Abstract: In recent years there has been significant interest in the deformation theory of stable $\infty$-categories. This has led to advances in computational results, as well as an increased structural understanding of these categories. In this talk we will survey some deformations relevant to the fundamental building blocks of stable homotopy theory -- $E(n)$-local and $K(n)$-local spectra -- and look at their interactions.
Outreach: The hat, the turtle and the spectre, 14.08.24
Event: Starting week for new bachelor students in mathematics at NTNU
Summary: For half a century mathematicians have been searching for the ellusive Einstein tile; a tile that can tesselate the plane, but only in a way that does not have any repetitions. Tilings of this nature are called aperiodic, and have applications for example in material sciences via the existence of quasi crystals. We will explore the history of periodic and non-periodic tesselations, and how one retired print technician pattern-enthusiast discovered the mythical aperiodic monotile last year.
Abstract: The theory of synthetic spectra has been critical for recent advances in homotopy theory, both theoretical and computational. They have several important features, most notably acting as a deformation of spectra, and as a categorification of the Adams spectral sequence. There is a deformation between $K(n)$-local spectra and certain derived-complete comodules, but this deformation is not simply the category of synthetic spectra based on $K(n)$. In this talk we will introduce a synthetic version of this deformation, which also categorifies the $K(n)$-local $E_n$-Adams spectral sequence. This is joint work with Marius Nielsen.
Abstract: Pstragowski’s category of hypercomplete $E_n$-based synthetic spectra, $\mathrm{Syn}_E$, acts as a one-parameter deformation between $\mathrm{Sp}_n$ and the derived category of $E_*E$-comodules. However, there is a more fundamental building block inside $\mathrm{Sp}_n$ – the category of $K(n)$-local spectra – and we can ask whether this also has an associated deformation. But, $K(n)$-based synthetic spectra deform to the 'wrong' category of comodules, and hence can be interpreted as an incorrect deformation for $\mathrm{Sp}_{K(n)}$. In this talk, we investigate a $K(n)$-local analog of $\mathrm{Syn}_E$ and prove that it has the correct one-parameter deformation properties. This is joint work with Marius Nielsen.
Seminar: Deformations of chromatic homotopy theory, 21.03.24
Abstract: Chromatic homotopy theory is the "quantum mechanics" of stable homotopy theory, where we split the theory down and study its smallest and most fundamental pieces. It has gotten the name "chromatic" as each piece operates at a certain fundamental frequency, or a certain wavelength. This gives a filtration of the pure white light $Sp$ into colors $Sp_{K(n)}$, and collections of colors $Sp_{E(n)}$.
Physics analogies aside, there are several well known deformations of $Sp$, for example Pstrągowski's category of synthetic spectra $Syn_E$, or Patchkoria–Pstrągowski's perfect derived category $D^\omega_H(Sp).$ In the talk we will survey the relevant deformations for $Sp_{E(n)}$ and discuss some possible models for deformations of $Sp_{K(n)}$.
Outreach: The hat, the turtle and the spectre, 05.03.24
Summary: For half a century mathematicians have been searching for the ellusive Einstein tile; a tile that can tesselate the plane, but only in a way that does not have any repetitions. Tilings of this nature are called aperiodic, and have applications for example in material sciences via the existence of quasi crystals. We will explore the history of periodic and non-periodic tesselations, and how one retired print technician pattern-enthusiast discovered the mythical aperiodic monotile last year.
Seminar: An exotic $\mathbb{A}_3$-structure on $L_1\mathbb{S}/3$?, 29.02.24
Seminar: Copenhagen cooloquium
University seminar: An obscure perspective on chromatic homotopy, 26.02.24
Abstract: The study of modules and comodules is ubiquitous throughout modern mathematics, and we have good recognition theorems for identifying them. Eilenberg and Moore defined in the 40’s a third type of module, which they called contramodules, but they were consequently forgotten and ignored until the early 2000’s. In this talk we will see some examples where contramodules naturally show up in stable homotopy theory, as well a present some ideas that I want to work on during my 4 months here at KU.
