http://www-math.umd.edu/research/seminars.html http://www-math.umd.edu/research/seminars.html Wed, 28 Aug 2024 14:00:00 EDT Where: Kirwan Hall 1313
Speaker: Ilya Goldsheid (Queen Mary University of London) - https://webspace.maths.qmul.ac.uk/i.goldsheid/
Abstract:
Let $(\xi_j)_{j\ge1} $, be a non-stationary Markov chain with phase
space $X$ and let $\mathfrak{g}_j:\,X\mapsto SL(m,R)$ be a sequence of
functions on $X$ with values in the unimodular group.
Set $g_j=\mathfrak{g}_j(\xi_j)$ and denote by $S_n=g_n\ldots g_1$, the
product of the matrices $g_j$.
We provide sufficient conditions for exponential growth of the norm
$\|S_n\|$ when the Markov chain is not supposed to be stationary. This
generalizes the classical theorem of Furstenberg on the exponential
growth of products of independent identically distributed matrices as
well as its extension by Virtser to products of stationary
Markov-dependent matrices.
]]> http://www-math.umd.edu/research/seminars.html Wed, 04 Sep 2024 14:00:00 EDT Where: Kirwan Hall 1313
Speaker: Ron Peled (University of Maryland and Tel Aviv University) - https://www.tau.ac.il/~peledron/
Abstract: A minimal surface in a random environment (MSRE) is a surface which minimizes the sum of its elastic energy and its environment potential energy, subject to prescribed boundary conditions. Apart from their intrinsic interest, such surfaces are further motivated by connections with disordered spin systems, first-passage percolation models and minimal cuts in the $Z^D$ lattice with random capacities.
We wish to study the geometry of $d$-dimensional minimal surfaces in a $(d+n)$-dimensional random environment. Specializing to a model that we term harmonic MSRE, in an "independent" random environment, we rigorously establish bounds on the geometric and energetic fluctuations of the minimal surface, as well as a scaling relation that ties together these two types of fluctuations. In particular, we prove, for all values of $n$, that the surfaces are delocalized in dimensions $d\le 4$ and localized in dimensions $d\ge 5$. Moreover, the surface delocalizes with power-law fluctuations when $d\le 3$ and with sub-power-law fluctuations when $d=4$. Many of our results are new even for $d=1$ (indeed, even for $d=n=1$), corresponding to the well-studied case of (non-integrable) first-passage percolation.

No prior knowledge in the topic will be assumed.
Based on joint works with Barbara Dembin, Dor Elboim and Daniel Hadas and with Michal Bassan and Shoni Gilboa.
]]> http://www-math.umd.edu/research/seminars.html Wed, 11 Sep 2024 14:00:00 EDT Where: Brin Center
Speaker: No Seminar - Probability-related workshop in Brin Center (.) - https://brinmrc.umd.edu/programs/workshops/fall24/fall24-workshop-macroscopic.html
Abstract: .
]]> http://www-math.umd.edu/research/seminars.html Wed, 18 Sep 2024 14:00:00 EDT Where: Kirwan Hall 1313
Speaker: Tomasz Komorowski (Polish Academy of Sciences) - https://www.impan.pl/en/insitute/mathematicians/scientific-staff
Abstract: We consider the limit of a linear kinetic equation with a degenerate scattering kernel and a reflection-transmission-absorption condition at an interface. An equation of this type arises from the kinetic limit of a microscopic harmonic chain of oscillators whose dynamics is perturbed by a stochastic term, conserving energy and momentum. The chain is in contact, via one oscillator, with a heat bath, which, in the limit, generates the boundary condition at the
interface.

It is known that in the absence of the interface, the solution of the kinetic equation exhibits either superdiffusive, or diffusive behavior, in the proper long time - large scale limit, depending on the dispersion relation of the harmonic chain. We discuss how the presence of the interface influences the boundary condition for the limiting diffusion, or anomalous diffusion.

The presented results have been obtained in collaboration with G. Basile (Univ.
