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Seminars (ARA)

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Event When Speaker Title Presentation Material
ARA 18th March 2021
15:30 to 16:30
Jonathan Chapman Exponential asymptotics in applied mathematics
ARA 18th March 2021
17:00 to 18:00
Sergei Gukov The role of resurgence in QFT and in string theory
ARAW01 22nd March 2021
13:50 to 14:00
Director of INI’s Welcome Talk
ARAW01 22nd March 2021
14:00 to 16:00
Adri Olde Daalhuis, Christopher Howls Stokes Phenomena: From Dingle to Exponential Asymptotics
ARAW01 22nd March 2021
16:00 to 17:00
Gerald Dunne Resurgence in Differential Equations, and Effective Summation Methods
ARAW01 23rd March 2021
14:00 to 15:00
Adri Olde Daalhuis, Christopher Howls Stokes Phenomena: From Dingle to Exponential Asymptotics
ARAW01 23rd March 2021
15:00 to 17:00
Gerald Dunne Resurgence in Differential Equations, and Effective Summation Methods
ARAW01 23rd March 2021
17:00 to 19:00
Evening Tutorial
ARAW01 24th March 2021
14:00 to 15:00
Philippe Trinh, John King, Jonathan Chapman Exponential Asymptotics for Physical Applications
ARAW01 24th March 2021
15:00 to 16:00
Kemal Gokce Basar WKB, Eigenvalue Problems and Quantisation in QM
ARAW01 24th March 2021
16:00 to 17:00
Daniele Dorigoni Ecalle’s Theory of Resurgence
ARAW01 24th March 2021
17:00 to 19:00
Evening Tutorial
ARAW01 25th March 2021
10:00 to 12:00
Morning Tutorial
ARAW01 25th March 2021
14:00 to 15:00
Philippe Trinh, John King, Jonathan Chapman Exponential Asymptotics for Physical Applications
ARAW01 25th March 2021
15:00 to 16:00
Kemal Gokce Basar WKB, Eigenvalue Problems and Quantisation in QM
ARAW01 25th March 2021
16:00 to 17:00
Daniele Dorigoni Ecalle’s Theory of Resurgence
ARAW01 25th March 2021
17:00 to 19:00
Evening Tutorial
ARAW01 26th March 2021
10:00 to 12:00
Morning Tutorial
ARAW01 26th March 2021
14:00 to 15:00
Philippe Trinh, John King, Jonathan Chapman Exponential Asymptotics for Physical Applications
ARAW01 26th March 2021
15:00 to 16:00
Kemal Gokce Basar WKB, Eigenvalue Problems and Quantisation in QM
ARAW01 26th March 2021
16:00 to 17:00
Daniele Dorigoni Ecalle’s Theory of Resurgence
ARA 30th March 2021
16:00 to 17:00
Michael Berry Differentiations and Diversions
Asymptotic procedures, such as generating slowness

corrections to geometric phases, involve successive differentiations. For a

large class of functions, the universal attractor of the differentiation map is,

when suitably scaled, locally trigonometric/exponential; nontrivial examples

illustrate this. For geometric phases, the series must diverge, reflecting the

exponentially small final transition amplitude. Evolution of the amplitude

towards this final velue depends sensitively on the representation used. If

this is optimal, the transition takes place rapidly and universally across a

Stokes line emanating from a degeneracy in the complex time plane. But some

Hamiltonian ODE systems do not generate transitions; this is because the

complex-plane degeneracies have a peculiar structure, for which there is no

Stokes phenomenon.  Oscillating high

derivatives (asymptotic monochromaticity) and superoscillations (extreme

polychromaticity) are in a sense opposite mathematical phenomena.








ARA 1st April 2021
16:00 to 17:00
Ovidiu Costin Optimal reconstruction of functions from their truncated power series at a point





I will speak about the question of the mathematically

optimal reconstruction of a function from a finite number of terms of its power

series at a point, and on aditional data such as: as domain of analyticity,

bounds or others.



