Date/Time : 11 Jun 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Subhroshekhar Ghosh, National University of Singapore

The goal of this talk is to give an overview of my research over the past few years. Much of this talk is going to be devoted to the theory and applications of negative association. Two random variables are said to be negatively associated if an increase in one leads to a decrease in the other, in an appropriate stochastic sense. In statistical physics terms, it is a mathematical model that captures the phenomenon of repulsion between charged particles such as electrons. On the theoretical side, we will discuss the phenomena of rigidity and hyperuniformity, and their consequences for the behaviour of large scale particle systems. We will also discuss applications of such strongly correlated stochastic models in machine learning, and how these applications in turn naturally lead to phenomena and questions of interest in theoretical probability. Time permitting, we will also explore other recent results that, strictly speaking, fall outside the ambit of negative associations; a key direction being that of learning under latent group actions.

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Date/Time : 10 Jun 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Ionut Danaila, University of Rouen Normandy

We present several numerical tools using classical finite elements with mesh adaptivity to solve the Bogoliubov-de Gennes (BdG) eigenvalue problem, representing a linearization of the Gross-Pitaevskii (GP) equation. The BdG model is commonly used to assess the stability of stationary solutions of the GP equation. Revealing the existence of such solutions (wave configurations, exotic vortices or vortex structures) in Bose-Einstein condensates is an active and timely research topic in experimental and theoretical physics.

For the computation of the GP stationary solutions we use a Newton algorithm coupled with a continuation method exploring the parameter space (the chemical potential or the interaction constant). Bogoliubov-de Gennes equations are then solved using dedicatedlibraries (ARPACK or SLEPc) for the associated eigenvalue problem. For 3D configurations, we use domain decomposition techniques enabling parallel computing.

The programs are written as toolboxes for FreeFem++ (www.freefem.org), a free finite-element software, allowing to easily implement various numerical algorithms. Validations and illustrations are presented for computing difficult configurations with vortices observed in physical experiments: dark solitons, single-line vortex, dark/anti-dark solitons in one or two-component Bose-Einstein condensates.

References:

I. Danaila, M. A. Khamehchi, V. Gokhroo, P. Engels and P. G. Kevrekidis, Vector dark-antidark solitary waves in multicomponent
Bose-Einstein condensates, Phys. Rev. A 94, 053617, 2016.

G. Sadaka, V. Kalt, I. Danaila and F. Hecht, A finite element toolbox for the
Bogoliubov-de Gennes stability analysis of Bose-Einstein condensates, Computer Physics Communications, 294, 108948(1-17), 2024.

G. Sadaka, P. Jolivet, E. G. Charalampidis, I. Danaila, Parallel finite-element codes for the Bogoliubov-de Gennesstability analysis of Bose-Einstein condensates, submitted, 2024.

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Date/Time : 29 May 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Jaydeep Singh, Princeton University

An important class of finite-time blowup solutions to the Einstein-scalar field system was constructed by Christodoulou (’94), going by the name of k-self-similar naked singularities. In this talk we will motivate the study of these spacetimes from a PDE and general relativity perspective, and discuss the subtle issue of their (in-)stability. Our main results will show how the chief instability mechanism, a blue-shift along the past light-cone of the singularity, is highly sensitive to the space of perturbations considered.

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Date/Time : 29 May 2024 16:00
Venue : S17 #04-05 (SR2)
Speaker : Renbo Zhao, University of Iowa

We present and analyze a new Frank–Wolfe method for minimizing a -log-homogenous self-concordant barriers, with applications including positron emission tomography, D-optimal design, TV-regularized Poisson image de-blurring, quantum state tomography. The iteration complexity of our method is essentially , which recovers that obtained by Khachiyan (1996) on the D-optimal design problem. In addition, we also present and analyze an away-step variant of our proposed Frank–Wolfe method, and we show the global linear convergence of this method. When specialized to the D-optimal design problem, this settles an open problem in Ahipasaoglu, Sun and Todd (2008).

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Date/Time : 28 May 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Schwab Christoph, ETH

