# Mean estimation in high dimension

Shahar Mendelson

## Institution:

Australian National University

## Time:

Thursday, October 5, 2023 - 4:00pm to 5:00pm

## Location:

RH 306

Consider an unknown random vector $X$ that takes values in $R^d$. Is it possible to "guess" its mean accurately if the only information one is given consists of $N$ independent copies of $X$? More accurately, given an arbitrary norm on $R^d$, the goal is to find a mean estimation procedure upon receiving a wanted confidence parameter $\delta$ and $N$ independent copies $X_1,\cdots,X_N$ of an unknown random vector $X$ (that has a finite mean $\mu$ and finite covariance) the procedure returns $\hat{\,\,\mu}$ for which the norm of the error $\hat{\,\mu} - \mu$ is as small as possible, with high probability $1-\delta$.

This mean estimation problem has been studied extensively over the years. I will present some of the ideas that have led to its solution. An obvious choice is to set $\hat{\,\mu}$ to be the empirical mean, i.e. the arithmetic mean of the sample vectors $X_i$. Surprisingly, the empirical mean is a terrible option for small confidence parameters $\delta$ -- most notably, when X is "heavy-tailed". What is even more surprising is that one can find an optimal estimation procedure that performs as if the (arbitrary) random vector X were Gaussian. (A joint work with G. Lugosi)

# Nonexistence of Type II Blowups for Energy-Critical heat Equation in Large Dimensions

Jun-Cheng Wei

UBC

## Time:

Monday, April 17, 2023 - 3:00pm to 4:00pm

## Location:

RH306

In this talk I will  consider energy-critical nonlinear heat equation

$$u_t=\Delta u+ u^{\frac{n+2}{n-2}}, u\geq 0$$

We prove that for $n\geq 7$, any blow-up must be of Type I, i.e. the blow-up rate must be bounded by $(T-t)^{-\frac{n-2}{4}}$. The proof is built on several key ingredients: first we perform tangent ow analysis and study bubbling formation in this process; next we give a second order bubbling analysis in the multiplicity one case, where we use a reverse inner-outer gluing mechanism; finally, in the higher multiplicity case (bubbling tower/cluster), we develop Schoen's Harnack inequality and obtain next order estimates in Pohozaev identities for critical parabolic flows. (Joint work with Kelei Wang.)

The option to join via Zoom may be accessed through this link.

# Volumes of Hyperbolic Polytopes and the Goncharov Depth Conjecture

Daniil Rudenko

## Institution:

University of Chicago

## Time:

Thursday, February 23, 2023 - 4:00pm to 5:00pm

## Location:

RH 306

Lobachevsky started to work on computing volumes of hyperbolic polytopes long before the first model of the hyperbolic space was found. He discovered an extraordinary formula for the volume of an orthoscheme via a particular function called dilogarithm. We will discuss a generalization of the formula of Lobachevsky to higher dimensions. For reasons I do not fully understand, a mild modification of this formula leads to the proof of a conjecture of Goncharov about the depth of multiple polylogarithms. Moreover, the same construction leads to a functional equation for polylogarithms generalizing known equations of Abel, Kummer, and Goncharov.

# Infinite patterns in large sets of integers: Dynamical approaches

Bryna Kra

## Institution:

Northwestern University

## Time:

Friday, February 3, 2023 - 3:00pm

## Location:

RH 306

***Special Dynamical Systems and Ergodic Theory Seminar***

Furstenberg's proof of Szemeredi's theorem introduced the Correspondence Principle, a general technique for translating a combinatorial problem into a dynamical one. While the original formulation suffices for certain patterns, including arithmetic progressions and some infinite configurations, higher order generalizations have required refinements of these tools. We discuss the new techniques introduced in joint work with Moreira, Richter, and Robertson that are used to show the existence of infinite patterns in large sets of integers.

# Patterns in large sets of integers: Finite to infinite

Bryna Kra

## Institution:

Northwestern University

## Time:

Thursday, February 2, 2023 - 4:00pm

## Location:

Natural Sciences II, Room 1201

***Distinguished Visitor Colloquium***

Resolving a conjecture of Erdos and Turan from the 1930's, in the 1970's Szemerédi showed that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Soon thereafter, Furstenberg used ergodic theory to give a new proof of this result, leading to the development of combinatorial ergodic theory. These tools have led to uncovering new patterns that occur in any sufficiently large set of integers, but until recently all such patterns have been finite. Based on joint work with Joel Moreira, Florian Richter, and Donald Robertson, we discuss recent developments for infinite patterns, including the resolution of a conjecture of Erdos.

# On Finite Time Blowup of the 3D Euler Equations and Related Models Using Computer-Assisted Proofs

Thomas Hou

Caltech

## Time:

Thursday, December 1, 2022 - 4:00pm

RH 192

# Set theory and the continuum hypothesis

Ralf Schindler

## Institution:

University of M\"unster

## Time:

Thursday, April 21, 2022 - 4:00pm

## Location:

RH 306

This talk is about two prominent axioms of set theory which were introduced independently from one another in the late 80's/early 90's by Foreman-Magidor-Shelah and Woodin and which both decide the size of the continuum in the same way. Answering a long standing question, in a 2021 Annals paper D. Asperó and the speaker showed that these two axioms of set theory are compatible, in fact one implies the other. Both axioms are so-called forcing axioms which are also exploited in topology, algebra, the theory of operator algebras, and elsewhere. I am going to provide a soft hand, accessible introduction to our result.

# Seeing Through Space-Time

Gunther Uhlmann

## Institution:

University of Washington

## Time:

Thursday, November 4, 2021 - 3:00pm

## Location:

RH 306

The inverse problem we address is whether we can determine the structure of a region in space-time by measuring point light sources coming from the region. We can also observe gravitational waves since the LIGO detection in 2015. We will also consider inverse problems for nonlinear hyperbolic equations, including Einstein's equations, involving active measurements.

# Piecewise Polynomials by Neural Networks and Finite Elements

Jinchao Xu

## Institution:

Penn State University

## Time:

Thursday, December 2, 2021 - 4:00pm

## Location:

Zoom ID: 930 9003 8527

Polynomials and piecewise polynomials are most commonly used function classes in analysis and applications.  In this talk, I will use these function classes to motivate and to present some remarkable properties of the function classes given by (both shallow and deep) neural networks.  In particular, I will present some recent results on the connections between finite element and neural network functions and comparative studies of their approximation properties and applications to numerical solution of partial differential equations of high order and/or in high dimensions.

Meeting ID: 930 9003 8527
Passcode: 215830

Join Zoom Meeting

https://uci.zoom.us/j/93090038527?pwd=cGJaT0g2U1JKUHEvT1FuS1RaSldsdz09

# Quasiperiodicity and quasiintegrability

Igor Krasovsky

Imperial College

## Time:

Thursday, November 4, 2021 - 4:00pm

## Location:

RH 306

We will discuss the critical almost Mathieu operator: Azbel/Hofstadter/Harper model of an electron on the square lattice in a magnetic field. When the commensurability parameter between the lattice and the magnetic field is irrational, the spectrum of the model is a zero-measure Cantor set and its Hausdorff dimension is not larger than 1/2. We will emphasize the significance of the two-dimensionality of the problem, which was used in recent work of the speaker with S. Jitomirskaya. We will also discuss some similarities with integrable two-dimensional statistical models: the Ising model and the dimer problem.

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