 



Aside from its intrinsic mathematical interest, this

question is important in a variety of applications in mathematics and physics

such as the practical computation of the Painleve transcendents, which I will

use as an example, and the reconstruction of functions from resurgent

perturbative series in models of quantum field theory and string theory. Given

a class of functions which have a common Riemann surface and a common type of bounds

on it, we show that the optimal procedure stems from the uniformization

theorem. A priori Riemann surface information and bounds exist for the Borel

transform of asymptotic expansions in wide classes of mathematical problems

such as meromorphic systems of linear or nonlinear ODEs, classes of PDEs and

many others,  known, by mathematical

theorems,  to be resurgent.  I will also discuss some (apparently) new

uniformization methods and maps. Explicit uniformization in Borel plane is

possible for all linear or nonlinear second order meromorphic ODEs.



 



This optimal procedure is dramatically superior to the

existing (generally ad-hoc) ones, both theoretically and in their effective

numerical application, which I will illustrate. The comparison with Pade approximants

is especially interesting.



 



When more specific information exists, such as the nature

of the singularities of the functions of interest, we found methods based on

convolution operators to eliminate these singularities. The type of

singularities is known for resurgent functions coming from many problems in

analysis. With this addition, the accuracy is improved substantially with

respect to the optimal accuracy which would be possible in full generality.



 



Work in collaboration with G. Dunne, U. Conn.







ARA 8th April 2021
16:00 to 17:00
Gergő Nemes Realistic error bounds for asymptotic expansions via integral representations

We shall consider the problem of deriving realistic error

bounds for asymptotic expansions arising from integrals. It was demonstrated by

W. G. C. Boyd in the early 1990's that Cauchy-Heine-type representations for

remainder terms are quite suitable for obtaining such bounds. I will show that

the Borel transform can lead to a more globally valid expression for remainder

terms involving R. B. Dingle's terminant function as a kernel. We will see

through examples that such a representation is, in a sense,



optimal: it leads to error bounds that are valid in large

sectors and which cannot be improved in general. Building on the important

results of Sir M. V. Berry and C. J. Howls, I will provide analogous results

for asymptotic expansions arising from integrals with saddles.



Finally, I will show how a Cauchy-Heine-type argument can

be applied to implicit problems by outlining the recent proof of a conjecture

of F. W. J. Olver on the large negative zeros of the Airy function.

ARA 13th April 2021
16:00 to 17:00
Mithat Unsal From rainbow to mass gap: Resurgence and Lefschetz thimbles at work
ARA 15th April 2021
16:00 to 17:00
Marcel Vonk Matrix models; asymptotics, transesseries, theta functions and all that
ARA 20th April 2021
16:00 to 17:00
John King Some PDEs and relatives
I shall describe the role of exponentially small

terms in some PDEs and related differential difference equations, and ask

questions about possible commonalities with QFTs. While I am mainly interested

in dissipative (parabolic) cases, I shall - for reasons that are presumably

obvious - focus on properties shared by the corresponding time-reversible

(hyperbolic) models.
ARA 22nd April 2021
16:00 to 17:00
Christopher Howls A More Exotic Asymptotic Zoo: New Stokes Lines, Virtual Turning Points and the Higher Order Stokes Phenomenon
ARA 27th April 2021
16:00 to 17:00
Sheehan Olver Riemann–Hilbert problems and Stokes phenomena
Riemann–Hilbert problems consist of recovering a piecewise analytic function from information about jumps along branch cuts. To quote Wikipedia (as of today): "Stokes phenomenon, discovered by G. G. Stokes (1847, 1858), is that the asymptotic behaviour of functions can differ in different regions of the complex plane”. We demonstrate how Stokes phenomena can lead naturally to a Riemann–Hilbert problem which in fact uniquely determines the analytic function, beginning, of course, with everyone’s favourite example: the Airy function. We further review the applications to Painlevé equations, and finally show that integrals with coalescing saddles can also be reformulated as a Riemann–Hilbert problem, in a way that, perhaps, avoids the computational pitfalls of applying quadrature directly to integral reformulations along steepest descent contours
ARA 29th April 2021
08:00 to 09:00
Yoshitsugu Takei Global study of differential equations via the exact WKB - from Schrödinger and Panlevé

The exact WKB analysis provides a powerful tool for the


global study of differential equations. In this talk we would like to give a


brief review of this analysis and discuss an important problem related to it.





