Date/Time : 18 Apr 2024 16:30
Venue : S14 #06-19
Speaker : Ching Wei Ho, Academia Sinica

It is a classical problem to study the evolution of roots of polynomials under application of a differential operator. In this talk, I will discuss the heat evolution of random polynomials with a rotationally invariant root distribution on the complex plane. The limiting root distribution of the heat-evolved random polynomial can be completely determined in terms of its log potential. For example, when a Weyl polynomial, whose root distribution converges to the uniform distribution on the unit disk, undergoes heat flow, the limiting root distribution is uniform on some ellipse until time 1 at which it becomes exactly the semicircle law. This is joint work with Brian Hall, Jonas Jalowy, and Zakhar Kabluchko.

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Date/Time : 18 Mar 2024 16:00
Venue : S16 #03-04
Speaker : Stefan Junk, Gakushuin University

We consider the directed polymer model, which describes paths affected by a random space-time environment in spatial dimension d>2. The model undergoes a phase transition between a high-temperature, weak disorder phase and a low-temperature, strong disorder phase, which is characterized by whether the associated (normalized) partition function converges to zero. From the physical point of view, it is more natural to consider a very-strong-disorder regime, characterized by whether the partition function converges to zero exponentially fast. It has been a long-standing conjecture that these two notions are equivalent and we now give a proof of this. Moreover, our proof reveals that weak disorder holds at the critical value. Joint work with Hubert Lacoin.

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Date/Time : 13 Mar 2024 16:00
Venue : S16 #03-09
Speaker : Hampus Nyberg, KTH Stockholm

Deep zero problems consider holomorphic functions satisfying local properties at a small number of points. We take a look at one such problem about uniqueness in the Bargmann-Fock space and show that it reduces down to a question about products such as $\frac{\sin{1}}{\sin{2}}\frac{\sin{3}}{\sin{4}}\frac{\sin{5}}{\sin{6}}…$.

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Date/Time : 25 Jan 2024 15:00
Venue : S16-06-118, Seminar Room
Speaker : Zhou Fan, Yale University

In statistical and probabilistic models, “mean-field” behavior arises when the aggregate effect of many variables is well-approximated by an average or effective field. Ideas from the analysis of mean-field models in statistical physics have been influential in our understanding of various problems of high-dimensional statistics in recent years. In this talk, I will argue that statistical applications can conversely drive the basic study of mean-field phenomena.

As a motivating example, I will discuss an application of high-dimensional mean-field theory to a nonparametric empirical Bayes procedure for PCA. I will then describe several developments of the theory beyond i.i.d. models, including extensions of Approximate Message Passing algorithms and analyses of spin glass models where the couplings matrices are orthogonally invariant in law.

Joint work w/ Yufan Li, Subhabrata Sen, Chang Su, Tianhao Wang, Yihong Wu, Xinyi Zhong.

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Date/Time : 17 Aug 2023 10:30
Venue : S17 #05-11 (SR5)
Speaker : Sayan Das, Columbia University

Sticky Brownian Motions are a family of correlated Brownian motions that have a tendency to stick together. They can be realized as the scaling limit of random walks in random environments and can also be viewed as random motions in a continuum-random environment. In this talk, I will present some results related to the quenched density of the motion of a particle in this continuum-random environment. In particular, I will show that under moderate deviation regime (a particular regime in between large deviation regime and diffusive regime), this quenched density after rescaling weakly converges to the solution of the stochastic heat equation with multiplicative space-time white noise. I will explain how our results shed light on the extreme behavior of Sticky Brownian Motions. This is a joint work with Hindy Drillick and Shalin Parekh.

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Date/Time : 18 Apr 2023 11:00
Venue : S17 #04-05 (SR2)
Speaker : Simon Gabriel, University of Warwick

We study the Allen-Cahn equation, commonly used to describe convection phenomena in physics. It is natural to start the system from chaos, corresponding to an initial condition described by spatial white noise. In two spatial dimensions, this leads to a critical initial datum, for which the equation can not be made sense of a priori.

In this talk we give a gentle introduction into the methods used to study this equation using an infinite expansion of stochastic integrals. Both the convergence and truthfulness of such an approximation are non-trivial, since the criticality has the effect that each term in the infinite expansion contributes towards the solution. Thus, it is crucial to determine each term’s statistic exactly, for which we exploit a tree-like notation. This allows us to manipulate stochastic integrals using solely their underlying tree structure. After performing such manipulations, we can conclude convergence of the expansion to a solution with the help of a series expansion of an explicit ODE.

The talk is based on joint work with Tommaso Rosati and Nikos Zygouras.

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Date/Time : 29 Mar 2023 11:00
Venue : S17 #04-06 (SR1)
Speaker : George Marinescu, Universität zu Köln

We introduce a probability measure on the space of square integrable holomorphic sections of a prequantum line bundle with the help of a Toeplitz operator and we study the behaviour of random sections for high tensor powers
of the bundle. We can recover the zero set of the symbol of the Toeplitz operator from this asymptotic.

The talk is based on joint work with Alexander Drewitz and Bingxiao Liu (arXiv:2302.08426).

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Date/Time : 25 Nov 2022 13:30
Venue : S17 #04-06 (SR1)
Speaker : Gennady Samorodnitsky, Cornell University

This is a quick introduction to simplicial complexes, both deterministic and random, simplicial homology and connections to graphs.

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Date/Time : 23 Nov 2022 15:30
Venue : S17 #04-06 (SR1)
Speaker : Gennady Samorodnitsky, Cornell University

This is a quick introduction to simplicial complexes, both deterministic and random, simplicial homology and connections to graphs.

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