Department of

Mathematics

Seminar Calendar
for Probability Seminar events the year of Monday, May 17, 2021.

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events for the
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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, March 16, 2021

2:00 pm in Zoom Meeting (email daesungk@illinois.edu for info),Tuesday, March 16, 2021

Solution to Enflo's problem

Paata Ivanishvili (North Carolina State University)

Abstract: Pick any finite number of points in a Hilbert space. If they coincide with vertices of a parallelepiped then the sum of the squares of the lengths of its sides equals the sum of the squares of the lengths of the diagonals (parallelogram law). If the points are in a general position then we can define sides and diagonals by labeling these points via vertices of the discrete cube {0,1}^n. In this case the sum of the squares of diagonals is bounded by the sum of the squares of its sides no matter how you label the points and what n you choose. In a general Banach space we do not have parallelogram law. Back in 1978 Enflo asked: in an arbitrary Banach space if the sum of the squares of diagonals is bounded by the sum of the squares of its sides for all parallelepipeds (up to a universal constant), does the same estimate hold for any finite number of points (not necessarily vertices of the parallelepiped)? In the joint work with Ramon van Handel and Sasha Volberg we positively resolve Enflo's problem. Banach spaces satisfying the inequality with parallelepipeds are called of type 2 (Rademacher type 2), and Banach spaces satisfying the inequality for all points are called of Enflo type 2. In particular, we show that Rademacher type and Enflo type coincide.

Tuesday, March 23, 2021

2:00 pm in Zoom Meeting (email daesungk@illinois.edu for info),Tuesday, March 23, 2021

Phylogenomics: Inverting Random Trees

Abstract: Phylogenomic analysis, in particular the estimation of species phylogenies from genome-scale data, is a common step in modern evolutionary studies. This estimation is complicated by the fact that genes evolve under biological processes that produce discordant trees. Such processes include horizontal gene transfer (HGT), incomplete lineage sorting (ILS), and gene duplication and loss (GDL), all of which can be modeled using specialized random tree distributions. I will survey some recent results regarding the analysis of these probabilistic models. Specifically their identifiability, or "invertibility," will be discussed as well as the asymptotic properties of species tree estimation methods (time permitting). Based on joint works with Max Bacharach, Brandon Legried, Erin Molloy, Elchanan Mossel, Allan Sly, Tandy Warnow, Shuqi Yu.

Tuesday, March 30, 2021

2:00 pm in Zoom Meeting (email daesungk@illinois.edu for info),Tuesday, March 30, 2021

Exponential concentration of overlap, free energy, and replica symmetry breaking for inhomogeneous Sherrington-Kirkpatrick Spin glass

Qiang Wu (UIUC)

Abstract: Multi-species Sherrington-Kirkpatrick Spin glass model, as an inhomogeneous generalization of classical SK model, was first introduced by Barra etal in 2015, later Panchenko set up the Parisi formula to compute limiting free energy at all temperatures. However, all those results can only hold when the disorder variance matrix is positive semi-definite. For the indefinite MSK model, nearly nothing rigorous is known, physicists conjectured that this model has some intrinsic difference with the positive definite case. In this talk, we will discuss some fluctuation results for the general MSK model in replica symmetric regime. First, A unified argument for the exponential concentration of overlap will be presented, and this concentration result further enables one to prove a CLT of free energy. Besides that, we also introduce a new species-wise cavity approach to study the fluctuation of overlap vectors, and this approach does not require positive definite assumption. The fluctuation results also suggest the phase boundary (known as AT line) of replica symmetry and replica symmetry breaking for general MSK model, we will conclude with an explicit form of conjectured AT equation.

Tuesday, April 6, 2021

2:00 pm in Zoom Meeting (email daesungk@illinois.edu for info),Tuesday, April 6, 2021

Modified log-Sobolev inequalities, Beckner inequalities and moment estimates

Abstract: I will present recent results concerning the equivalence between the modified log-Sobolev inequality and a family of Beckner type inequalities with constants uniformly separated from zero. Next I will discuss moment estimates which can be derived from such inequalities, generalizing previous results due to Aida and Stroock, based on a stronger log-Sobolev inequality due to Federbush and Gross. If time permits I will present examples to moment estimates for certain Cauchy type measures, for invariant measures of Glauber dynamics and on the Poisson path space. Based on joint work with B. Polaczyk and M. Strzelecki.

