Seminarium i matematik
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Tidigare seminarier
Some mathematical structures of Feynman integrals
Datum: 23 oktober 2024
Tid: 15.30-16.30
Plats: D209
Föreläsare: Felix Tellander, Oxford University
Some mathematical structures of Feynman integrals
Abstrakt: The Large Hadron Collider (LHC) in Geneva is the biggest physics experiment ever build on earth and is made to study the most fundamental aspects of reality by colliding protons near the speed of light. Zooming out to outside our own galaxy, many light years away, black holes collide in some of the most energetic processes in the entire universe sending waves through space-time itself. What does these seemingly different processes have in common? They can be described using Feynman integrals.
These integrals are not only of physical interest but have many non-trivial mathematical properties and serve as a connective tissue between many different areas of mathematics. In this talk, I will introduce these integrals from a physical perspective, give their mathematical definition and highlight some recent results from analysis, algebra, and numerics.
The strong Lefschetz property via Gröbner bases
Datum: 11 september 2024
Föreläsare: Filip Jonsson Kling – Stockholms universitet
Abstrakt: In this talk, we will sketch part of a new proof that an algebra defined by an ideal generated by the squares of all variables of a polynomial ring do have the so called strong Lefschetz property. This will be done by finding some nicely described Gröbner bases for a family of related ideals. One surprising fact about these Gröbner basis is that all elements have 0,1-coefficinets, allowing for combinatorial interpretations to be made. In particular, we will show how the initial terms of the Gröbner basis elements enumerate a special kind of lattice paths and how this enumeration will aid us in establishing the strong Lefschetz property.
This is based on joint work with Samuel Lundqvist, Fatemeh Mohammadi, Matthias Orth and Eduardo Sáenz-de-Cabezón. No prior knowledge of Lefschetz properties or Gröbner bases will be assumed.
The mathematics of "hearing the shape of a drum"
Datum: 2 maj 2024
Föreläsare: Julie Rowlett, Chalmers Tekniska Högskola
Abstrakt: Have you heard the question, ``Can one hear the shape of a drum?" Do you know the answer?
In 1966, M. Kac's article of the same title popularized the inverse isospectral problem for planar domains. Twenty six years later, Gordon, Webb, and Wolpert demonstrated the answer, but many naturally related problems remain open today. We will discuss results inspired by ``hearing the shape of a drum'' which involve geometry, mathematical physics, (microlocal) analysis, and number theory. Along the way, we will visit several currently open problems.
Binomial complete intersections
Datum: 26 mars 2024
Föreläsare: Lisa Nicklasson, Mälardalens universitet
Abstrakt: Consider a collection C of n homogeneous polynomials in n variables. The polynomials C form a complete intersection if their only common zero is the origin, or equivalently if the induced quotient ring is a finite dimensional vector space. A third equivalent condition is that a polynomial in the coefficients of C, called the resultant, is nonzero.
Unfortunately, the resultant is in general difficult to compute. In this talk we will discuss the special case when C is a collection of binomials. We will see that drawing a directed graph helps us compute the resultant, and at the same time proves a natural vector space basis for the induced quotient ring of a binomial complete intersection.
The talk is based on a joint work with Filip Jonsson Kling and Samuel Lundqvist.
Using Mathematical Optimization to Enhance Cancer Treatment with Brachytherapy
Datum: 6 december 2023
Föreläsare: Björn Morén, Linköping University
Using Mathematical Optimization to Enhance Cancer Treatment with Brachytherapy
Abstrakt: Brachytherapy is a type of radiation therapy that has been used for cancer treatments for more than 100 years. Advancements in medical imaging have created both the need and opportunity for enhanced treatment planning.
Brachytherapy is an invasive treatment where hollow needles are used to place the radiation source, commonly Iridium, inside or close to the tumour. The aim of a treatment is to deliver a sufficiently high dose to the tumour while minimizing radiation exposure to nearby healthy organs at risk. I will present the treatment planning problem from a mathematical perspective, providing an overview of its formulation and solution through mathematical optimization.
I will also introduce a new delivery technique for brachytherapy, called intensity modulated brachytherapy. With this technique, shields are inserted in the hollow needles to better shape the dose distribution and spare organs at risk. Potential benefits and robustness of the treatment will be discussed.
Towards entropy conservative numerical methods for partial differential equations
Datum: 4 oktober 2023
Föreläsare: Viktor Linders, Lunds Universitet
Abstrakt: There are essentially four ingredients necessary to successfully solve a time dependent partial differential equation with numerical methods. In this talk, we remind ourselves about what they are, their relations to properties of the governing equation, and how - despite our best efforts - we tend to violate most of them. Special attention is paid to the final step in many algorithms; solving a large system of nonlinear equations using iterative methods.