 First we review


the exact WKB analysis for one-dimensional stationary Schrödinger


equations. In this case the exact WKB analysis gives a quite satisfactory


answer, that is, global behavior of solutions such as the monodromy group, the


exact quantization condition, etc are described by contour integrals of


logarithmic derivative of WKB solutions. Next we consider its generalization to

Panlevé equations. Even for such nonlinear equations the exact WKB analysis


is successful and we can obtain an explicit connection formula for Stokes


phenomena of Panlevé equations in terms of their formal power series


solutions and transseries solutions. However, to complete the global study of

Panlevé equations, we need to deal with the instanton-type solutions (or


two-parameter transseries solutions), which are purely formal and whose


behavior are much wilder than transseries solutions. It is really a big and


important problem to give an analytic interpretation to instanton-type


solutions in the exact WKB analysis. In the latter half of the talk we discuss


our recent trial to attack this challenging problem.










 

ARA 3rd May 2021
08:00 to 09:00
Yasuyuki Hatsuda Quantization conditions and Seiberg-Witten theory
ARA 4th May 2021
08:00 to 09:00
Katsushi Ito WKB periods for higher order ODE and TBA equations
ARA 5th May 2021
06:00 to 07:00
Exact -WKB vs resurgence in semiclassics -Dr. Naohisa Sueishi
ARA 5th May 2021
08:00 to 09:00
Degenerate Weber Stokes graph and its application - Dr. Syo Kamata
ARA 6th May 2021
10:00 to 11:00
Gergő Nemes Resurgent methods in exact WKB analysis - Discussion Leader Dr Gergő Nemes
ARA 6th May 2021
16:00 to 17:00
Andrew Neitzke Exact WKB and abelianization of flat connections
 I will discuss a



geometric approach to the exact WKB method via "abelianization" of



flat connections -- replacing SL(N)-connections over a Riemann surface by



GL(1)-connections over an N-fold branched cover -- and relations to



Donaldson-Thomas invariants and cluster algebras. Most of the story is on solid



footing for equations of order N=2 (the case of Schrodinger equations), and



conjectural for higher N.








ARA 11th May 2021
16:15 to 17:15
Michal P. Heller Resurgence in Relativistic Hydrodynamics: Bjorken Flow Results
Relativistic hydrodynamics is a classical field

theory with dissipation. Significant contemporary interest in relativistic

hydrodynamics comes from its central role in modelling nuclear collisions at

RHIC and LHC, robust connections with gravity uncovered in the course of the

past two decades, as well as more recent applications to neutron stars mergers.

The ideas of transseries and resurgence in relativistic hydrodynamics appeared

for the first time as a by-product of studies of Bjorken flow in the context of

nuclear collisions at RHIC and LHC. The talk will provide a pedagogical review

of these developments, as well as it will try to put them in the context of

other topics to be discussed during the focus week on resurgence in

relativistic hydrodynamics.
ARA 13th May 2021
16:00 to 17:00
Benjamin Withers From quasinormal modes to constitutive relations
I will review recent developments in large-order




relativistic hydrodynamics. Starting with properties gleaned from holographic




theories and black hole physics, I will show that quasinormal mode dispersion




relations have a finite radius of convergence set by branch points




corresponding to 'mode collisions' in the complex momentum plane. I will




explore the consequences of this observation for the hydrodynamic gradient




expansion of the one-point functions of currents, including the role played by




initial data, the impact of nonlinearities, and the structure of transseries.
ARA 18th May 2021
14:00 to 15:00
Alba Grassi title tba
ARA 20th May 2021
11:00 to 12:00
Nalini Joshi title tba
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons