Department of

# Mathematics

Seminar Calendar
for Probability Seminar events the year of Friday, October 22, 2021.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
    September 2021          October 2021          November 2021
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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

###### Sebastien Roch (University of Wisconsin-Madison)

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.

Tuesday, August 24, 2021

2:00 pm in 347 Altgeld Hall,Tuesday, August 24, 2021

#### Heat kernel upper bounds for symmetric Markov semigroups

###### Panki Kim (Seoul National University)

Abstract: It is well known that Nash-type inequalities for symmetric Dirichlet forms are equivalent to on-diagonal heat kernel upper bounds for the associated symmetric Markov semigroups. In this talk, we discuss the equivalence among these and off-diagonal heat kernel upper bounds under some mild assumptions. Our approach is based on a new generalized Davies' method. This talk is based on a joint work with Zhen-Qing Chen, Takashi Kumagai and Jian Wang.

Tuesday, August 31, 2021

2:00 pm in 347 Altgeld Hall,Tuesday, August 31, 2021

#### Semilinear equations for non-local operators: Beyond the fractional Laplacian

###### Zoran Vondracek (University of Zagreb)

Abstract: In this talk I will present some results on semilinear problems in general bounded open sets for non-local operators with exterior and boundary conditions. The operators are more general than the fractional Laplacian. I will also give results in case of bounded $C^{1,1}$ open sets. Joint work with I. Biocic (Biov{c}i\'c) and V.Wagner.

Tuesday, September 7, 2021

2:00 pm in 347 Altgeld Hall,Tuesday, September 7, 2021

#### Information Projections On Banach Spaces With Applications To KL Weighted Control And A Feynman-Kac Formula For ODEs

###### Zachary Selk (Purdue University)

Abstract: In this talk, we discuss a portmanteau theorem establishing the equivalence between information projections on a Banach space, constrained Kullback-Leibler weighted control, finding the mode of a measure through Onsager-Machlup formalism and in the classical Wiener space case, an Euler-Lagrange equation. As one example of an application of our theorem, we discuss a Feynman-Kac type formula, showing that the solution to a second order linear ODE (or system of ODEs) is the mode of a particular diffusion. Our portmanteau theorem along with our Feynman-Kac result provides numerics and insight for solving these ODEs. Joint work with William Haskell and Harsha Honnappa.

Thursday, September 23, 2021

2:00 pm in Altgeld Hall 347,Thursday, September 23, 2021

#### Coupling from the Past 1

###### Aditya Suresh Gopalan (UIUC ISE )

Abstract: Abstract: The Coupling from the Past algorithm of Propp and Wilson is a good technique for sampling from and proving the existence of stationary distributions for Markov chains. In this talk, we discuss the proof of the technique. Briefly, the system works by constructing sequences sample paths whose initial states are determined by the previous element in the sequence, so that the original states are pushed to minus infinity'', thus establishing the coupling and convergence. This technique has applications to a wide variety of fields; one such application to interference queueing will be discussed in the next talk.

Thursday, September 30, 2021

2:00 pm in Altgeld Hall 347,Thursday, September 30, 2021

#### Coupling from the Past 2

###### Aditya S. Gopalan (UIUC ISE)

Abstract: We give a coupling from the past proof for the existence of a stationary distribution for interference queueing networks on grids. This talk highlights the use of this technique for purposes other than sampling, for which it was developed.

Thursday, October 7, 2021

2:00 pm in Altgeld Hall 347,Thursday, October 7, 2021

#### Use of couplings in distributional convergence

###### Grigory Terlov (UIUC MATH)

Abstract: It was mentioned in previous talks that bounding distance between two distributions in appropriate metric is often equivalent to finding a good coupling of random variables. In this talk we will build up on that idea and study the use of couplings in bounding total variation distance between two counting random variables. We will apply these techniques to head runs in a sequence of independent coin tosses and occupancy problem as well as other models if time permits. This talk serves as introduction to the Chen-Stein method for Poisson approximation, on which we will focus in the second talk.

Thursday, October 14, 2021

2:00 pm in Altgeld Hall 347,Thursday, October 14, 2021

#### Use of couplings in distributional convergence (Part2: Stein-Chen method)

###### Grigory Terlov (UIUC Math )

Abstract: At the end of the last lecture I presented two results due to C. Stein and L.H.Y. Chen that utilize couplings to prove convergence to Poisson distribution. These techniques allow for dependence among random variables and give explicit rate of convergence. In this talk I will explain the main idea behind Stein-Chen method and prove both of the theorems stated last time.

Thursday, October 21, 2021

2:00 pm in Altgeld Hall 347,Thursday, October 21, 2021

#### Coupling method for diffusion processes

###### Peixue Wu (UIUC Math)

Abstract: In my first part of my talk, first I will introduce the notion of ergodicity for continuous-time stochastic processes. Then I will briefly discuss the relation between the ergodicity of a one-dimensional diffusion process and its coupling under some assumptions on the diffusion process.

2:00 pm in Altgeld Hall 347,Thursday, October 21, 2021

#### Coupling method for diffusion processes

###### Peixue Wu (UIUC Math)

Abstract: In my first part of my talk, first I will introduce the notion of ergodicity for continuous-time stochastic processes. Then I will briefly discuss the relation between the ergodicity of a one-dimensional diffusion process and its coupling under some assumptions on the diffusion process.

Thursday, October 28, 2021

2:00 pm in Altgeld Hall 347,Thursday, October 28, 2021

#### Coupling for diffusion processes II

###### Peixue Wu (UIUC math )

Abstract: First I will continue talking about the examples of ergodic&non-ergodic one-dimensional diffusion processes, using the fundamental theorem of ergodicity and coupling. Second, I will introduce some functional inequalities using coupling method, which is undergoing active research nowadays.

Thursday, November 4, 2021

2:00 pm in Altgeld Hall 347,Thursday, November 4, 2021

#### introduction to optimal transport

###### Kesav Krishnan (UIUC math )

Abstract: I will introduce the notion of optimal transport and the interpretation of coupling of measure in this regard.

Thursday, November 11, 2021

2:00 pm in Altgeld Hall 347,Thursday, November 11, 2021

#### Intro to optimal transport II

###### Kesav Krishnan (UIUC math )

Abstract: Having introduced the notion of optimal transport, I will discuss some of its uses, notably in proving the isoperimetric inequality.

Thursday, November 18, 2021

2:00 pm in Altgeld Hall 347,Thursday, November 18, 2021

#### Non-concentration of the chromatic number of a random graph

###### Bob Krueger (UIUC math )

Abstract: In a 2019 breakthrough, Heckel proved that the chromatic number of a uniformly random graph on n vertices is not concentrated on $n^{1/4-o(1)}$ values for all sufficiently large n. I will outline an improvement by Heckel and Riordan which improves the 1/4 to an optimal 1/2, focusing on their use of coupling.