Fri, 14 Jun 2024

12:00 - 13:00
Quillen Room

Different Approaches to the Borel-Weil-Bott Theorem

Xuanzuo Chen
(University of Oxford)
Abstract

It is well-known that the set of irreducible (finite-dimensional) representations of a semisimiple complex Lie algebra g can be indexed by the dominant weights. The Borel-Weil theorem asserts that they can be seen geometrically as the global sections of line bundles over the flag variety. The Borel-Weil-Bott theorem computes the higher sheaf cohomology groups. There are several ways to prove the Borel-Weil-Bott theorem, which we will discuss. The classical idea is to study how the Casimir operator acts on the sheaf of sections of line bundles. Instead of this, the geometric idea is trying to compute the Doubeault cohomology, transferring the sheaf cohomology to the Lie algebra cohomology. The algebraic idea is to realize that the sheaf cohomology group can be computed by the derived functor of the induction, by using the Peter-Weyl the Borel-Weil theorem can be shown immediately.

Fri, 21 Jun 2024
13:30
Lecture Room 6

Groups and Geometry in South England

Luis Jorge Sánchez Saldaña, Rachael Boyd, Mladen Bestvina
(University of Oxford)
Abstract

Dimensions of mapping class groups of orientable and non-orientable surfaces

1:30pm

Luis Jorge Sánchez Saldaña (UNAM)

Mapping class groups have been studied extensively for several decades. Still in these days these groups keep being studied from several point of views. In this talk I will talk about several notions of dimension that have been computed (and some that are not yet known) for mapping class groups of both orientable and non-orientable manifolds. Among the dimensions that I will mention are the virtual cohomological dimension, the proper geometric dimension, the virtually cyclic dimension and the virtually abelian dimension. Some of the results presented are in collaboration with several colleagues: Trujillo-Negrete, Hidber, León Álvarez and Jimaénez Rolland.

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Diffeomorphisms of reducible 3-manifolds

2:45pm

Rachael Boyd (Glasgow)

I will talk about joint work with Corey Bregman and Jan Steinebrunner, in which we study the moduli space B Diff(M), for M a compact, connected, reducible 3-manifold. We prove that when M is orientable and has non-empty boundary, B Diff(M rel ∂M) has the homotopy type of a finite CW-complex. This was conjectured by Kontsevich and previously proved in the case where M is irreducible by Hatcher and McCullough.

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Nonunique ergodicity in strata of geodesic laminations and the boundary of Outer space

4:00pm

Mladen Bestvina (Utah)

It follows from the work of Gabai and Lenzhen-Masur that the maximal number of projectively distinct ergodic transverse measures on a filling geodesic lamination on a hyperbolic surface is equal to the number of curves in a pants decomposition. In a joint work with Jon Chaika and Sebastian Hensel, we answer the analogous question when the lamination is restricted to have specified polygons as complementary components. If there is enough time, I will also talk about the joint work with Elizabeth Field and Sanghoon Kwak where we consider the question of the maximal number of projectively distinct ergodic length functions on a given arational tree on the boundary of Culler-Vogtmann's Outer space of a free group.
 

Thu, 30 May 2024
16:00
Lecture Theatre 5, Mathematical Institute

Large values of Dirichlet polynomials, and primes in short intervals

James Maynard
(University of Oxford)
Abstract

One can get fairly good estimates for primes in short
intervals under the assumption of the Riemann Hypothesis. Weaker
estimates can be shown unconditionally by using a 'zero density
estimate' in place of the Riemann Hypothesis. These zero density
estimates are typically proven by bounding how often a Dirichlet
polynomial can take large values, but have been limited by our
understanding of the number of zeros with real part 3/4. We introduce a
new method to prove large value estimates for Dirichlet polynomials,
which improves on previous estimates near the 3/4 line.

This is joint work (still in progress) with Larry Guth.

Thu, 17 Oct 2024
16:00
Lecture Room 3

Primes of the form $x^2 + ny^2$ with $x$ and $y$ prime

Ben Green
(University of Oxford)
Abstract

If $n$ is congruent to 0 or 4 modulo 6, there are infinitely many primes of the form $x^2 + ny^2$ with both $x$ and $y$ prime. (Joint work with Mehtaab Sawhney, Columbia)

Thu, 30 May 2024

11:00 - 12:00
C3

Axiomatizing monodromy

Ehud Hrushovski
(University of Oxford)
Abstract

Consider definable sets over the family of finite fields $\mathbb{F}_q$. Ax proved a quantifier-elimination result for this theory, in a reasonable geometric language. Chatzidakis, Van den Dries and Macintyre showed that to a first-order approximation, the cardinality of a definable set $X$ is definable in a very mild expansion of Ax's theory.  Can such a statement be true of the next higher order approximation, i.e. can we write $|X(\mathbb{F}_q)| = aq^{d} + bq^{d-1/2} + o(q^{d-1/2})$, with $d,a,b$ varying definably with $X$ in a tame theory?    Here $b$ must be viewed as real-valued so continuous logic is needed. I will report on joint work in progress with Will Johnson.

