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Laura DeMarco: Intersection Theory and the Mandelbrot Set

5th October, 2023

One of the most famous – and still not fully understood – objects in mathematics is the Mandelbrot set. It is defined as the set of complex numbers c for which the polynomial fc(z)=z2+c has a connected Julia set. But the Mandelbrot set turns out to be related to many different areas of mathematics. Inspired by recent results in arithmetic geometry, I will describe how the tools of arithmetic intersection theory can be applied in the setting of these complex dynamical systems to give new information about the Mandelbrot set.

Stevell Muller: On symplectic transformations of OG10-type hyperkähler manifolds via cubic fourfolds

15th September, 2023

We know thanks to the work of L. Giovenzana, A. Grossi, C. Onorati and D. Veniani that OG10-type hyperkähler manifolds do not admit any non-trivial symplectic automorphisms. What about non-regular symplectic birational transformations? Given a cubic fourfold V, one can construct a hyperkähler manifold XV of OG10-type following a construction of R. Laza, G. Saccà, C. Voisin. Such manifolds are known as LSV manifolds. It can be shown that any symplectic automorphism on V induces a symplectic birational transformation on XV. In a couple of works with L. Marquand, we study and classify all possible cohomological actions on the OG10-lattice which can be realised as symplectic birational transformations. By investigating further the induced action on cohomology, we exhibit a criterion to decide which of these actions can be realised as induced from a cubic fourfold on an associate LSV manifold.

Sasha Viktorova: The defect of a cubic threefold

13th September, 2023

In this talk, we relate the defect σ(X) := b4(X) − b2(X) of a singular cubic threefold X to various geometric properties of X. The question is motivated by the construction of the exceptional example of a Hyperkähler manifold of type O'Grady 10 from a cubic fourfold by Laza, Saccà and Voisin. By a result of Brosnan, the defect of hyperplane sections of the cubic fourfold is an obstruction for the LSV construction to work. The talk is based on a joint work in progress with Lisa Marquand.

Ana Balibanu: Moment maps, multiplicative reduction, and Dirac geometry

31st August, 2023

We develop a general approach to reduction along strong Dirac maps, which are a broad generalization of Poisson moment maps. We recover a number of familiar constructions and we give several new reduction procedures, including a multiplicative analogue of Whittaker reduction.

Peng Lu: Conformal Bach flow

10th August, 2023

We introduce conformal Bach flow and establish its well-posedness on closed manifolds. We also obtain its backward uniqueness. To give an attempt to study the long-time behaviour of conformal Bach flow, assuming that the curvature and the pressure function are bounded, global and local Shi's type L2-estimate of derivatives of curvatures are derived. To make the talk more accessible, we will spend some time to survey on high-order parabolic curvature flow.

Alex Iosevich: Some number-theoretic aspects of finite point configurations

29th June, 2023

We are going to survey some recent and less recent results pertaining to the study of finite point configurations in Euclidean space and vector spaces over finite fields, centred around the Erdős/Falconer distance problems. We shall place particular emphasis on number-theoretic ideas and obstructions that arise in this area.

Alina Vdovina: Higher structures in Algebra, Geometry and C*-algebras

23rd June, 2023

We present buildings as universal covers of certain infinite families of CW-complexes of arbitrary dimension. We will show several different constructions of new families of k-rank graphs and C*-algebras based on these complexes, for arbitrary k. The underlying building structure allows explicit computation of the K-theory as well as the explicit spectra computation for the k-graphs.

Jeremy Brent Hume: The K-theory of a rational function

22nd June, 2023

The dynamics of iterating a rational function exhibits complicated and interesting behaviour when restricted to points in its Julia set. Kajiwara and Watatani constructed a C*-algebra from a rational function restricted to its Julia set in order to study its dynamics from an operator algebra perspective. They showed the C*-algebras are Kirchberg algebras that satisfy the UCT, and are therefore classified by K-theory. The K-theory groups of these algebras have been computed in some special cases, for instance by Nekrashevych in the case of a hyperbolic and post-critically finite rational function. We compute the K-theory groups for a general rational function using methods different to those used before. In this talk, we discuss our methods and results.

Efthymios Sofos: The second moment method for rational points

22nd June, 2023

In a joint work with Alexei Skorobogatov we used a second-moment approach to prove asymptotics for the average of the von Mangoldt function over the values of a typical integer polynomial. As a consequence, we proved Schinzel's Hypothesis in 100% of the cases. In addition, we proved that a positive proportion of Châtelet equations have a rational point. I will explain subsequent joint work with Tim Browning and Joni Teräväinen that develops the method and establishes asymptotics for averages of an arithmetic function over the values of typical polynomials. Part of the new ideas come from the theory of averages of arithmetic functions in short intervals. One of the applications is that the Hasse principle holds for 100% of Châtelet equations. This agrees with the conjecture of Colliot-Thélène stating that the Brauer-Manin obstruction is the only obstruction to the Hasse principle for rationally connected varieties.

