I will give a leisurely overview of parametrized and highersemiadditivity. In particular I will motivate this concept by giving avariety of examples. As one such example, I will explain how it gives aconceptual interpretation of definitions in (globally) equivariantalgebra and homotopy theory. I will then finish by discussing the closeconnection between generalized semiadditivity and the construction oftransfer maps.

In relation to his work on Hilbert'sthird problem, Dehn proved in 1903 that a rectangle can be tiled by squares ifand only if the ratio of its sides is a rational number. In 1994-95Freiling-Rinne and Laczkovich-Szekeres solved a converse problem: for whichreal numbers r can one tile a square by rectangles with side ratio r. Theanswer is much more interesting: the property holds if and only if r is analgebraic number all of whose conjugates have positive real part. Soonafterwards the result was generalised to tilings of arbitrary rectangles byrectangles with a given side ratio. Unfortunately, in this case thecorresponding criterion was somewhat mysterious, and it was not clear whetherit was algorithmically decidable.

In the talk, we give a geometric interpretation of this condition and we showthat it is decidable. Using these methods, we generalise the above results totilings of squares by finitely many rectangles with possibly different sideratios under the extra assumption that all these ratios are algebraic numbers.We also classify all pairs of rectangles which are mutually tileable (there arenontrivial examples!). This is joint work with Radomił Baran.

 

Seminar webpage https://utrechtgeometrycentre.nl/ugc-seminar/

Abstract:  (coming soon) (joint work with J. Commelin and P. Habbegger)

AG seminar webpage: https://webspace.science.uu.nl/~marse004/ag_seminar.html

Homotopy theory has proven to be a robust tool for studying non-homotopical questions about manifolds; for example, surgery theory addresses manifold classification questions using homotopy theory. In joint work with Sarah Yeakel, we are developing a program to study manifold topology via isovariant homotopy theory. I'll explain what isovariant homotopy theory is and how it relates to the study of manifolds via their configuration spaces, and talk about an application to fixed point theory.

Website of the DDTG seminar: https://www.few.vu.nl/~trt800/ddtg.html

Unstable Periodic Homotopy Theory

There will be:
Tea (15:30): Pangea
Drinks (17:30): Library, Hans Freudenthal Building

Abstract: Continued fractions (CF) are a beautiful chapter of classical analysis.
Remarkably, some algebraic aspects of CF find their analogs in the very
foundations of category theory: in relation to the concept of adjoint functors
introduced by D. Kan in 1958. More precisely, Euler's continuants (the universal
polynomials expressing the numerators and denominators of convergents of
a CF) lift to certain complexes built out of a given functor and its iterated
adjoints. Requiring exactness of some of these complexes leads to the
concept of an N-spherical functor which encompasses ordinary spherical functors
for N=4. Such functors describe N-periodic semi-orthogonal decompositions
of (enhanced) triangulated categories.

The lectures will present the classical continuant formalism, its categorical lifting
and the point of view on spherical and N-spherical functors as regular and irregular
perverse schobers (categorified perverse sheaves) on the complex plane. They
are based on joint work with T. Dyckerhoff, V. Schechtman and Y. Soibelman.

TBD

Zeros of the Riemann zeta function and eigenvalues of quantized chaotic Hamiltonians appears to have something in common. Namely, they both seem to be ruled by random matrix theory and consequently should exhibit "repulsion" in the sense that small gaps between elements are very rare.  More mysteriously, while zeros of different L-functions (i.e., generalizations of the Riemann zeta function) are "mostly independent" they also exhibit subtle repulsion effects on zeros of other L-functions.
We will give a survey of the above phenomena.  Time permitting we will also discuss repulsion between eigenvalues of "arithmetic Seba billiards", a certain singular perturbation of the Laplacian on the 3D torus R^3/Z^3.  The perturbation is weak enough to allow for arithmetic features from the unperturbed system to be brought into play, yet strong enough to provably induce repulsion.

 
Seminar webpage https://utrechtgeometrycentre.nl/ugc-seminar/

TBA

Webpage of the UGC seminar: https://utrechtgeometrycentre.nl/ugc-seminar/

TBA

Seminar webpage: https://utrechtgeometrycentre.nl/ugc-seminar/

TBA

UGC seminar webpage https://utrechtgeometrycentre.nl/ugc-seminar/