Legendrian knots and Lagrangian cobordisms
Datum: 8 mars 2023
Föreläsare: Noémie Legout, Uppsala Universitet
Abstrakt: Legendrian knots are knots satisfying some tangential constraints in a 3-dimensional manifold 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 (smooth deformation of the knot among Legendrian knots). In order to distinguish and classify Legendrian knots up to Legendrian isotopy, many algebraic invariants have been developed in the last two decades. Contact manifolds are the odd-dimensional siblings of the perhaps better-known symplectic manifolds. Beyond Legendrian isotopy, another relation one can study between Legendrian knots is that of Lagrangian cobordism, i.e. given two Legendrian knots, do they appear as the boundary components of a Lagrangian surface in an appropriate symplectic manifold?
In this talk I will define the mathematical objects introduced above, describe some algebraic invariants of Legendrian knots on one side, and Lagrangian cobordisms on the other side, and explain how these are related.
Chain algebras of finite distributive lattices
Datum: 8 februari 2023
Föreläsare: Oleksandra Gasanova, University of Duisburg-Essen, Tyskland
Abstrakt: This talk is based on my ongoing project with Professor Jürgen Herzog and Rodica Dinu. Let L be a finite distributive lattice and let t_1,...,t_n denote the elements of its ground set. To each maximal chain C of L one can associate a squarefree monomial m in K[t_1,...,t_n] which equals the product of all t_i belonging to C. We then consider the subalgebra K[m_1,...,m_s], generated by all such monomials, and call it the chain algebra of L. In my talk I will discuss some properties of such algebras and their connection to combinatorial properties of the corresponding lattices. The main result of this talk will be an algebraic characterization of finite planar distributive lattices.
Generative Adversarial Networks applied to Geosciences
Datum: 25 januari 2023
Föreläsare: Charlie Garayt och Ferdinand Bhavsar, Mines Paris, Frankrike
Abstrakt: Geomodeling, also known as geosciences modeling, is a complex inverse problem with a high degree of uncertainty. One of the main challenges in using deep learning techniques for geomodeling is the limited availability of geomodeling datasets. However, the knowledge of physical and chemical processes can be used to create simulators, but these simulations can be difficult to condition with real-world data.
To address this challenge, an alternative approach is to view geomodeling as the generation of 2D/3D images, using an ensemble approach rather than a single solution. In this context, Generative Adversarial Networks (GANs) is an interesting tool because they can generate an ensemble of images, and they can be more easily conditioned to real-world data than simulators.
Two examples are presented to illustrate the proposed approach: the first of structural geological geomodels (2D) and the second of fluvial geomodels (2D and 3D).
Introduction to finite element approximation of stochastic evolution equations
Datum: 7 december 2022
Föreläsare: Stig Larsson, Chalmers
Abstrakt: I will give a rather elementary introduction to how stochastic evolution PDE, such as the stochastic heat equation or the stochastic wave equation, can be given a rigorous meaning and how to show existence and regularity of solutions. If time permits, I will also talk about finite element approximation of the solutions.
Urban snow removal problem: models and solutions
Datum: 23 november 2022
Föreläsare: Roghayeh Hajizadeh, Linköpings Universitet
Abstrakt: Snow removal is an important and unavoidable problem in some countries like Sweden. A number of streets in a city need to be cleared of snow by a limited number of vehicles. Since it is hard to predict the snow amount for each year, it is not possible to use one year’s plan for the next year and new tours must be planned each time. The problem is to plan tours for vehicles such that the time and/or cost is minimized. We consider urban snow removal, which can be more difficult, with all details such as precedences and turning penalties.
The problem can be formulated as a very large mixed integer programming model, which is practically unsolvable. In order to find a feasible solution for the whole original problem, first we break down the work into smaller parts and find which streets a vehicle shall take care of. Based on the allocation obtained, we consider snow removal problems for single vehicles and find their feasible solutions based on a reformulation to asymmetric traveling salesman problems in extended graphs. Then, we use an iterative procedure to combine the tours, and a procedure for constructive coordination of the tours. We also have new improvement procedures for the combined solution. We have implemented the methods and applied them to real life city networks. The numerical results show that the methods obtain feasible tours for large problems within a reasonable time.