Francesco Genovese: Deforming t-structures

21st June, 2023

A guiding principle of non-commutative algebraic geometry is that geometric objects (i.e. rings and schemes) are replaced by categories of modules/sheaves thereof. In order to keep track of the homological information, we actually take derived categories of such modules/sheaves. From this point of view, we are now interested in understanding typical geometric concepts directly in this categorical framework. A key example is given by deformations. In this talk, I will report on joint work with W. Lowen and M. Van den Bergh, where we attempt to define and study deformations categorically, in the framework of (enhanced) triangulated categories with a t-structure. This will also shed light on Hochschild cohomology.

Joé Brendel: Local exotic tori

26th May, 2023

We discuss exotic Lagrangian tori in dimension greater than or equal to six. First, we give another approach to Auroux's result that there are infinitely many tori in ℝ6 which are distinct up to symplectomorphisms of the ambient space. The exotic tori we construct naturally appear in a two-​parameter family, some of which are not monotone. Small enough tori in this family can be embedded by a Darboux chart into any tame symplectic manifold and one can show that they are still distinct up to symplectomorphisms.

Vinicius Gripp Barros Ramos: The Toda Lattice, Billiards and the Viterbo Conjecture

19th May, 2023

The Toda lattice is one of the earliest examples of non-linear completely integrable systems. Under a large deformation, the Hamiltonian flow can be seen to converge to a billiard flow in a simplex. In the 1970s, action-angle coordinates were computed for the standard system using a non-canonical transformation and some spectral theory. In this talk, I will explain how to adapt these coordinates to the situation to a large deformation and how this leads to new examples of symplectomorphisms of Lagrangian products with toric domains. In particular, we find a sequence of Lagrangian products whose symplectic systolic ratio is one and we prove that they are symplectic balls. This is joint work with Y. Ostrover and D. Sepe.

Federico Binda: Motivic monodromy and p-adic cohomologies 

16th May, 2023

In this talk, I will discuss some recent advances in the theory of motives in the context of rigid analytic geometry. Building on work of Ayoub, Bondarko, we provide an equivalence between the category of “unipotent” rigid analytic motives over a non-archimedean field and the category of “monodromy maps” M → M (−1) of algebraic motives over the residue field. This allows us to build a unified framework for the study of monodromy operators and weight filtrations of cohomology theories for varieties over a local field. As an application, we give a streamlined definition of Hyodo–Kato cohomology without recourse to log-geometry, as predicted by Fontaine, and we produce an induced Clemens–Schmid chain complex.

Patrick Lank: High Frobenius pushforwards generate the bounded derived category

9th May, 2023

This talk is concerned with generators for the bounded derived category of coherent sheaves over a noetherian scheme X of prime characteristic p when the Frobenius morphism is finite. It is shown that for any compact generator G of D(X), the e-th Frobenius pushforward of G classically generates the bounded derived category whenever pe is larger than the codepth of X, an invariant that is a measure of the singularity of X. From this, we can establish a canonical choice of strong generator when X is separated. The work is joint with Matthew R. Ballard, Srikanth B. Iyengar, Alapan Mukhopadhyay, and Josh Pollitz.

Michael Entov: Kähler-type and Tame Embeddings of Balls into Symplectic Manifolds

5th May, 2023

A symplectic embedding of a disjoint union of domains into a symplectic manifold M is said to be of Kähler type (respectively tame) if it is holomorphic with respect to some (not a priori fixed) integrable complex structure on M which is compatible with (respectively tamed by) the symplectic form. I'll discuss when Kähler-type embeddings of disjoint unions of balls into a closed symplectic manifold exist and when two such embeddings can be mapped into each other by a symplectomorphism. If time permits, I'll also discuss the existence of tame embeddings of balls, polydisks and parallelepipeds into tori and K3 surfaces.

Michael Bialy: Locally Maximizing Orbits and Rigidity for Convex Billiards

28th April, 2023

Given a convex billiard table, one defines the set ℳ swept by locally maximizing orbits for convex billiard. This is a remarkable closed invariant set which does not depend (under certain assumptions) on the choice of the generating function. I shall show how to get sharp estimates on the measure of this set, recovering as a corollary rigidity result for centrally symmetric convex billiards. Also I shall discuss rigidity of Mather β-function.

Angela Wu: On Lagrangian Quasi-Cobordisms

24th April, 2023

A Lagrangian cobordism between Legendrian knots is an important notion in symplectic geometry. Many questions, including basic structural questions about these surfaces are yet unanswered. For instance, while it is known that these cobordisms form a preorder, and that they are not symmetric, it is not known if they form a partial order on Legendrian knots. The idea of a Lagrangian quasi-cobordism was first defined by Sabloff, Vela-Vick, and Wong. Roughly, for two Legendrians of the same rotation number, it is the smooth composition of a sequence of alternatingly ascending and descending Lagrangian cobordisms which start at one knot and ends at the other. This forms a metric monoid on Legendrian knots, with distance given by the minimal genus between any two Legendrian knots. In this talk, I will discuss some new results about Lagrangian quasi-cobordisms, based on some work in progress with Sabloff, Vela-Vick, and Wong.