Outreach: Symmetri og kunst, 10.10.23
Event: IMF Språkcafé
Outreach: Symmetry: A journey through dimensions, 22.09.23
Summary: The world around us is filled to the brim with symmetry – from rotational symmetry in flowers to mirror symmetry in a calm mountain lake. Through inspiration from nature symmetry also unfolds itself in human art, for example the rose window in the Nidaros cathedral. Many paintings are also symmetric, but there are several ways in which they can be so. A curious person could then ask: how many different ways can a painting be symmetric? What happens when we don’t restrict ourselves to a flat painting but a three-dimensional sculpture? What about hypothetical shapes in any possible dimension?
Throughout the talk we will venture through the world of symmetry, exploring their simple mathematical descriptions and their uses in the arts, mainly focusing on the works of M. C. Escher.
Outreach: Mathematical art for a staircase, 01.09.23
Abstract: Chromatic homotopy theory views the category of spectra as certain nicely behaved chromatic layers glued together along formal neighborhoods — described respectively using Morava E-theory and Morava K-theory. This theory is a beautiful cacophony of colors, but can be very difficult to make sense of. Restricting to just one of these layers we get a category of monocromatic spectra, which are the focus of the talk. We look into some properties, especially focusing on approximations using algebraic information.
Abstract: The classical Barr-Beck theorem gives sufficient criteria for an adjunction being monadic. In a monoidal setting this gives criteria for checking when a category is equivalent to modules over a ring object. During the talk we will explore some adjacent results, more specifically a local, co-local and dual version of the theorem, as well as some consequences and examples.
Seminar: Torsion in topology and algebra, 10.05.23
Abstract: Torsion is a concept used in algebra, topology and their topology-focused intersection: algebraic topology. One can compare topological and algebraic structures using abstract homology theories, and one can ask how torsion interacts under these comparisons. During the talk we will set up a general framework for these questions and put forward several ideas about how this should behave — particularily focusing on ideas trying to relating TTF triples, chromatic homotopy theory and (co)monads.
University seminar: Derived completion in chromatic homotopy theory, 27.04.23
Seminar: Aarhus Topology seminar
Seminar: An introduction to the chromatic nullstellensatz, 25.03.23
Abstract: Chromatic homotopy theory views the stable homotopy category as certain nicely behaved layers glued together along formal neighborhoods. These are respectively described by the famous Morava E-theories $E(n)$ and Morava K-theories $K(n)$. We can single out one of these layers – in some sense reducing the entire colorful spectrum down to a single chroma – giving us monochromatic homotopy theory. During the talk we will see some properties of this theory, in particular focusing on its local duality with $K(n)$-local spectra and exotic algebraicity.
University seminar: Algebraicity in monochromatic homotopy theory, 28.11.22
Abstract: Chromatic homotopy theory views the category of spectra as certain nicely behaved chromatic layers glued together along formal neighbourhoods — described respectively using Morava E-theory and Morava K-theory. Using Franke’s algebraicity theorem we know that these chromatic layers can be approximated using algebraic information. In this talk we will try to explain these approximations, as well as describe some progress on constructing similar approximations for the formal neighbourhoods.
Outreach: Knots and links, 19.08.22
Event: Starting week for new students in Physics and Mathematics at NTNU
Outreach: Hanging a picture using algebraic topolgy, 17.08.22
Event: Starting week for new master students in mathematics at NTNU
Abstract (in norwegian): Homotopigruppene er en av de mest fundamentale invariantene i fagfeltet algebraisk topologi. For å forstå disse må vi forstå de for enkle rom vi kjenner godt — for eksempel sfærene. Sfærenes homotopigrupper består av homotopiklasser av kontinuerlige funksjoner mellom de, og disse gruppene er noen av de vanskeligste og mest intrikate gruppene å finne. Gjennom foredraget skal vi prøve å motivere hvorfor de er så kompliserte, hvorfor vi bryr oss om de, samt forklare noe av det vi allerede vet om de.