Thu, 23 May 2024
17:00
Lecture Theatre 1, Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, OX2 6GG

Infinite Jesters: what can philosophers learn from a puzzle involving infinitely many clowns? - Ofra Magidor and Alexander Kaiserman

Ofra Magidor and Alexander Kaiserman
(University of Oxford)
Further Information

Ofra and Alexander consider a simple but intriguing mathematical argument, which purports to show how infinitely many clowns appear to have some surprising powers. They'll discuss what conclusions philosophers can and cannot draw from this case, and connect the discussion to a number of key philosophical issues such as the problem of free will and the Grandfather Paradox for time travel.

Ofra Magidor is Waynflete Professor of Metaphysical Philosophy at the University of Oxford and Fellow of Magdalen College. Alex Kaiserman is Associate Professor of Philosophy at the University of Oxford and Fairfax Fellow and Tutor in Philosophy at Balliol College. While they are both philosophers, Ofra holds a BSc in Philosophy, Mathematics, and Computer Science and Alex holds an MPhysPhil in Physics and Philosophy, so they are no strangers to STEM subjects.

Please email @email to register to attend in person.

The lecture will be broadcast on the Oxford Mathematics YouTube Channel on Thursday 13 June at 5-6pm and any time after (no need to register for the online version).

The Oxford Mathematics Public Lectures are generously supported by XTX Markets.

Fri, 03 May 2024

12:00 - 13:00
Quillen Room

The canonical dimension of depth-zero supercuspidal representations

Mick Gielen
(University of Oxford)
Abstract

Associated to a complex admissible representation of a p-adic group is an invariant known is the "canonical dimension". It is closely related to the more well-studied invariant called the "wavefront set". The advantage of the canonical dimension over the wavefront set is that it allows for a completely different approach in computing it compared to the known computational methods for the wavefront set. In this talk we illustrate this point by finding a lower bound for the canonical dimension of any depth-zero supercuspidal representation, which depends only on the group and so is independent of the representation itself. To compute this lower bound, we consider the geometry of the associated Bruhat-Tits building.

Wed, 12 Jun 2024

16:00 - 17:00
L6

The relation gap and relation lifting problems

Marco Linton
(University of Oxford)
Abstract

If \(F\) is a free group and \(F/N\) is a presentation of a group \(G\), there is a natural way to turn the abelianisation of \(N\) into a \(\mathbb ZG\)-module, known as the relation module of the presentation. The images of normal generators for \(N\) yield \(\mathbb ZG\)-module generators of the relation module, but 'lifting' \(\mathbb ZG\)-generators to normal generators cannot always be done by a result of Dunwoody. Nevertheless, it is an open problem, known as the relation gap problem, whether the relation module can have strictly fewer \(\mathbb ZG\)-module generators than \(N\) can have normal generators when \(G\) is finitely presented. In this talk I will survey what is known and what is not known about this problem and its variations and discuss some recent progress for groups with a cyclic relation module.

Thu, 13 Jun 2024

11:00 - 12:00
C3

The Ultimate Supercompactness Measure

Wojciech Wołoszyn
(University of Oxford)
Abstract

Solovay defined the inner model $L(\mathbb{R}, \mu)$ in the context of $\mathsf{AD}_{\mathbb{R}}$ by using it to define the supercompactness measure $\mu$ on $\mathcal{P}_{\omega_1}(\mathbb{R})$ naturally given by $\mathsf{AD}_{\mathbb{R}}$. Solovay speculated that stronger versions of this inner model should exist, corresponding to stronger versions of the measure $\mu$. Woodin, in his unpublished work, defined $\mu_{\infty}$ which is arguably the ultimate version of the supercompactness measure $\mu$ that Solovay had defined. I will talk about $\mu_{\infty}$ in the context of $\mathsf{AD}^+$ and the axiom $\mathsf{V} = \mathsf{Ultimate\ L}$.

https://woloszyn.org/

Thu, 06 Jun 2024

11:00 - 12:00
C3

Demushkin groups of infinite rank in Galois theory

Tamar Bar-On
(University of Oxford)
Abstract
Demushkin groups play an important role in number theory, being the maximal pro-$p$ Galois groups of local fields containing a primitive root of unity of order $p$. In 1996 Labute presented a generalization of the theory for countably infinite rank pro-$p$ groups, and proved that the $p$-Sylow subgroups of the absolute Galois groups of local fields are Demushkin groups of infinite countable rank. These results were extended by Minac & Ware, who gave necessary and sufficient conditions for Demushkin groups of infinite countable rank to occur as absolute Galois groups.
In a joint work with Prof. Nikolay Nikolov, we extended this theory further to Demushkin groups of uncountable rank. Since for uncountable cardinals, there exists the maximal possible number of nondegenerate bilinear forms, the class of Demushkin groups of uncountable rank is much richer, and in particular, the groups are not determined completely by the same invariants as in the countable case.  
Additionally, inspired by the Elementary Type Conjecture by Ido Efrat and the affirmative solution to Jarden's Question, we discuss the possibility of a free product over an infinite sheaf of Demushkin groups of infinite countable rank to be realizable as an absolute Galois group, and give a necessary and sufficient condition when the free product is taken over a set converging to 1.
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