Fri, 17 Nov 2023

14:00 - 15:00
L2

Self-similar solutions to two-dimensional Riemann problems involving transonic shocks

Mikhail Feldman
(University of Wisconsin)
Abstract

In this talk, we discuss two-dimensional Riemann problems in the framework of potential flow
equation and isentropic Euler system. We first review recent results on the existence, regularity and properties of
global self-similar solutions involving transonic shocks for several 2D Riemann problems in the
framework of potential flow equation. Examples include regular shock reflection, Prandtl reflection, and four-shocks
Riemann problem. The approach is to reduce the problem to a free boundary problem for a nonlinear elliptic equation
in self-similar coordinates. A well-known open problem is to extend these results to a compressible Euler system,
i.e. to understand the effects of vorticity. We show that for the isentropic Euler system, solutions have
low regularity, specifically velocity and density do not belong to the Sobolev space $H^1$ in self-similar coordinates.  
We further discuss the well-posedness of the transport equation for vorticity in the resulting low regularity setting.

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Fri, 09 Jun 2023
16:00
L2

North meets South

Dr Thomas Karam (North Wing) and Dr Hamid Rahkooy (South Wing)
Abstract

North Wing talk: Dr Thomas Karam
Title: Ranges control degree ranks of multivariate polynomials on finite prime fields.

Abstract: Let $p$ be a prime. It has been known since work of Green and Tao (2007) that if a polynomial $P:\mathbb{F}_p^n \mapsto \mathbb{F}_p$ with degree $2 \le d \le p-1$ is not approximately equidistributed, then it can be expressed as a function of a bounded number of polynomials each with degree at most $d-1$. Since then, this result has been refined in several directions. We will explain how this kind of statement may be used to deduce an analogue where both the assumption and the conclusion are strengthened: if for some $1 \le t < d$ the image $P(\mathbb{F}_p^n)$ does not contain the image of a non-constant one-variable polynomial with degree at most $t$, then we can obtain a decomposition of $P$ in terms of a bounded number of polynomials each with degree at most $\lfloor d/(t+1) \rfloor$. We will also discuss the case where we replace the image $P(\mathbb{F}_p^n)$ by for instance $P(\{0,1\}^n)$ in the assumption.

 

South Wing talk: Dr Hamid Rahkooy
Title: Toric Varieties in Biochemical Reaction Networks

Abstract: Toric varieties are interesting objects for algebraic geometers as they have many properties. On the other hand, toric varieties appear in many applications. In particular, dynamics of many biochemical reactions lead to toric varieties. In this talk we discuss how to test toricity algorithmically, using computational algebra methods, e.g., Gröbner bases and quantifier elimination. We show experiments on real world models of reaction networks and observe that many biochemical reactions have toric steady states. We discuss complexity bounds and how to improve computations in certain cases.

Tue, 25 Apr 2023
15:30
L2

HKKP Theory for algebraic stacks

Andres Ibanez Nunez (Oxford)
Abstract

In work of Haiden-Katzarkov-Konsevich-Pandit (HKKP), a canonical filtration, labeled by sequences of real numbers, of a semistable quiver representation or vector bundle on a curve is defined. The HKKP filtration is a purely algebraic object that depends only on a lattice, yet it governs the asymptotic behaviour of a natural gradient flow in the space of metrics of the object. In this talk, we show that the HKKP filtration can be recovered from the stack of semistable objects and a so called norm on graded points, thereby generalising the HKKP filtration to other moduli problems of non-linear origin.

 

Tue, 16 May 2023
15:30
L2

Topological recursion, exact WKB analysis, and the (uncoupled) BPS Riemann-Hilbert problem

Omar Kidwai
(University of Birmingham)
Abstract
The notion of BPS structure describes the output of the Donaldson-Thomas theory of CY3 triangulated categories, as well as certain four-dimensional N=2 QFTs. Bridgeland formulated a certain Riemann-Hilbert-like problem associated to such a structure, seeking functions in the ℏ plane with given asymptotics whose jumping is controlled by the BPS (or DT) invariants. These appear in the description of natural complex hyperkahler metrics ("Joyce structures") on the tangent bundle of the stability space,and physically correspond to the "conformal limit". 
 
Starting from the datum of a quadratic differential on a Riemann surface X, I'll briefly recall how to associate a BPS structure to it, and explain, in the simplest examples, how to produce a solution to the corresponding Riemann-Hilbert problem using a procedure called topological recursion, together with exact WKB analysis of the resulting "quantum curve". Based on joint work with K. Iwaki.
Fri, 27 Jan 2023
15:00
L2

TDA Centre Meeting

Various Speakers
(Mathematical Institute (University of Oxford))
Thu, 06 Oct 2022

12:00 - 13:00
L2

Some Entropy Rate Approaches in Continuum Mechanics

Prof. Hamid Said
(Kuwait University)
Abstract

Irreversible processes are accompanied by an increase in the internal entropy of a continuum, and as such the entropy production function is fundamental in determining the overall state of the system. In this talk, it will be shown that the entropy production function can be utilized for a variational analysis of certain dissipative continua in two different ways. Firstly, a novel unified Lagrangian-Hamiltonian formalism is constructed giving phase space extra structure, and applied to the study of fluid flow and brittle fracture.  Secondly, a maximum entropy production principle is presented for simple bodies and its implications to the study of fluid flow discussed. 

Thu, 06 Oct 2022

11:00 - 12:00
L2

Second-order regularity properties of solutions to nonlinear elliptic problems

Prof. Andrea Cianchi
(Universita' di Firenze)
Abstract

Second-order regularity results are established for solutions to elliptic equations and systems with the principal part having a Uhlenbeck structure and square-integrable right-hand sides. Both local and global estimates are obtained. The latter apply to solutions to homogeneous Dirichlet problems under minimal regularity assumptions on the boundary of the domain. In particular, if the domain is convex, no regularity of its boundary is needed. A critical step in the approach is a sharp pointwise inequality for the involved elliptic operator. This talk is based on joint investigations with A.Kh.Balci, L.Diening, and V.Maz'ya.

Fri, 11 Nov 2022

12:00 - 15:45
L2

Centre for Topological Data Analysis Centre Meeting

Adam Brown, Heather Harrington, Živa Urbančič, David Beers.
(University of Oxford, Mathematical Institute)
Further Information

Details of speakers and schedule will be posted here nearer the time. 

Abstract

Here is the program.

Thu, 16 Jun 2022

14:00 - 15:00
L2

Factorization in AdS/CFT

Carmen Jorge Diaz
((Oxford University))
Abstract
Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome
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