Location: Å4003
Title: Kernel regression
Abstract:
When we are interested in explaining how some response, e.g. mouse weight depends on some independent variable(s) such as mouse tail length, the classical approach, dating all the way back to Legendre and Gauss in the early 1800's, is to—in the simplest sense—"fit a line to the data". More specifically, we are modelling the conditional expectation of the response, given the independent variables, by limiting the mean value function to be restricted to some linear surface.

This approach has worked well for two hundred years, and will most likely continue to do so for many more. However, with the advent of more computational power, new approaches to estimating the mean value function have become available to us—with some of them requiring fewer restrictions on our model(such as linearity) than the parametric linear regression models of Gauss and Legendre.

We will instead explore the so-called kernel regression approach: a non-parametric regression modelling strategy, requiring only some smoothness assumptions on the regression curve. First we will pose the regression problem, then we take a detour into non-parametric density estimation, before finally returning with the tools needed to formulate the kernel regression model.

Last modified: 2023-05-17