Roma I), A. Bobrowski (Lublin Univ. of Techn.), K. Bogdan (Wrocław Univ. of Sci. and
Techn.), L. Arino (Ensta, Paris), S. Olla (Univ. Paris-Dauphine and GSSI, L’Aquila),
L. Ryzhik (Stanford Univ.), H. Spohn (TU, Munich).
]]> http://www-math.umd.edu/research/seminars.html Wed, 25 Sep 2024 14:00:00 EDT Where: Kirwan Hall 1313
Speaker: Arka Adhikari (University of Maryland) -
Abstract: We find a natural four-dimensional analog of the moderate deviation results for the capacity of the random walk, which corresponds to Bass, Chen, and Rosen concerning the volume of the random walk range for $d=2$. We find that the deviation statistics of the capacity of the random walk can be related to the optimal constant of generalized Gagliardo-Nirenberg inequalities.

This is based on joint work with I. Okada.
]]> http://www-math.umd.edu/research/seminars.html Wed, 02 Oct 2024 14:00:00 EDT Where: Kirwan Hall 1313
Speaker: Alexei Novikov (Penn State University) -
Abstract: The Lucas-Moll system is a mean-field game type model describing the growth of an economy by means of diffusion of knowledge. The individual agents in the economy advance their knowledge by learning from each other and via internal innovation. The main result is a long time convergence theorem. Namely, the solution to the Lucas-Moll system behaves in such a way that at large times an overwhelming majority of the agents spend no time producing at all and are only learning. In particular, the agents density propagates at the Fisher-KPP speed. We name this type of solutions a lottery society.
]]> http://www-math.umd.edu/research/seminars.html Wed, 09 Oct 2024 14:00:00 EDT Where: Kirwan Hall 1313
Speaker: Kazuo Yamazaki (University of Nebraska - Lincoln) - https://sites.google.com/view/kazuo-yamazaki/home/
Abstract: We will describe ideas of the convex integration technique and its recent developments for stochastic partial differential equations in fluid mechanics. Then, we will specifically consider the momentum formulation of the two-dimensional surface quasi-geostrophic equations forced by random noise, of both additive and linear multiplicative types. By using the technique of convex integration, for any prescribed deterministic function satisfying suitable conditions, we construct solutions to each system whose energy coincides with the fixed function. Consequently, we prove the non-uniqueness of almost sure Markov selections of a suitable class of weak solutions associated with the momentum surface quasi-geostrophic equations in both cases of noise. Part of this talk is based on a joint work with Elliott Walker.
]]> http://www-math.umd.edu/research/seminars.html Wed, 16 Oct 2024 14:00:00 EDT Where: Kirwan Hall 1313
Speaker: Doron Puder (Tel Aviv University) - https://sites.google.com/site/doronpuder/
Abstract: Let w be a word in a free group. A few years ago, Magee and I, relying on a work of Calegari, discovered that the stable commutator length of w, which is a well-studied topological invariant, can also be defined in terms of certain Fourier coefficients of w-random unitary matrices. 
But there are very natural ways to tweak the random-matrix side of this story: one may consider, for example, w-random permutations or w-random orthogonal matrices, and apply the same definition to obtain other "stable" invariants of w. Are these invariants interesting? Do they have, too, alternative topological/combinatorial definitions?
In a joint work with Yotam Shomroni, we present concrete conjectures and begin to answer some of them.

No background is assumed - I will define all notions, including what a w-random element is.
]]> http://www-math.umd.edu/research/seminars.html Wed, 23 Oct 2024 14:00:00 EDT Where: Kirwan Hall 1313
Speaker: Zdzislaw Brzezniak (University of York) - https://www.york.ac.uk/maths/people/zdzislaw-brzezniak/
Abstract: I will talk about the local and global well-posedness theory in $L^1$, inspired by the approach of Keel and Tao from the 1998 paper "Local and global well-posedness of wave maps on $\mathbb{R}^{1+1}$ for rough data", for the forced wave map equation in the "external'' formalism. In this context, the target manifold is treated as a submanifold of an Euclidean space. As a byproduct, we can reprove Y. Zhou's uniqueness result from the 1999 paper "Uniqueness of weak solutions of $1+1$ dimensional wave maps",  leading to the uniqueness of weak solutions with locally finite energy. Additionally, we achieve the scattering of such solutions through a conformal compactification argument. This talk is based on a joint paper with Jacek Jendrej (Paris) and Nimit Rana (York) of the same title, arXiv:2404.09195.
