# Zeros of harmonic functions and Laplace eigenfunctions

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We will discuss geometrical and analytic properties of zero sets of harmonic functions and eigenfunctions of the Laplace operator. For harmonic functions on the plane there is an interesting relation between local length of the zero set and the growth of harmonic functions. The larger the zero set is, the faster the growth of harmonic function should be and vice versa. A curious object is Laplace eigenfunctions on two-dimensional sphere, which are restrictions of homogeneous harmonic polynomials of three variables onto 2-dimensional sphere. They are called spherical harmonics. Zero sets of such functions are unions of smooth curves with equiangular intersections. Topology of zero set could be quite complicated, but the total length of the zero set of any spherical harmonic of degree n is comparable to n. Though the Laplace eigenfunctions are known for ages, we still don't understand them well enough (even the spherical harmonics).

# Fibers of maps to totally nonnegative spaces

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The space of totally nonnegative real matrices, namely the real n by n matrices with all minors nonnegative, intersected with the ``unipotent radical'' of upper triangular matrices with 1's on the diagonal carries important information related to Lusztig's theory of canonical bases in representation theory. This space of matrices (and generalizations of it beyond type A) is naturally stratified according to which minors are positive and which are 0, with the resulting stratified space described combinatorially by a well known partially ordered set called the Bruhat order. I will tell the story of these spaces and in particular of a map from a simplex to these spaces that has recently been used to better understand them. The fibers of this map encode exactly the nonnegative real relations amongst exponentiated Chevalley generators of a Lie algebra. This talk will especially focus on recent joint work with Jim Davis and Ezra Miller uncovering overall combinatorial and topological structure governing these fibers. Plenty of background, examples, and pictures will be provided along the way.

# Asymptotics: the unified transform, a new approach to the Lindelöf Hypothesis, and the ultra-relativistic limit of the Minkowskian approximation of general relativity

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Employing standard, as well as novel techniques of asymptotics, three different problems will be discussed: (i) The computation of the large time asymptotics of initial-boundary value problems via the unified transform (also known as the Fokas Method, www.wikipedia.org/wiki/Fokas_method)[1]. (ii) The evaluation of the large t-asymptotics to all orders of the Riemann zeta function [2], and the introduction of a new approach to the Lindelöf Hypothesis [3]. (iii) The proof that the ultra-relativistic limit of the Minkowskian approximation of general relativity [4] yields a force with characteristics of the strong force, including confinement and asymptotic freedom [5].

[1] J. Lenells and A. S. Fokas. The Nonlinear Schrödinger Equation with t-Periodic Data: I. Exact Results, Proc. R. Soc. A 471, 20140925 (2015).

J. Lenells and A. S. Fokas, The Nonlinear Schrödinger Equation with t-Periodic Data: II. Perturbative Results, Proc. R. Soc. A 471, 20140926 (2015).

[2] A.S. Fokas and J. Lenells, On the Asymptotics to All Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function, Mem. Amer. Math. Soc. (to appear).

[3] A.S. Fokas, A Novel Approach to the Lindelof Hypothesis, Transactions of Mathematics and its Applications (to appear).

[4] L. Blanchet and A.S. Fokas, Equations of Motion of Self-Gravitating N-Body Systems in the First Post-Minkowskian

Approximation, Phys. Rev. D 98, 084005 (2018).

[5] A.S. Fokas, Super Relativistic Gravity has Properties Associated with the Strong Force, Eur. Phys. J. C (to appear).

# The Sperner property

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In 1927 Emanuel Sperner proved that if $S_1,\dots,S_m$ are distinct

subsets of an $n$-element set such that we never have $S_i\subset S_j$,

then $m\leq \binom{n}{\lfloor n/2\rfloor}$. Moreover, equality is achieved

by taking all subsets of $S$ with $\lfloor n/2\rfloor$ elements. This

result spawned a host of generalizations, most conveniently stated in the

language of partially ordered sets. We will survey some of the highlights

of this subject, including the use of linear algebra and the cohomology of

certain complex projective varieties. An application is a proof of a

conjecture of Erd\H{o}s and Moser, namely, for all integers $n\geq 1$ and

real numbers $\alpha\geq 0$, the number of subsets with element sum

$\alpha$ of an $n$-element set of positive real numbers cannot exceed the

number of subsets of $\{1,2,\dots,n\}$ whose elements sum to $\lfloor

\frac 12\binom n2\rfloor$. We will conclude by discussing two recent

proofs of a 1984 conjecture of Anders Bj\"orner on the weak Bruhat order

of the symmetric group $S_n$.

# Higher algebra and arithmetic

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This talk concerns a twenty-thousand-year old mistake: The natural numbers record only the result of counting and not the process of counting. As algebra is rooted in the natural numbers, the higher algebra of Joyal and Lurie is rooted in a more basic notion of number which also records the process of counting. Long advocated by Waldhausen, the arithmetic of these more basic numbers should eliminate denominators. Notable manifestations of this vision include the Bökstedt-Hsiang-Madsen topological cyclic homology, which receives a denominator-free Chern character, and the related Bhatt-Morrow-Scholze integral *p*-adic Hodge theory, which makes it possible to exploit torsion cohomology classes in arithmetic geometry. Moreover, for schemes smooth and proper over a finite field, the analogue of de Rham cohomology in this setting naturally gives rise to a cohomological interpretation of the Hasse-Weil zeta function by regularized determinants, as envisioned by Deninger.

# test

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# Boundary rigidity and the local inverse problem for the geodesic X-ray transform on tensors

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In this talk, based on joint work with Plamen Stefanov and Gunther Uhlmann, I discuss the boundary rigidity problem on manifolds with boundary (for instance, a domain in Euclidean space with a perturbed metric), i.e. determining a Riemannian metric from the restriction of its distance function to the boundary. This corresponds to travel time tomography, i.e. finding the Riemannian metric from the time it takes for solutions of the corresponding wave equation to travel between boundary points. A version of this relates to finding the speed of seismic waves inside the Earth from travel time data, which in turn permits a study of the structure of the inside of the Earth.

This non-linear problem in turn builds on the geodesic X-ray transform on such a Riemannian manifold with boundary. The geodesic X-ray transform on functions associates to a function its integral along geodesic curves, so for instance in domains in Euclidean space along straight lines. The X-ray transform on symmetric tensors is similar, but one integrates the tensor contracted with the tangent vector of the geodesics. I will explain how, under suitable convexity assumptions, one can invert the geodesic X-ray transform on functions, i.e. determine the function from its X-ray transform, in a stable manner, as well as the analogous tensor result, and the connection to the full boundary rigidity problem.

# Geometric Partial Differential Equations from M Theory

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Since the mid 1990’s, the leading candidate for a unified theory of all fundamental physical interactions has been M Theory.

A full formulation of M Theory is still not available, and it is only understood through its limits in certain regimes, which are either one of five 10-dimensional string theories, or 11-dimensional supergravity. The equations for these theories are mathematically interesting in themselves, as they reflect, either directly or indirectly, the presence of supersymmetry. We discuss recent progresses and open problems about two of these theories, namely supersymmetric compactifications of the heterotic string and of 11-dimensional supergravity. This is based on joint work of the speaker with Sebastien Picard and Xiangwen Zhang, and with Teng Fei and Bin Guo.

# (Distinguished Lecture)

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(Distinguished Lecture)