Time: 14:00 (GMT+9) 

Location: Zoom

Abstract: Computing the ground state and energy of a quantum Hamiltonian is of significant importance in quantum chemistry, and quantum information. In this talk, we propose Quantum random power method for ground state computation. Our method is simularable with a fixed-depth quantum circuit, and a temporary access to a quantum computer.  In the method, a matrix polynomial of a Hamiltonian is randomly estimated using the Fourier expansion, a polynomial filtering and Hamiltonian simulation on a quantum computer. The cost of estimating the matrix element has a logarithmic dependence on the system size. After the estimation of matrix polynomial is finished, our random power method is applied for ground state approximation.  Due to the use of polynomial filtering, we obtain an enlarged spectral gap, and stable convergence properties despite the presence of biased noise involved in the matrix polynomial estimation. Under a small device noise assumption, we show the convergence of our method to a ground state with probabilty one. Numerical experiments with the one-dimensional transverse-field Ising model and Hubbard models validate our findings.

Time: 10:00 (GMT+9)

Location: Zoom

Abstract: We study asymptotic expansions of the Euclidean Φ4-measure in the low-temperature regime. In particular, this extends the asymptotic expansions of Gaussian function space integrals developed in Schilder (1966) and Ellis and Rosen (1982) to the singular setting, where the field is no longer a function, but just a distribution. As a consequence, we obtain limit theorems, specifically the law of large numbers and the central limit theorem for the Φ4-measure in the low-temperature limit.  

We also study the focusing Φ4-measure, namely, focusing Gibbs measure associated with the Schrödinger and wave system, initiated by Lebowitz, Rose, and Speer (1988). Through the process of taking the infinite volume limit, we analyze that the focusing Φ4-measure with appropriate scaling exhibits Gaussian fluctuation around a scaled solitary wave, namely, the central limit theorem.

The first part of the talk is based on a joint work with Benjamin Gess (Bielefeld, Max Planck) and Pavlos Tsatsoulis (Bielefeld), while the second part is based on a joint work with Philippe Sosoe (Cornell).

Time: 10:00 (GMT+9)

Location: Zoom

Abstract: We study asymptotic expansions of the Euclidean Φ4-measure in the low-temperature regime. In particular, this extends the asymptotic expansions of Gaussian function space integrals developed in Schilder (1966) and Ellis and Rosen (1982) to the singular setting, where the field is no longer a function, but just a distribution. As a consequence, we obtain limit theorems, specifically the law of large numbers and the central limit theorem for the Φ4-measure in the low-temperature limit.  

We also study the focusing Φ4-measure, namely, focusing Gibbs measure associated with the Schrödinger and wave system, initiated by Lebowitz, Rose, and Speer (1988). Through the process of taking the infinite volume limit, we analyze that the focusing Φ4-measure with appropriate scaling exhibits Gaussian fluctuation around a scaled solitary wave, namely, the central limit theorem.

The first part of the talk is based on a joint work with Benjamin Gess (Bielefeld, Max Planck) and Pavlos Tsatsoulis (Bielefeld), while the second part is based on a joint work with Philippe Sosoe (Cornell).

Time: 15:00 (GMT+9) 

Location: Zoom

Abstract: We consider parabolic equations with critical drifts, commonly referred to as drift-diffusion equations:

∂tu−∆u+b·Du=f

We focus on the case where the drift term b∈Lt,\infty(Lx,d) satisfies divergence-free condition, and discuss Holder regularity of the solutions to the above equations. We also discuss its application to weak uniqueness of corresponding martingale problem.

Time: 14:00 (GMT+9) 

Location: Zoom

Abstract: This talk is based on the paper "A Furstenberg-type problem for circles, and a Kaufman-type restricted projection theorem in R3" by M. Pramanik, T. Yang, and J. Zahl. Their main tool is so-called maximal function estimates, which is Theorem 1.7 in the paper. The speaker explains what the maximal functions estimates are, contents of Theorem 1.7, and presents a sketch of the proof in which the authors used a result from topological graph theory.

[1] M. Pramanik, T. Yang, J. Zahl , A Furstenberg-type problem for circles, and a Kaufman-type restricted projection theorem in R3, arXiv preprint, arXiv:2207.02259.

Time: 14:00 (GMT+9) 

Location: Zoom

Abstract: We consider elliptic and parabolic equations with (critical) drifts, commonly referred to as drift-diffusion equations:

∂tu−∆u+b·Du=f

Here, the drift b belongs to critical Lebesgue spaces, such as b∈Lq(Lp) with d/p + 2/q = 1. We discuss boundedness, Holder regularity to the solutions of drift-diffusion equations. In particular, Aleksandrov-Fabes-Stroock estimate and De Giorgi theory are still hold even in the presence of singular drifts. Furthermore, if time permits, we introduce applications of these results to related fields.

Time: 14:00 (GMT+9) 

Location: Zoom

Abstract: Focus on the following question initiated by Riemann and Cantor: "For which open set G does it hold that if a trigonometric series is zero on G, then this series is identically zero?". In this presentation, we introduce some results of when the complement of G is a self-similar set, reviewing the paper [1].

[1] Jialun Li, Tuomas Sahlsten, Trigonometric series and self-similar sets, J. Eur. Math. Soc. 24 (2022), no. 1, pp. 341–368.

Time: 14:00 (GMT+9) 

Location: Zoom

Abstract: In this talk, we examine the results related to the boundedness of Levy operators with measurable coefficients within Lebesgue spaces. Multiple approaches exist for investigating Levy operators, and we focus on the martingale transform approach. 

References

[1] R. Bañuelos, K. Bogdan, Levy processes and Fourier multipliers, Journal of Functional Analysis, Volume 250,  Issue 1, Pages 197-213, 2007.

[2] R. Bañuelos, A. Bielaszewski, K. Bogdan, Fourier multipliers for non-symmetric Levy processes, Banach Center Publications, Volume 95 Issue 1, Pages 9-25, 2011. 

Time: 14:00 (GMT+9) 

Location: Zoom

Abstract: In this presentation, we review the paper [1], which deals with the small-time estimation of spectral heat content of stable processes in smooth domains. In short, the authors provided two-sided estimations of spectral heat content of isotropic stable processes for small time. The main idea was estimations for stopping time and geometric aspect of domain. Considering the method given in the paper, we might obtain a glimpse of approaches for nonsmooth domains. Further references will be given during the presentation.

References

[1] H. Park, R. Song, Spectral heat content for $\alpha$-stable processes in $C^{1,1}$ open sets, Electronic Journal of Probability, Article no.27, Pages 1--19, 2022.

Time: 14:00(GMT+9) 

Location: Zoom

Abstract: In this talk, we review the paper “A generalization of Bourgain’s circular maximal function” by W. Schlag. The purpose of the paper is to present a Fourier transform-free approach to the maximal functions. Hence, focus on methods from combinatorics derived from the three circles lemma.

References

[1] W. Schlag, A generalization of Bourgain's circular maximal function, Journal of the American Mathematical Society, Volume 10, Number 1, Pages 103-122, 1997.

Time: 14:00(GMT+9) 

Location: Zoom

Abstract: In this talk, we review the paper “A generalization of Bourgain’s circular maximal function” by W. Schlag. The purpose of the paper is to present a Fourier transform-free approach to the maximal functions. Hence, focus on methods from combinatorics derived from the three circles lemma.

References

[1] W. Schlag, A generalization of Bourgain's circular maximal function, Journal of the American Mathematical Society, Volume 10, Number 1, Pages 103-122, 1997.