]]> http://www-math.umd.edu/research/seminars.html Wed, 30 Oct 2024 14:00:00 EDT Where: Kirwan Hall 1313
Speaker: Duncan Dauvergne (University of Toronto) - https://www.math.toronto.edu/ddauver/
Abstract: The directed landscape is a random directed metric on the plane that is the scaling limit for models in the KPZ universality class. In this metric, typical pairs of points are connected by a unique geodesic. However, certain exceptional pairs are connected by more exotic geodesic networks. The goal of this talk is to describe a full classification for these exceptional pairs. I will also discuss some connections with other models of random geometry.
]]> http://www-math.umd.edu/research/seminars.html Wed, 06 Nov 2024 14:00:00 EST Where: Kirwan Hall 1313
Speaker: Will Perkins (Georgia Tech) - https://willperkins.org/
Abstract: A classic problem in probability theory and combinatorics is to estimate the probability that the random graph $G(n,p)$ contains no triangles. This problem can be viewed as a question in "non-linear large deviations".
The asymptotics of the logarithm of this probability (and related lower tail probabilities) are known in two distinct regimes.  When $p\gg 1/\sqrt{n}$, at this level of accuracy the probability matches that of $G(n,p)$ being bipartite; and when $p\ll 1/\sqrt{n}$, Janson's Inequality gives the asymptotics of the log.  I will discuss a new approach to estimating this probability in the "critical regime": when $p = \Theta(1/\sqrt{n})$.  The approach uses ideas from statistical physics and algorithms and gives information about the typical structure of graphs drawn from the corresponding conditional distribution.  Based on joint work with Matthew Jenssen, Aditya Potukuchi, and Michael Simkin.
]]> http://www-math.umd.edu/research/seminars.html Wed, 13 Nov 2024 14:00:00 EST Where: Kirwan Hall 1313
Speaker: Hindy Drillick (Columbia University) - https://www.math.columbia.edu/~hdrillick/
Abstract: In this talk, I will discuss joint work with Levi Haunschmid-Sibitz where we construct the stochastic six-vertex model speed process. We will first define the stochastic six-vertex model with step initial data and show that a second-class particle started at the origin converges almost surely to an asymptotic speed. We will then generalize this by assigning each particle in the model a different class, allowing us to simultaneously track the speeds of all the particles. The speed process is obtained as the joint limit of these speeds. Along the way, we will develop a stochastic domination result for second- and third-class particles, as well as moderate deviation tail bounds for the stochastic six-vertex model.
]]> http://www-math.umd.edu/research/seminars.html Wed, 20 Nov 2024 14:00:00 EST Where: Kirwan Hall 1313
Speaker: Kenneth Alexander (University of Southern California) - https://dornsife.usc.edu/kenneth-alexander/
Abstract: It is not known (and even physicists disagree) whether first passage percolation (FPP) on $\mathbb{Z}^d$ has an upper critical dimension $d_c$, such that the fluctuation exponent $\chi=0$ in dimensions $d>d_c$. In part to facilitate study of this question, we may nonetheless try to understand properties of FPP in such dimensions should they exist, in particular how they should differ from $d0$ must be false if $\chi=0$.  A particular one of the three is most plausible to fail, and we explore the consequences if it is indeed false.  These consequences support the idea that when $\chi=0$, passage times are ``local'' in the sense that the passage time from $x$ to $y$ is primarily determined by the configuration near $x$ and $y$. Such locality is manifested by certain ``disc--to--disc'' passage times, between discs in parallel hyperplanes, being typically much faster than the fastest mean passage time between points in the two discs.
]]> http://www-math.umd.edu/research/seminars.html Wed, 27 Nov 2024 14:00:00 EST Where: Kirwan Hall 1313
Speaker: No Seminar - Thanksgiving recess (.) -
Abstract: .
]]> http://www-math.umd.edu/research/seminars.html Wed, 04 Dec 2024 14:00:00 EST Where: Kirwan Hall 1313
Speaker: Nathanaël Berestycki (University of Vienna) - https://homepage.univie.ac.at/nathanael.berestycki/
Abstract: Liouville quantum gravity (LQG) is a certain canonical random geometry in two dimensions, which physicists have developed for the last 40 years in order to solve or approach a large variety of problems. Recently a rigorous construction has been proposed, leading to a number of breakthroughs in probability. 