Polynom, system och osäkerhetsrelationer
Datum: 19 oktober 2022
Föreläsare: Bartosz Malman, KTH
Abstrakt: Påståendet “ett idealt bandpassfilter existerar inte” är något som en student på ett ingenjörsprogram må ha hört en eller annan gång under utbildningen. Möjligen har studenten också hört en intuitiv förklaring till varför ett sådant instrument inte finns. I den matematiska teorin kan frågor av denna typ studeras på abstrakt nivå, och formuleras som matematiska satser. Problem av denna typ är en stor inspirationskälla för matematisk forskning. I mitt föredrag kommer jag att relatera ovanstående påstående till flera matematiska satser, där “algebrans fundamentalsats” är den kanske mest kända, och där den djupaste är en osäkerhetsrelation inom Fourieranalysen. I alla dessa satser spelar analytiska polynom en roll, ibland i förgrunden, och ibland i bakgrunden.
Frogs and branching random walks on grids - does the winner take it all?
Datum: 13 april 2022
Föreläsare: Timo Vilkas, Göteborgs universitet
Abstrakt: The topic of this seminar are two fairly well-known interacting particle systems: the so-called frog model and branching random walk. We are going to look at competition versions of these models, evolving on the lattice Zd, in which two types race to mark as many sites as possible by visiting them first. The key question we investigate is in which regime both types can coexist, in the sense that they both markan infinite number of sites and in which regimes the winner takes it all. Despite being simplistic, these processes can be seen as toy models for two brands competing on the market with a new product or two different opinions/variants of a disease spreading on a network.
Tidsfrekvensanalys inom matematik och musik
Datum: 23 mars 2022
Föreläsare: Joachim Toft, Linnéuniversitetet
Abstrakt: Vi undersöker begrepp som faltningar, Fouriertransformer, lokala Fouriertransformer (korttidsfouriertransformer) och dess betydelser. Bl a kopplas vissa frågeställningar ihop inom signalanalys och differentialekvationer. Inom dessa angreppssätt lyfts även vissa transformeringar med rötter i kvantmekaniken fram. Bl a visas hur den s k Born-Jordantransformen kan utnyttjas för att bättre momentant undersöka tidsfrekvensspektrat för signaler.
Applied linear algebra: matrix equations and factorizations
Datum: 26 november 2021
Föreläsare: Andrii Dmytryshyn, Örebro universitet
Abstrakt: I will present an overview of our research on two classical topics in Matrix Analysis and Applied Linear Algebra, namely challenging problems regarding the solution matrix equations and the analysis of the complete eigenstructure of matrices under perturbations. Then I will focus on our results about coupled matrix equations and block diagonalization:
In 1952 Roth revealed the connection between the existence of a solution (i.e., the consistency) for a Sylvester matrix equation AX-XB=C and the similarity (having the same eigenvalues and their multiplicities) of the particular block-matrices constructed from the matrix coefficients of the considered Sylvester matrix equation. Since then, similar results have been published for a number of other Sylvester-type matrix equations as well as for some systems of matrix equations. I will present the general Roth’s type theorem for systems of matrix equations consisting of an arbitrary mix of Sylvester-type equations.
Black hole physics – through the Legacy of Einstein and Hawking
Datum: 17 november 2021
Föreläsare: Narit Pidokrajt, Högskolan i Borås
Black hole physics---through the Legacy of Einstein and Hawking
Abstrakt: We know that black holes do exist and they are mysterious in a variety of ways. Serious studies of black holes have been carried out for the past 50 years by physicists, astronomers as well as mathematicians. In 2020 three Laureates share Nobel Prize in Physics for their discoveries about one of the most exotic phenomena in the universe, the black hole--through their theoretical and observational works. Three years earlier the Nobel Prize in Physics of 2017 went to three physicists for their decisive contributions to detection and observation of gravitational waves as predicted by Einstein (the waves were generated as a consequence of a massive black hole collision). Hawking's famous prediction that black holes do radiate thermally is yet to be observed but his legacy has so far been shaping our understanding of the fundamental laws of Nature with the black hole at the center of the stage.
In this talk I will give a brief introduction to Einstein's theory of General Relativity (GR), its predictions and the current status of the research field of GR. I will focus in particular on some aspects of black hole thermodynamics and how it connects the classical with quantum physics. A short description of application of information geometry (my research field) in the field of black hole physics will also be given.
Wignerdistributionen och dess tillämpningar i kvantmekanik och signalbehandling
Datum: 6 oktober 2021
Föreläsare: Patrik Wahlberg, Linnéuniversitetet och Politecnico i Turin
Abstrakt: Wignerdistributionen ger en tidsfrekvensrepresentation av en signal som är en funktion av tiden. Som sådan har den många goda egenskaper varav inte minst den höga upplösningen. I kvantmekanisk tolkning ger den en kvasisannolikhetsfördelning över fasrummet för en partikel. Det är dock inte en riktig sannolikhetsfördelning eftersom Wignerdistributionen sällan är icke-negativ. I föredraget diskuteras Wignerdistributionens egenskaper och metoder att hantera dess nackdelar.