seminars:anal

## The Analysis Seminar

The seminar meets Wednesdays in WH-100E at 3:30-4:30 p.m. There are refreshments and snacks in WH-102 at 3:15.

The seminar is partly funded as one of Dean's Speaker Series in Harpur College (College of Arts and Sciences) at Binghamton University.

Organizers: Paul Loya, David Renfrew, Xiangjin Xu and Gang Zhou

### Spring 2023

* January 25th, Wednesday (3:30-4:30pm)

Speaker : organizational meeting
Topic: organizational meeting

Abstract: organizational meeting

* February 1st, Wednesday (3:30-4:30pm)

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* February 8th, Wednesday (3:30-4:30pm)

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* February 15th , Wednesday (3:30-4:30pm)

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* February 22nd, Wednesday (3:30-4:30pm)

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* March 2nd, Thursday (3:30-4:30pm)

Speaker: Jacob Shapiro (Princeton)
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* March 8th, Wednesday (3:30-4:30pm)

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* March 15th, Wednesday (3:30-4:30pm)

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* March 22nd, Wednesday (3:30-4:30pm)

Speaker: Mihai Stoiciu (Williams College)
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* March 29th, Wednesday (3:30-4:30pm)

Speaker: David Renfrew (Binghamton University)
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* April 5th, Wednesday (3:30-4:30pm) (Spring break)

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* April 12th, Wednesday (3:30-4:30pm)

Speaker:Rongwei Shen (University at Albany)
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* April 19th, Wednesday (3:30-4:30pm)

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* April 26th, Wednesday (3:30-4:30pm)

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* May 3rd, Wednesday (3:30-4:30pm)

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### Fall 2022

* September 7th, Wednesday (3:30-4:30pm)

Speaker :
Topic: organizational meeting

* September 21st, Wednesday (3:30-4:30pm)

Speaker : Xiangjin Xu (Binghamton University)
Topic: Sharp Hamilton's Gradient and Laplacian Estimates for Heat Kernels on complete manifolds

Abstract: We first extend the gradient and Laplacian estimates of R. Hamilton for positive solutions of the heat equation on closed manifolds, to bounded positive solutions on complete non-compact manifolds with $Ric(M)\geq -k$ for constant $k\geq 0$. An application of our results, together with the two side Gaussian bounds of our recent work on the heat kernel, yields sharp estimates on the gradient and Laplacian of the heat kernel for complete manifolds with $Ric(M)\geq -k$, which are sharp with the same leading term in the short time asymptotic for all manifolds.

* September 28th, Wednesday (3:30-4:30pm)

Speaker : Gang Zhou (Binghamton University)
Topic: “Random currents and continuity of Ising model’s spontaneous magnetization” by M. Aizenman, H. Duminil-Copin and V. Sidoravicius

Abstract: I will present the paper “Random currents and continuity of Ising model’s spontaneous magnetization” by M. Aizenman, H. Duminil-Copin and V. Sidoravicius.

In the paper they considered three dimensional antiferromagnetic Ising model. It is known that at the high temperature, the system is at disorder; at the low temperature, the system exhibits ferromagnetic order, or magnetization. They proved that at the critical temperature, the magnetization is continuous, which was a long standing conjecture.

A crucial technique is the so-called switching lemma. It establishes a bijection between undirected graphs generated by the random current representation. In many important papers this was used, including the ones helping Hugo Duminil-Copin to win a Fields medal in 2022.

However this technique does not work for the other spin models, for example, XY model or most of the quantum models.

Any input is welcome.

* October 5th , Wednesday (3:30-4:30pm)

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* October 12th, Wednesday (3:30-4:30pm)

Speaker: Marius Beceanu (University at Albany)

Topic: Spectral multipliers and decay estimates

Abstract: I will present some recent results about spectral multipliers for $-\Delta+V$, where V is a scalar potential in an optimal or almost optimal class of potentials. The results are used to prove new estimates for some partial differential equations. All results are in three space dimensions. This is joint work with Gong Chen and, separately, Michael Goldberg.

* October 19th, Wednesday (3:30-4:30pm)

Speaker: Calvin Chin (Binghamton University)
Topic: Deriving the central limit theorem from the de Moivre-Laplace theorem

Abstract: The de Moivre-Laplace theorem says that binomial distributions, when correctly rescaled, resemble normal distributions. This is arguably the simplest non-trivial special case of the central limit theorem. Given the fact that the de Moivre-Laplace theorem can be proved by direct computation, it is natural to ask whether the general version of the central limit theorem follows from it. In this talk, I will briefly go over existing proofs of the (Lindeberg-Lévy) central limit theorem to provide a context, and derive the central limit theorem from the de Moivre-Laplace theorem using relatively elementary arguments. In particular, this proof will avoid the use of characteristic functions and Brownian motions.

* October 26th, Wednesday (3:30-4:30pm)

Speaker: Paul Barber (Binghamton University)
Topic: Blowup dynamics of some nonlinear heat equations

Abstract: In this talk I will present the papers “Asymptotically Self-similar Blow-up of Semilinear Heat Equations” by Yoshikazu Giga and Robert V. Kohn, and “Refined Asymptotics for the Blowup of $u_t-\Delta u = u^p$” by Stathis Filippas and Robert V. Kohn. The authors study the blowup dynamics of semilinear heat equation $u_t-\Delta u = u^p, p>1$ in $\mathbb{R}^n$ in both papers, with the first paving the way for the second. This nonlinear heat equation has structure similar to many other nonlinear equations, in particular several which arise in geometric analysis, and so often the techniques used to study the dynamics of spacial blow up of this equation may be used to study the blowup of curvature in geometric flows.

Giga and Kohn show that for blowup solutions u which satisfy a weak blowup growth restriction, a version of u rescaled in time and space approaches its steady state solution asymptotically. Fillipas and Kohn then study the long time behavior of the rescaled solution from a dynamical systems point of view: by projecting the equation satisfied by v onto suitably chosen subspaces, one can show that the long time behavior is dominated by the neutral mode, whose dynamics may be obtained exactly.

Some more recent and current work will be briefly discussed at the end.

* November 2nd, Wednesday (3:30-4:30pm)

Speaker: Gang Zhou (Binghamton University)
Topic: About two postulates for the quantum measurement

Abstract: In this talk I will present a progress we made on the quantum measurement. After a measurement on the quantum system, it will collapse into the observed state. There are two postulates for this, one was made by Von Neumann for the density matrices, the other one was made by Luder for the wave function. Based on a model proposed by Gisin, we prove the equivalence between these two.

This is a joint work with Juerg Froehlich.

This is a new direction for me. I will try to answer the questions.

November 9th, Wednesday (3:30-4:30pm)

Speaker: Hans Emil Oscar Mickelin (Princeton University)
Topic: An optimal scheduled learning rate for a randomized Kaczmarz algorithm

Abstract: The Kaczmarz algorithm is a classical iterative numerical method for solving large and overdetermined linear systems. It has received increasing attention over the last decade, starting with a proof in 2009 of a convergence rate that applies to general matrices, for a variation of the algorithm known as the randomized Kaczmarz algorithm. This talk will outline extensions of the algorithm to deal with systems perturbed by noise, with applications to machine learning and medical imaging.

* November 16th, Wednesday (3:30-4:30pm)

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* November 23rd, Wednesday (3:30-4:55pm)

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* November 30th, Wednesday (3:30-4:30pm)

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* December 7th, Wednesday (3:30-4:30pm)

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