In this talk we will discuss the spectral geometry of the associated Laplace-Beltrami operator. In particular we will show that the eigenvalues a.s. obey a Weyl law. This result (joint work with Mo Dick Wong) comes from a fine analysis of the LQG heat trace, which homogenises despite overwhelming pointwise fluctuations.

We will also present a number of conjectures, notably suggesting a connection with the field of "quantum chaos".
]]> http://www-math.umd.edu/research/seminars.html Wed, 29 Jan 2025 14:00:00 EST Where: Kirwan Hall 1313
Speaker: Sky Cao (MIT) - https://mit.edu/~skycao/
Abstract: There has been much recent progress on the local solution theory for geometric singular SPDEs. However, the global theory is still largely open. In my talk, I will discuss the global well-posedness of the stochastic Abelian-Higgs model in two dimensions, which is a geometric singular SPDE arising from gauge theory. The proof is based on a new covariant approach, which consists of two parts: First, we introduce covariant stochastic objects, which are controlled using covariant heat kernel estimates. Second, we control nonlinear remainders using a covariant monotonicity formula, which is inspired by earlier work of Hamilton. Joint work with Bjoern Bringmann.
]]> http://www-math.umd.edu/research/seminars.html Wed, 05 Feb 2025 14:00:00 EST Where: Kirwan Hall 1313
Speaker: Hao Shen (University of Wisconsin-Madison) - https://people.math.wisc.edu/~hshen3/
Abstract: Lattice Yang-Mills or lattice gauge theory are natural lattice models where the field takes values in a matrix group. There are some important questions, such as exponential decay of correlations (mass gap), uniqueness of infinite volume limit. The Langevin dynamics, or so called stochastic quantization, can be exploited to obtain results in these directions, in a large coupling regime. If time permitted, I will also discuss more general models, such as lattice Yang-Mills coupled with Higgs fields. Based on joint work with Rongchan Zhu and Xiangchan Zhu.
]]> http://www-math.umd.edu/research/seminars.html Wed, 12 Feb 2025 14:00:00 EST Where: Kirwan Hall 1313
Speaker: Ramon van Handel (Princeton University) - https://web.math.princeton.edu/~rvan/
Abstract: It was conjectured by Alon in the 1980s that random d-regular graphs have
the largest possible spectral gap (up to negligible error) among all
d-regular graphs. This conjecture was proved by Friedman in 2004 in major
tour de force. In recent years, deep generalizations of Friedman's
theorem, such as strong convergence of random permutation matrices due to
Bordenave and Collins, have played a central role in a series of
breakthrough results on random graphs, geometry, and operator algebras.

In joint work with Chen, Garza-Vargas, and Tropp, we recently discovered a
surprisingly simple new approach to such results that is almost entirely
based on soft arguments. This approach makes it possible to address
previously inaccessible questions: for example, it enables a sharp
understanding of the large deviation probabilities in Friedman's theorem,
and establishes optimal spectral gaps of random Schreier graphs that
require many fewer random bits than ordinary random regular graphs. I will
aim to explain some of these results and some intuition behind the proofs.
]]> http://www-math.umd.edu/research/seminars.html Wed, 19 Feb 2025 14:00:00 EST Where: Kirwan Hall 1313
Speaker: Cheng Ouyang (UIC) - https://homepages.math.uic.edu/~couyang/
Abstract: We introduce a family of intrinsic Gaussian noises on compact manifolds that we call “colored noise” on manifolds. With this noise, we study the parabolic Anderson model (PAM) on compact manifolds. Under some curvature conditions, we show the well-posedness of the PAM and provide some preliminary (but sharp) bounds on the second moment of the solution. It is interesting to see that global geometry enters the play in the well-posedness of the equation.