Outreach: Knot theory, 19.08.21
Event: Starting week for new bachelor students in mathematics at NTNU
Experiences: Experiences from being a master student in mathematics, 18.08.21
Event: Starting week for new master students in mathematics at NTNU
Abstract (in norwegian): Noe man lærer på universitetet er at matematikk omfatter mye mer en man tror, og er ofte veldig annerledes enn det man er vandt til. Matematikk er et ufattelig stort, rikt og diverst fagfelt, med hundrevis av underfagfelt. En av disse underfagfeltene har veldig få hørt om før man begynner på universitetet, nemlig knuteteori. Ofte tenker man på matematikk som formler, regler og likninger, men i dette foredraget skal vi vise at dette absolutt ikke alltid er tilfellet. Vi skal gjøre rigorøse beviser, men også tegne oss til alle svarene. Knuteteori er en matematisk gren innenfor fagfeltet topologi, og bruker teknikker og metoder som man mest sannsynlig aldri har sett i sammenheng med matematikk før. Gjennom fordraget skal vi utforske noe veldig elementært, noe som vi alle kjenner fra før, nemlig knuter. Vi skal se på hvordan vi kan jobbe med knuter matematisk, og vi skal bevise at knuter faktisk eksisterer.
Speach: Graduation speach, 04.06.21
Event: Graduation seremony for bachelor and master students at the IE faculty at NTNU
Abstract (in norwegian): På slutten av det andre årtusenet publiserte CLAY instituttet for matematikk en liste over syv uløste problemer som ville være de viktigste og vanskeligste problemene i matematikk å løse i de neste årtusenet. Problemene på denne listen er nå de syv mest berømte problemene i matematikkens verden, og det med god grunn. Tre år senere ble ett bevis av et av problemene, kalt Poincaré-formodningen, publisert på ArXiV, og det er nettopp dette problemet som vil være fokuset for dette foredraget.
Poincaré-formodningen (egentlig Poincaré-teoremet, men navnet forble av historiske årsaker) er et veldig fundamentalt og grunnleggende problem innen fagfeltet topologi, og var den siste brikken i puslespillet for å forstå hvilke rom som faktisk er sfærer. Vi vil dekke litt grunnleggende differensialtopologi, formuleringen av formodningen, overfladisk forståelse av beviset og til slutt problemets generaliseringer og relaterte problemer.
Abstract: A sad story one quickly learns in topology is that we can’t fully classify topological spaces by their cohomology. The cochain complex gives much better knowledge about the space we are working with, but it is often much harder to compute compared to the cohomology. Instead of topological spaces we can work more generally with differential graded algebras (dg-algebras). Since cohomology is easier to compute, a natural question to ask is; for which dg-algebras do we have “all information” if we know its cohomology? These dg-algebras are called formal, because we can “formally reconstruct them from their cohomology”. In this talk I will attempt to define properly the above situation, look at some properties of these dg-algebras, as well as how we can abstract an important part of the theory to so-called $A_\infty$-algebras.
Abstract (in norwegian): En av tingene man ser relativt fort gjennom et matematikkstudie er at det ofte er viktigere å studere avbildninger mellom objekter enn objektene selv. Altså må man studere objekter relativt til andre objekter av samme type. Dette er grunnlaget for kategoriteori. En fundamental innsikt er da at likhet mellom to objekter, foreksempel to grupper, er en for sterk sammenligning, og at isomorfi er en mye bedre metode for å si at to objekter er «like nok til å være det samme». Dette er også heldigvis innebygd i kategoriteorien. Men hvis avbildningene er de viktige, hva skjer når man prøver å behandle disse som objekter selv?
Foredraget skal skape en grunnleggende forståelse av hvorfor vi bryr oss om høyere kategoriteori. Vi skal også definere flere av de viktigste objektene som brukes i denne teorien, samt se at vi allerede kjenner mange av de uten å kanskje vite det. Foredraget skal være grunnleggende så det vil ikke bli antatt mye forkunnskaper da mesteparten vil foregå i det intuitive planet og ikke blodig rigorøst.
Seminar: Stable model categories, 09.11.20
Seminar: Seminar on triangulated categories
Crash course: An introductory course in $\LaTeX{}$, 20.10.20
Event: Giving bachelor students an introductioin to using $\LaTeX{}$ for writing their thesis.