Recent years have seen the appearance of a plethora of possible metrics on spaces of Lagrangian submanifolds. Indeed, on top of the better-known Lagrangian Hofer metric and spectral norm, Biran, Cornea, and Shelukhin have constructed families of so-called weighted fragmentation metrics on these spaces. I will explain how — under the presence of bounds coming from Riemannian geometry — all these metrics behave well with respect to the set-theoretic Hausdorff metric.
In this talk we consider the links of simple singularities, which are contactomoprhic to S3/G for finite subgroups G of SU2(ℂ). We explain how to compute the cylindrical contact homology of S3/G by means of perturbing the canonical contact form by a Morse function that is invariant under the corresponding rotation subgroup. We prove that the ranks are given in terms of the number of conjugacy classes of G, demonstrating a form of the McKay correspondence. We also explain how our computation realizes the Seifert fiber structure of these links.
We consider the standard L-function attached to a cuspidal automorphic representation of a general linear group. We present a proof of a subconvex bound in the t-aspect. More generally, we address the spectral aspect in the case of uniform parameter growth.
Mackey functors play a central role in equivariant homotopy theory, where homotopy groups are replaced by homotopy Mackey functors. In this talk, I will discuss joint work with Dan Dugger and Christy Hazel classifying perfect chain complexes of constant Mackey functors for G=ℤ/2. Our decomposition leads to a computation of the Balmer spectrum of the derived category. We extend these results to classify all finite modules over the equivariant Eilenberg-MacLane spectrum Hℤ/2.
Let G be a finite simple group and S a normal subset of G. If |G| is large enough in terms of |S|/|G|, can we deduce that every element of G can be expressed as x y-1 for x and y elements of S? Shalev, Tiep, and I have proven that this is true assuming G is an alternating group or a group of Lie type in bounded rank, but the question remains open for classical groups of high rank over small fields. I will say something about the methods of proof, which involve both character methods and geometric ideas and also say something about the more general question of covering G by ST where S and T are large normal subsets.
The linear sieve is a powerful tool to tackle problems related to the primes, when combined with equidistribution estimates for the remainder. In 1977 Iwaniec introduced a well-factorable modification of the linear sieve to prove there are infinitely many integers n such that n2+1 has at most two prime factors. Furthermore, the (well-factorable) linear sieve leads to the best known upper bounds for twin primes. These bounds use work of Bombieri, Friedlander, and Iwaniec from 1986, showing these sieve weights equidistribute primes of size x in arithmetic progressions to moduli up to x4/7. This level was recently increased to x7/12 by Maynard.We introduce a new modification of the linear sieve whose weights equidistribute primes of size x to level x10/17. As an application we refine a 2004 upper bound for twin primes of Wu, which gives the largest percent improvement since the work of Bombieri, Friedlander, and Iwaniec.
We introduce the notion of a (signed) 𝜏-exceptional sequence for a finite dimensional algebra. We show that there is a bijection between the set of complete signed exceptional sequences and ordered basic support 𝜏-tilting objects.
Suppose that G is a finite group and k is a field of characteristic p>0. Let M be a finitely generated kG-module. We consider the complete cohomology ring ℰ̂M∗=Σn∈ℤ Ex̂tkGn(M, M). We show that the ring has two distinguished ideals I∗ ⊆ J∗ ⊆ ℰ̂M∗ such that I∗ is bounded above in degrees, ℰ̂M∗/J∗ is bounded below in degree and J∗/I∗ is eventually periodic with terms of bounded dimension. We prove that if M is neither projective nor periodic, then the subring of all elements in negative degrees in ℰ̂M∗ is a nilpotent algebra.
In the seventies, Feldman and Moore studied Cartan pairs of von Neumann algebras. These pairs consist of an algebra A and a maximal commutative subalgebra B with B sitting "nicely" inside of A. They showed that all such pairs of algebras come from twisted groupoid algebras of quite special groupoids (in the measure theoretic category) and their commutative subalgebras of functions on the unit space, and that moreover the groupoid and twist were uniquely determined (up to equivalence). Kumjian and Renault developed the C*-algebra theory of Cartan pairs. Again, in this setting all Cartan pairs arise as twisted groupoid algebras, this time of effective etale groupoids, and again the groupoid and twist are unique (up to equivalence).
In recent years, Matsumoto and Matui exploited this to give C*-algebraic characterizations of continuous orbit equivalence and flow equivalence of shifts of finite type using graph C*-algebras and their commutative subalgebras of continuous functions on the shift space (which form a Cartan pair under mild assumptions on the graph). The key point was translating these dynamical conditions into groupoid language. The ring theoretic analogue of graph C*-algebras are Leavitt path algebras. Leavitt path algebras are also connected to Thompson's group V and some related simple groups considered by Matui and others. Since the Leavitt path algebra associated to a graph is the "Steinberg" algebra of the same groupoid (a ring theoretic version of groupoid C*-algebras whose study was initiated by the speaker), this led people to wonder whether these dynamical invariants can be read off the pair consisting of the Leavitt path algebra and its subalgebra of locally constant maps on the shift space. The answer is yes, and it turns out in the algebraic setting one doesn’t even need any conditions on the graph. Initially work was focused on recovering a groupoid from the pair consisting of its "Steinberg" algebra and the algebra of locally constant functions on the unit space. But no abstract theory of Cartan pairs existed and twists had not yet been considered. Our work develops the complete picture.
