In this talk we will introduce the basic definitions and tools for understanding One-dimensional Stochastic Differential Equations. We will discuss well-posedness, the scale function, the speed measure, hitting times, and Feller's Test: the definitive characterization of explosion for these equations. Relevant examples will be shown to motivate. If time permits, we will introduce Feller's boundary classification.

We investigate infinite highly arc transitive digraphs with two additional properties, descendant-homogeneity and Property $Z$.
A digraph $D$ is {\itshape {highly arc transitive}} if for each $s \geq 0$ the automorphism group of $D$ is transitive on the set of directed paths of length $s$; and $D$ is {\itshape {descendant-homogeneous}} if any isomorphism between finitely generated subdigraphs of $D$ extends to an automorphism of $D$. A digraph is said to have {\itshape {Property $Z$}} if it has a homomorphism onto a directed line.
We show that if $D$ is a highly arc transitive descendant-homogeneous digraph with Property $Z$ and $F$ is the subdigraph spanned by the descendant set of a directed line in $D$, then $F$ is a locally finite 2-ended digraph with equal in- and out-valencies. If, moreover, $D$ has prime out-valency then $F$ is isomorphic to the digraph $\Delta_p$. This knowledge is then used to classify the highly arc transitive descendant-homogeneous digraph of prime out-valency which have Property $Z$.

The asymptotics of the second-largest eigenvalue in random regular graphs (also referred to as the ``Alon conjecture'') have been computed by Joel Friedman in his celebrated 2004 paper. Recently, a new proof of this result has been given by Charles Bordenave, using the non-backtracking operator and the Ihara-Bass formula. In the same spirit, we have been able to translate Bordenave's ideas to bipartite biregular graphs in order to calculate the asymptotical value of the second-largest pair of eigenvalues, and obtained a similar spectral gap result. Applications include community detection in equitable graphs or frames, matrix completion, and the construction of channels for efficient and tractable error-correcting codes (Tanner codes). This work is joint with Gerandy Brito and Kameron Harris.

A 2-ton interactive sculpture came to life at Burning Man 2018, the world's most influential large-scale sculpture showcase. Rising 12 feet tall with an 18-foot wingspan in the Nevada desert, the unfolding dodecahedron was illuminated by 16,000 LEDs, requiring 6,500 person hours and \$50,000 in funds.

Its interior, large enough to hold 15 people, was fully lined with massive mirrors, alluding to a possible shape of our universe. The unfolding exterior points to the 500-year-old work of Albrecht D$\ddot{\text{u}}$rer, and the tantalizing open problem of discovering a geometric unfolding for every convex polyhedron. We discuss the state-of-the-art, and consider higher-dimensional unfolding analogs, with elegant geometric and combinatorial relationships.

In this talk, we prove sharp lower bound of the first nonzero eigenvalue of the weighted $p$-Laplacian on compact Bakry-Emery manifolds, without boundary or with convex boundary and Neuman boundary condition. This is joint work with Kui Wang.

Sampling Gibbs measures at low temperature is a very important task but computationally very challenging. Numeric evidence suggest that the infinite-swapping algorithm (isa) is a promising method. The isa can be seen as an improvement of replica methods which are very popular. We rigorously analyze the ergodic properties of the isa in the low temperature regime deducing Eyring-Kramers formulas for the spectral gap (or Poincar\'e constant) and the log-Sobolev constant. Our main result shows that the effective energy barrier can be reduced drastically using the isa compared to the classical over-damped Langevin dynamics. As a corollary we derive a deviation inequality showing that sampling is also improved by an exponential factor. Furthermore, we analyze simulated annealing for the isa and show that isa is again superior to the over-damped Langevin dynamics. This is joint work with Georg Menz, Andr\'e Schlichting and Tianqi Wu.

Let $(X, T^{1, 0}X)$ be a compact CR manifold and $(L, h)$ be a Hermitian CR line bundle over $X$. When $X$ is Levi-flat and $L$ is positive, Ohsawa and Sibony constructed for every $\kappa\in\mathbb N$ a CR projective embedding of $C^\kappa$-smooth
of the Levi-flat CR manifold. Adachi constructed a counterexample to show that the $C^k$-smooth can not be generalized to $C^\infty$-smooth. The difficulty comes from the fact that the Kohn Laplacian is not hypoelliptic on Levi flat manifolds.

In this talk, we will consider CR manifold $X$ with a transversal CR $G$-action where $G$ is a compact Lie group and $G$ can be lifted to a CR line bundle $L$ over $X$.
The talk will be divided into two parts. In the first part, we will talk about the Morse inequalities for the Fourier components of Kohn-Rossi cohomology on CR manifolds with transversal CR $S^1$-action.
By studying the partial Szeg\"o kernel on $(0

In this series of lectures we will describe the construction
of a $G$-equivariant $L$-function $Theta^E_{K/F}(s)$ associated to an abelian
extension $K/F$ of characteristic $p$ global fields of Galois group $G$ and a
suitable Drinfeld module $E$ defined over $F$, as well as state and sketch the
proof of a theorem linking the special value
$Theta^E_{K/F}(0)$ to a quotient of volumes of certain compact topological
spaces canonically associated to the pair $(K/F, E)$.

In lecture I (1-2pm), Cristian will define the $L$--function, give an
arithmetic interpretation of its special value at $s=0$ and state the
main theorem.

In lecture II (2-3pm), Joe will introduce the main ingredients involved
in the proof of the main theorem and sketch the main ideas of proof.