]]> http://www-math.umd.edu/research/seminars.html Wed, 26 Feb 2025 14:00:00 EST Where: Kirwan Hall 1313
Speaker: Roger van Peski (Columbia University) - https://www.math.columbia.edu/~rv2549/
Abstract: I will discuss recent work with Hoi Nguyen (https://arxiv.org/abs/2409.03099 ) on products of discrete random matrices with IID integer entries from any non-degenerate distribution. Roughly speaking, the analogues of singular values for these products converge to the single-time marginal distribution of an interacting particle system, the reflecting Poisson sea. I will also discuss the technique we developed to show this, which is a general-purpose 'rescaled moment method' for fluctuations of random abelian groups.
]]> http://www-math.umd.edu/research/seminars.html Wed, 05 Mar 2025 14:00:00 EST Where: Kirwan Hall 1313
Speaker: Christopher Hoffman (University of Washington) - https://sites.math.washington.edu/~hoffman/
Abstract: One of the most cited physics papers of the 1980s was Bak, Tang and Wiesenfeld's introduction of self-organized criticality. This theory said that diverse physical systems such as earthquakes, forest fires and avalanches are all governed by similar underlying dynamics and thus they share certain fundamental behaviors. They also proposed a mathematical model that should have all of the properties that define self-organized criticality. Such a model would be called a universal model for self-organized criticality. Abelian networks are a class of models that contain several good candidates for a universal model, including activated random walk and the stochastic sandpile model. We will discuss recent results about these models which give some progress to finding a universal model for self-organized criticality.This is joint work with Maddy Brown, Toby Johnson, Matt Junge, Joshua Meisel and Hyojeong Son.
]]> http://www-math.umd.edu/research/seminars.html Wed, 12 Mar 2025 14:00:00 EDT Where: Kirwan Hall 1313
Speaker: Reza Gheissari (Northwestern University) - https://sites.northwestern.edu/gheissari/
Abstract: Many low-temperature dynamics in high-dimensional landscapes are expected to exhibit a sharp form of metastability (akin to that of fixed-dimensional small-noise diffusions), where the state space can be partitioned into wells, such that the equilibration time within each well is much faster than the transit time between wells, and the process tracking which well the Markov chain belongs to, itself is asymptotically Markovian. We overview this predicted picture for spin system dynamics, and then describe recent results with Curtis Grant proving this for Glauber dynamics for mean-field heavy-tailed spin glasses.
]]> http://www-math.umd.edu/research/seminars.html Wed, 26 Mar 2025 14:00:00 EDT Where: Kirwan Hall 1313
Speaker: Ross Pinsky (Technion) - https://pinsky.net.technion.ac.il/
Abstract: A $\textit{cluster}$ of length $l$ in a permutation from $S_n$ is a set of $l$ consecutive numbers that appear in any order in $l$ consecutive positions in the permutation.For $\eta\in S_3$, let $S_n^{\text{av}(\eta)}$ denote the set of permutations in $S_n$ that avoid the pattern $\eta$, and let $E_n^{\text{av}(\eta)}$ denote the expectationwith respect to the uniform probability measure on $S_n^{\text{av}(\eta)}$.For $n\ge l\ge2$ and $\tau\in S_l^{\text{av}(\eta)}$,let $ N^{(n)}_{l}(\sigma)$ denote the number ofclusters of length $l$ in $\sigma$ and let $ N^{(n)}_{l;\tau}(\sigma)$ denote the number of clusters of length $l$ whose order is the pattern $\tau$.We obtain explicit formulas for$E_n^{\text{av}(\eta)} N^{(n)}_{l;\tau}$ and $E_n^{\text{av}(\eta)}N^{(n)}_{l}$.These exact formulas yield asymptotic formulasas $n\to\infty$ with $l$ fixed, and as $n\to\infty$ with $l=l_n\to\infty$.In particular, for fixed $l$, depending on $\eta$ and $\tau$, the expectation $E_n^{\text{av}(\eta)} N^{(n)}_{l;\tau}$ either grows linearly in $n$ or is bounded and bounded away from zero.The expectation $E_n^{\text{av}(\eta)}N^{(n)}_{l}$ always has linear growth in $n$, and the maximum threshold size of $l_n$ for which $E_n^{\text{av}(\eta)}N^{(n)}_{l_n}$ remains bounded away from zero depends on $\eta$. We also obtain a weak law of large numbers.Analogous results are obtained for $S_n^{\text{av}(\eta_1,\cdots,\eta_r)}$, the permutations in $S_n$ avoiding the patterns $\{\eta_i\}_{i=1}^r$, where $\eta_i\in S_{m_i}$, in the case that $\{\eta_i\}_{i=1}^n$ are all \it simple\rm\ permutations. A particular case of this, where we can calculate the asymptotic behavior explicitly, is the set of \it separable\rm\ permutations, which corresponds to $r=2$, $\eta_1=2413,\eta_2=3142$.