It turns out that a twist on a groupoid gives rise to a Cartan pair when the algebra satisfies a groupoid analogue of the Kaplansky unit conjecture. In particular, if the groupoid has a dense set of objects whose isotropy groups satisfy the Kaplansky unit conjecture (e.g., are unique product property groups or left orderable), then the groupoid gives rise to a Cartan pair. This is what happens in the case of Leavitt path algebras where the isotropy groups are either trivial or infinite cyclic and hence left orderable.
Some recent improvements of Wigner's unitary-antiunitary theorem will be presented. A connection with Gleason's theorem will be explained.
I will discuss some recent progress in analytic number theory for polynomials over finite fields, giving strong new estimates for the number of primes in arithmetic progressions, as well as for sums of some arithmetic functions in arithmetic progressions. The strategy of proof is fundamentally geometric, and I will explain some of the geometric ideas in the proof, including how we can use the representation of the symmetric group to handle many different arithmetic functions in a uniform way.
Factorization systems (both weak and strong) are commonly defined as consisting of two classes of maps satisfying a certain orthogonality relation and a factorization axiom. The standard definition of algebraic weak factorization system, involving comonads and monads, is rather different. The goal of this talk will be to describe an equivalent definition of algebraic weak factorization system emphasising orthogonality and factorization.
The Bruhat order on a Weyl group has a representation theoretic interpretation in terms of Verma modules. The talk concerns resulting interactions between combinatorics and homological algebra. I will present several questions around the above realization of the Bruhat order and answer them based on a series of recent works, partly joint with Volodymyr Mazorchuk and Rafael Mrden.
In this talk, I will show how to develop a general non-commutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (MΔC). Insights from non-commutative ring theory are used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of an MΔC, K, and then to associate to K a topological space: the Balmer spectrum Spc(K). We develop a general framework for (noncommutative) support data, coming in three different flavors, and show that Spc(K) is a universal terminal object for the first two notions (support and weak support). The first two types of support data are then used in a theorem that gives a method for the explicit classification of the thick (two-sided) ideals and the Balmer spectrum of an MΔC. The third type (quasi support) is used in another theorem that provides a method for the explicit classification of the thick right ideals of K, which in turn can be applied to classify the thick two-sided ideals and Spc(K). Applications will be given for quantum groups and non-cocommutative finite-dimensional Hopf algebras studied by Benson and Witherspoon.
Actions on trees are ubiquitous in group theory. The standard approach to describing them is known as Bass–Serre theory, which presents the group acting on the tree as assembled from its vertex and edge stabilizers. However, a different approach emerges if instead of considering vertex and edge stabilizers as a whole, we focus on local actions, that is, the action of a vertex stabilizer only on the immediate neighbours of that vertex. Groups acting on trees defined by their local actions are especially important as a source of examples of simple totally disconnected locally compact groups, with a history going back to a 1970 paper of Tits. I will go through some highlights of this theory and then present some recent joint work with Simon Smith: we develop a counterpart to Bass–Serre theory for local actions, which describes all possible local action structures of group actions on trees.
Given a log Calabi-Yau pair (X,D), consisting of a smooth projective variety X together with a normal crossings anti-canonical divisor D, we first provide a combinatorial algorithm for solving the enumerative problem of computing rational stable maps to (X,D) touching D at a single point. We then explain how to use the solution to write explicit equations for mirrors to such pairs at
arbitrary dimensions.
This will be an attempt to summarize what one might expect about Homological Mirror Symmetry in the presence of an anticanonical divisor... and the (much smaller, but more reliable) subset of things I can prove about that situation.
Consider a positively monotone (Fano) closed symplectic manifold M and a symplectic simple crossings divisor D in it. Assume that the Poincare dual of the anti-canonical class is a positive
rational linear combination of the classes [Di], where Di are the components of D with their symplectic orientation. A choice of such coefficients, called the weights, (roughly speaking) equips M - D with a Liouville structure. I will start by discussing results relating the components of D with their symplectic orientation. A choice of such coefficients, called the weights, (roughly speaking) equips M - D with a Liouville structure. I will start by discussing results relating the symplectic cohomology of M - D with quantum cohomology of M. These results are particularly sharp when the weights are all at most 1 (hypothesis A). Then, I will discuss certain rigidity results (inside M) for
skeleton type subsets of M - D, which will also demonstrate the geometric meaning of hypothesis A in examples.