These lectures describe joint work of J. Ferrara, N. Green, Z. Higgins and C.
Popescu. The results within generalize to the Galois equivariant setting
earlier work of L. Taelman on special values of Goss zeta functions associated
to Drinfeld modules (Taelman, Annals of Math. 2010).

A sunflower with $r$ petals is a collection of $r$ sets so
that the intersection of each pair is equal to the intersection of
all. Erdos and Rado in 1960 proved the sunflower lemma: for any fixed
$r$, any family of sets of size $w$, with at least about $w^w$ sets,
must contain a sunflower. The famous sunflower conjecture is that the
bound on the number of sets can be improved to $c^w$ for some constant
$c$. Despite much research, the best bounds until recently were all of
the order of $w^{cw}$ for some constant c. In this work, we improve
the bounds to about $(log w)^w$.

There are two main ideas that underlie our result. The first is a
structure vs pseudo-randomness paradigm, a commonly used paradigm in
combinatorics. This allows us to either exploit structure in the given
family of sets, or otherwise to assume that it is pseudo-random in a
certain way. The second is a duality between families of sets and DNFs
(Disjunctive Normal Forms). DNFs are widely studied in theoretical
computer science. One of the central results about them is the
switching lemma, which shows that DNFs simplify under random
restriction. We show that when restricted to pseudo-random DNFs, much
milder random restrictions are sufficient to simplify their structure.

Joint work with Ryan Alweiss, Kewen Wu and Jiapeng Zhang.

Suppose that an infinite subset $A$ of the natural numbers is
partitioned into finitely many subsets. A property of $A$ that is always
inherited by at least one of the elements of the partition is known as a
partition regular property. Suppose we have a method of measuring the size
of an infinite subset of the natural numbers. A property $P$ is said to be
a density property if every infinite subset of the natural numbers which
is large enough, according to our yardstick, satisfies $P$. In the
qualitative sense, partition regular properties guarantee that order is
conserved and density properties guarantee that order is always achieved
at a threshold size. We will provide some interesting examples and
applications of partition regular properties, density properties, and
related ideas.

We establish non-asymptotic concentration bound and Bahadur representation for quantile regression estimator in the random design setting. Smoothed quantile regression is then proposed with fast computation and high estimation accuracy. Finally, we provide rigorous theoretical guarantees for the validity of inference via multiplier bootstrap.

Consider a rational curve, described by a map f :$P^1 \to P^n$. The
Shapiro--Shapiro conjecture says the following: if all the inflection
points of the curve (the roots of the Wronskian of f) are real, then the
curve itself is defined by real polynomials (up to change of
coordinates). An equivalent statement is that certain real Schubert
varieties in the Grassmannian intersect transversely -- a fact with
broad combinatorial and topological consequences. The conjecture, made
in the 90s, was proven by Mukhin--Tarasov--Varchenko in '05/'09 using
methods from quantum mechanics.

I will present a generalization of the Shapiro--Shapiro conjecture,
joint with Kevin Purbhoo, where we allow the Wronskian to have complex
conjugate pairs of roots. We decompose the real Schubert cell according
to the number of such roots and define an orientation of each connected
component. For each part of this decomposition, we prove that the
topological degree of the restricted Wronski map is given by a symmetric
group character. In the case where all the roots are real, this implies
that the restricted Wronski map is a topologically trivial covering map;
in particular, this gives a new proof of the Shapiro-Shapiro conjecture.

Consider a rational curve, defined by a map $f:\mathbf{P}^1\rightarrow\mathbf{P}^n$. The Shapiro-Shapiro conjecture says the following: if all the inflection points of the curve (the roots of the Wronskian of $f$) are real, then the curve itself is defined by real polynomials (up to change of coordinates). An equivalent statement is that certain real Schubert varieties in the Grassmannian intersect transversely - a fact with broad combinatorial and topological consequences. The conjecture, made in the 90s, was proven by Mukhin-Tarasov-Varchenko in '05/'09 using methods from quantum mechanics. I will present a generalization of the Shapiro-Shapiro conjecture, joint with Kevin Purbhoo, where we allow the Wronskian to have complex conjugate pairs of roots. We decompose the real Schubert cell according to the number of such roots and define an orientation of each connected component. For each part of this decomposition, we prove that the topological degree of the restricted Wronski map is given by a symmetric group character. In the case where all the roots are real, this implies that the restricted Wronski map is a topologically trivial covering map; in particular, this gives a new proof of the Shapiro-Shapiro conjecture.

The cohomology groups of complex algebraic
varieties come equipped with a powerful but intrinsically analytic
invariant called a Hodge structure. The fact that Hodge structures of
certain very special algebraic varieties are nonetheless parametrized
by algebraic varieties has led to many important applications in
algebraic and arithmetic geometry. While this fails in general,
recent joint work with Y. Brunebarbe, B. Klingler, and J. Tsimerman
shows that parameter spaces of Hodge structures always admit a "tame"
analytic structure in a sense made precise using ideas from model
theory. A salient feature of the tame analytic category is that it
allows for the local flexibility of the full analytic category while
preserving the global behavior of the algebraic category.

In this talk I will explain this perspective as well as some important
applications, including an easy proof of a celebrated theorem of
Cattani--Deligne--Kaplan on the algebraicity of Hodge loci and the
resolution of a longstanding conjecture of Griffiths on the
quasiprojectivity of the images of period maps.