]]> http://www-math.umd.edu/research/seminars.html Wed, 09 Apr 2025 14:00:00 EDT Where: Kirwan Hall 1313
Speaker: Cheng Ouyang (University of Illinois at Chicago) - https://homepages.math.uic.edu/~couyang/
Abstract: We introduce a family of intrinsic Gaussian noises on compact manifolds that we call “colored noise” on manifolds. With this noise, we study the parabolic Anderson model (PAM) on compact manifolds. Under some curvature conditions, we show the well-posedness of the PAM and provide some preliminary (but sharp) bounds on the second moment of the solution. It is interesting to see that global geometry enters the play in the well-posedness of the equation.
]]> http://www-math.umd.edu/research/seminars.html Wed, 16 Apr 2025 14:00:00 EDT Where: Kirwan Hall 1313
Speaker: Jake Maranzatto (University of Maryland) - https://terpconnect.umd.edu/~tmaran/
Abstract: The Age of Information (AoI) is an important metric for measuring quality of data, and has recently seen considerable applications in wireless, queuing, and sensor networks. This talk will study the AoI in so-called $\textit{gossip networks}$. Nodes in a round-based gossip network autonomously communicate by randomly selecting a neighbor at each round and transmitting their stored data. I'll be discussing the natural continuous-time analog of this setup where rounds are replaced with Poisson Processes. It turns out that in this setup, the AoI of a node is identical in distribution to a certain first-passage percolation process on a 'dual graph'. I'll discuss the proof of this equivalence, as well as present a new recursive formula for computing the expected first-passage time between nodes in any locally finite network. Joint work with Marcus Michelen.
]]> http://www-math.umd.edu/research/seminars.html Wed, 23 Apr 2025 14:00:00 EDT Where: Kirwan Hall 1313
Speaker: Xiao Shen (University of Utah) - https://www.xshen.org/
Abstract: Many two-dimensional random growth models, including first-passage and last-passage percolation, are conjectured to fall within the Kardar–Parisi–Zhang (KPZ) universality class under mild assumptions on the underlying noise. In recent years, researchers have focused on a subset of exactly solvable models, where these conjectures can be rigorously verified. This talk discusses a specific line of research that focuses on advancing the understanding of both exactly solvable and non-solvable KPZ models using general probabilistic methods, such as percolation and coupling.
]]> http://www-math.umd.edu/research/seminars.html Wed, 30 Apr 2025 14:00:00 EDT Where: Kirwan Hall 1313
Speaker: Alex Dunlap (Duke) - https://services.math.duke.edu/~ajd91/
Abstract: I will discuss the ergodic behavior of multiple coupled solutions, with different means, to both inviscid and viscous stochastic Burgers equation on the circle. In particular, I will describe a relationship between the derivative of the solution with respect to the mean and a particle moving in the Burgers flow. Based in part on joint work with Yu Gu.
]]> http://www-math.umd.edu/research/seminars.html Wed, 07 May 2025 14:00:00 EDT Where: Kirwan Hall 1313
Speaker: Benjamin Schweinhart (George Mason University) - https://mason.gmu.edu/~bschwei/
Abstract: Graphical representations have proven to be a powerful tool in the study of lattice spin models. We will present an extension of this idea to higher dimensions wherein Potts lattice gauge theory --- which assigns spins to edges rather than vertices --- is coupled with a dependent plaquette percolation called the plaquette random cluster model. One application is a proof that Wilson loop expectations for Potts lattice gauge theory on Z3 undergo a sharp phase transition from area law to perimeter law behavior. We also show that the self-dual point for the plaquette random cluster model on the four-dimensional torus is the threshold for the existence of giant surfaces in the sense of homological percolation. Based on joint work with Paul Duncan.
]]>