Centered Gaus­sian ran­dom fields (GRFs) in­dexed by com­pacta such as smooth, bounded do­mains in Eu­c­lidean space or by smooth, com­pact and ori­ent­able man­i­folds are de­term­ined by their co­v­ari­ance op­er­at­ors. We con­sider centered GRFs given sample-​wise as vari­ational solu­tions to col­or­ing op­er­ator equa­tions driven by spa­tial white noise, with pseudodif­fer­en­tial col­or­ing op­er­ator be­ing el­liptic, self-​adjoint and pos­it­ive from the Hörmander class. This in­cludes the Mat\’ern class of GRFs as a spe­cial case. Us­ing mi­cro­local tools and biortho­gonal mul­tiresol­u­tion ana­lyses on the man­i­fold, we prove that the pre­ci­sion and co­v­ari­ance op­er­at­ors, re­spect­ively, may be iden­ti­fied with bi-​infinite matrices. Fi­nite sec­tions of the covariances may be di­ag­on­ally pre­con­di­tioned ren­der­ing the con­di­tion num­ber in­de­pend­ent of the di­men­sion p of the sec­tion. We prove that a taper­ing strategy by threshold­ing as e.g. in [Bickel, P.J. and Lev­ina, E. Co­v­ari­ance reg­u­lar­iz­a­tion by threshold­ing, Ann. Stat­ist., 36 (2008), 2577-​2604] ap­plied on fi­nite sec­tions of the bi-​infinite pre­ci­sion and co­v­ari­ance matrices res­ults in op­tim­ally nu­mer­ic­ally sparse ap­prox­im­a­tions.

Nu­mer­ical sparsity here sig­ni­fies that only asymp­tot­ic­ally lin­early many nonzero mat­rix entries are suf­fi­cient to ap­prox­im­ate the ori­ginal sec­tion of the bi-​infinite co­v­ari­ance or pre­ci­sion mat­rix us­ing this taper­ing strategy to ar­bit­rary pre­ci­sion. The taper­ing strategy is non-​adaptive and the loc­a­tions of these nonzero mat­rix entries are known a pri­ori. The tapered co­v­ari­ance or pre­ci­sion matrices may also be op­tim­ally di­ag­onally pre­con­di­tioned.

Ana­lysis of the re­l­at­ive size of the entries of the tapered co­v­ari­ance matrices mo­tiv­ates novel, mul­ti­level Monte Carlo (MLMC) or­acles for co­v­ari­ance es­tim­a­tion, in sample com­plex­ity that scales log-​linearly with re­spect to the num­ber p of para­met­ers.  This ex­tends [Bickel, P.J. and Lev­ina, E. Reg­u­lar­ized Es­tim­a­tion of Large Co­v­ari­ance Matrices, Ann. Stat., 36 (2008), pp. 199-​227] to es­tim­a­tion of (fi­nite sec­tions of) pseudodif­fer­en­tial co­v­ari­ances for GRFs by this fast MLMC method.

As­sum­ing at hand sec­tions of the bi-​infinite co­v­ari­ance mat­rix in wave­let co­ordin­ates, we pro­pose and ana­lyze a novel com­press­ive al­gorithm for sim­u­lat­ing and kri­ging of GRFs. The com­plex­ity (work and memory vs. ac­cur­acy) of these three al­gorithms scales near-​optimally in terms of the num­ber of para­met­ers p of the sample-​wise ap­prox­im­a­tion of the GRF in So­bolev scales. 

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Date/Time : 13 May 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Sheng Meng, East China Normal University

A critical point x of a differentiable self-map f of X=R^n or manifolds means the determinant of the Jacobian |J_f(x)|=0. We collect these points and call it the critical set C_f. This set C_f also carries algebraic data, denoted as R_f which is nothing but the zeros (counting multiplicities) of the equation |J_f(x)|=0, called the ramification.

In the algebraic geometric setting, R_f turns out to be an effective divisor, but not necessarily smooth anymore. In this talk, I shall show you the behaviour of the coefficients of R_f (multiplicity in codimension 1), and also the much more complicated, the singularities of R_f (multiplicity in higher codimension).
Some applications will be discussed if time permits.

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Date/Time : 10 May 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Erez Lapid, Weissman Institute of Science

Prehomogeneous vector spaces are ubiquitous in the theory of nilpotent orbits of semisimple Lie algebras. Their study was initiated by M. Sato in the 1960s. Zeta functions attached to them are useful for the asymptotics of arithmetic invariants. They are also important in the geometric side of Arthur’s trace formula. I will discuss some recent work on the analysis of these zeta functions.

Joint work with Tobias Finis.

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Date/Time : 08 May 2024 16:30
Venue : S17 #04-05 (SR2)
Speaker : Junfu Yao, Johns Hopkins University

Minimal surfaces are known as critical points to the area functional. In this talk, I’ll focus on the question of finding minimal surfaces with prescribed behavior at the asymptotic infinity. This is called the asymptotic Plateau problem. I’ll briefly review previous results and present a min-max theory for finding solutions to this problem. As a corollary of the mountain-pass theorem, I’ll also discuss a topological uniqueness result.

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Date/Time : 08 May 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Xiaofan Yuan, Arizona State University

Resolving a recent problem of Bell, Frieze, and Marbach, we establish the threshold result of Frankston–Kahn–Narayanan–Park in the rainbow setting. This is joint work with Jie Han.

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