Wednesday, April 14, 2021

3:00 pm in Zoom Meeting (email daesungk@illinois.edu for info),Wednesday, April 14, 2021

Random Graph Matching with Improved Noise Robustness

Konstantin Tikhomirov (Georgia Institute of Technology)

Abstract: Graph matching, also known as network alignment, refers to finding a bijection between the vertex sets of two given graphs so as to maximally align their edges. This fundamental computational problem arises frequently in multiple fields such as computer vision and biology. In this work we will discuss a new algorithm for exact matching of correlated Erdos-Renyi graphs. Based on joint work with Cheng Mao and Mark Rudelson.

Tuesday, April 20, 2021

2:00 pm in Zoom Meeting (email daesungk@illinois.edu for info),Tuesday, April 20, 2021

Geodesic Length in First-Passage Percolation

Firas Rassoul-Agha (University of Utah)

Abstract: We study first-passage percolation through related optimization problems over paths of restricted length. The path length variable is in duality with a shift of the weights. This puts into a convex duality framework old observations about the convergence of the normalized Euclidean length of geodesics due to Hammersley and Welsh, Smythe and Wierman, and Kesten, and leads to new results about geodesic length and the regularity of the shape function as a function of the weight shift. For points far enough away from the origin, the ratio of the geodesic length and the $\ell^1$ distance to the endpoint is uniformly bounded away from one. The shape function is a strictly concave function of the weight shift. Atoms of the weight distribution generate singularities, that is, points of nondifferentiability, in this function. We generalize to all distributions, directions and dimensions an old singularity result of Steele and Zhang for the planar Bernoulli case. When the weight distribution has two or more atoms, a dense set of shifts produce singularities. The results come from a combination of the convex duality, the shape theorems of the different first-passage optimization problems, and modification arguments. This is joint work with Arjun Krishnan and Timo Seppalainen.

Monday, April 26, 2021

3:00 pm in Zoom Meeting (email daesungk@illinois.edu for info),Monday, April 26, 2021

Freidlin-Wentzell type Large Deviations

Wenqing Hu (Missouri University of Science and Technology)

Abstract: We will introduce some basics of the Freidlin-Wentzell type large deviations principle.

Tuesday, April 27, 2021

2:00 pm in Zoom Meeting (email daesungk@illinois.edu for info),Tuesday, April 27, 2021

Wave front propagation for FKPP reaction-diffusion equation on a class of infinite random trees

Wenqing Hu (Missouri University of Science and Technology)

Abstract: The asymptotic wave speed for FKPP type reaction-diffusion equations on a class of infinite random metric trees is considered in this talk. We show that a travelling wave front emerges, provided that the reaction rate is large enough. The wave travels at a speed that can be quantified via a variational formula involving the random branching degrees and the random branch lengths of the tree. This speed is slower than that of the same equation on the real line, and we can estimate this slow-down in terms of the structure of the tree. Our key idea is to project the Brownian motion on the tree onto a one-dimensional axis along the direction of the wave propagation. This idea, combined with the Feynman-Kac formula, connect our analysis of the wave front propagation to the Large Deviations Principle (LDP) of the multi-skewed Brownian motion with random skewness and random interface set. Our LDP analysis for this multi-skewed Brownian motion involves delicate estimates for an infinite product of 2 by 2 random matrices parametrized by the structure of the tree and for hitting times of a random walk in random environment. Joint work with Wai-Tong (Louis) Fan (Indiana University) and Grigory Terlov (UIUC).

Tuesday, May 4, 2021

2:00 pm in Zoom Meeting (email daesungk@illinois.edu for info),Tuesday, May 4, 2021

Towards optimal gradient bounds for torsion functions in the plane

Jeremy Hoskins (University of Chicago)

Abstract: In this talk we consider a natural question going back to Saint Venant in 1856: if a beam of constant cross-section is twisted, how large is the maximum shear stress? As it turns out, this problem can be formulated as a basic question about elliptic PDEs which arises in a number of settings, including electrostatics, constrained maximization of the lifetime of Brownian motion started close to the boundary, and optimal Hermite-Hadamard inequalities for subharmonic functions on convex domains. In this talk we will present upper and lower bounds for the largest possible shear stress for convex domains, and discuss extremal shapes.