Date/Time : 20 Jun 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Sunil Chebolu, Illinois State University

More than 50 years ago, Fuchs asked which abelian groups can be the group of units of a commutative ring? Though progress has been made, the question remains open. I will present an overview of my joint work with Keir Lockridge on this problem. We answered this question for various classes of groups including indecomposable abelian groups, dihedral groups, quaternion groups and some p-groups. This work also gave us several new characterizations of Mersenne primes and Fermat primes.

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Date/Time : 19 Jun 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Yuan Yancheng, Hong Kong Polytechnic University

Convex clustering model has attracted growing attention in the machine learning community. In this talk, we will introduce some recent progress of convex clustering model on model performance and numerical computation. Moreover, we will introduce a newly developed solver for solving the convex clustering model with GPU acceleration.  

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Date/Time : 18 Jun 2024 16:00
Venue : S17 #04-05 (SR2)
Speaker : Weng Kee Wong, University of California at Los Angeles

This talk first reviews mathematical techniques for constructing various types of optimal experimental designs before I present applications to design early phase clinical trials. For more complex problems, the methodology is increasingly combined with nature-inspired metaheuristics, which are assumptions free, general-purpose optimization algorithms motivated by nature, like animal behavior. After a quick overview of metaheuristics, I use MATLAB and demonstrate how these algorithms can efficiently find multiple-objective optimal designs for a biomedical problem. E-optimal designs are a minimax type of optimal designs that pose particular mathematical challenges and I discuss some of them, including a continuing effort to find a practically useful algorithm for generating E-optimal designs and has a proof of convergence.

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Date/Time : 14 Jun 2024 15:00
Venue : S17 #04-04 (SR3)
Speaker : Greg McShane, Institut Fourier

The modular surface $\mathbb{H}/\Gamma,\, \Gamma= \math rm{SL}(2,\mathbb{Z})$ has many covers -for example the three punctured torus $\mathbb{H}/\Gamma(2)$ and the once punctured torus $\mathbb{H}/\Gamma’$.

We will discuss how classical Diophantine approximation can be interpreted in terms of the behavior of geodesics on the once-punctured torus and a geometric reformulation of the Frobenius uniqueness conjecture. We will then give an account of two theorems of Fermat in terms of the automorphisms of $\mathbb{H}/\Gamma(2)$:
– if $p$ is a prime such that $4|(p-1)$ then can be written as a sum of squares $p = c^2 + d^2$
– if $p$ is a prime such that $3|(p-1)$ then can be written as $ p = c^2 +cd + d^2$

Finally, we will discuss possible extensions to surfaces of the form $\mathbb{H}/\Gamma_0(N)$.

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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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