Enumerative combinatorics is an area in combinatorics that mainly focuses on enumerating the number of ways to form certain configurations. With the help from computers and the internet, researches in enumerative combinatorics has developed dramatically in the past decades. In this talk, we will introduce signed permutations and their structures. Then we will use signed permutations as an example of how one might use a computer to come up with interesting research problems to solve. Then, we will end the talk by presenting some open problems in enumerative combinatorics.

Fusion categories appear in many areas of mathematics. They are realized by topological quantum field theories, representations of finite groups and Hopf algebras, and invariants for knots and Murray-von Neumann subfactors. An important numerical invariant of these categories are the Frobenius-Schur indicators, which are generalized versions of those for finite group representations. It is thought that these indicators should provide a complete invariant for a fairly wide class of fusion categories; in this talk we will discuss new families of so-called near-group fusion categories (i.e. those with only one non-invertible indecomposable object) which satisfy this property.

In many specialized imaging systems, an unknown signal $x$ $C^d$ produces
measurements of the form
$y_i = | a_i , x |2 + \eta_i$ , where ${a_i}\subset C^d$
are known measurement vectors and is an arbitrary noise term.
Because this system seems to erase the phases of the entries of $x \in C^d$ , the
problem of reconstructing $x$
from $y$ is known as the phase retrieval problem. The first approaches to this
problem were ad hoc iterative
methods which still have no theoretical guarantees on convergence. Recent
advancements including
gradient descent and convex relaxation have supplied some theoretical promises,
but often require such
conditions on the system ${a_i}$ that they cannot be used by scientists in
practice. In particular, they tend
to require some randomness to be used in the choice of ${a_i}$ that does not
reflect the physical systems
that typically yield the phase retrieval problem. Our work contributes a
solution to this problem which
features (a) a deterministic, practicable construction for ${a_i}$ (b) numerical
stability with respect to noise
(c) a reconstruction algorithm with competitive runtimes. Our most recent
result is an improvement on
the robustness gained by leveraging the graph structure induced by our
measurement scheme.

This lecture will be about a remarkable law of nature discovered by George Polya. Consider a particle initially situated at a given point of the d-dimensional integer lattice. Suppose that, at each tick of the clock, the particle jumps to a neighboring lattice site, with equal probability of jumping in any direction. Polya's law states that the particle returns to its initial position with probability one in dimensions d = 1,2, but with probability strictly less than one in all higher dimensions. Thus, a drunk person wandering a city grid will always return to their starting point, but if the drunkard can fly s/he might never come back.

Suppose $\Gamma_1,\ldots,\Gamma_n$ are hyperbolic ICC groups and denote by $\Gamma =\Gamma_1\times \cdots \times \Gamma_n$. We show whenever $\Lambda$ is an arbitrary discrete group such that $L(\Gamma)\cong L(\Lambda)$ then $\Lambda =\Lambda_1\times \cdots \times \Lambda_n$ and up to amplifications $L(\Gamma_i)\cong L(\Lambda_i) $ for all $i$; in other words the von Neumann algebra $L(\Gamma)$ completely remembers the product structure of the underlying group. In addition, we will show that some of the techniques used to prove this product rigidity result can also be successfully applied to produce new examples of prime factors. In particular, we significantly generalize the primeness results obtained earlier by I. Chifan, Y. Kida and S. Pant for the factors arising poly-hyperbolic and surface braid groups. These are joint works with I. Chifan and T. Sinclair, and S. Pant, respectively.

Conformal and CR geometries are among the class of "parabolic geometries" which posses a canonical Cartan connection characterizing the geometry. Replacing the Levi-Civita connection with the Cartan connection we develop submanifold theory in parallel with the classical Riemannian case. This allows us to apply tools developed for conformal and CR invariant theory to develop a theory of submanifold invariants and invariant operators, relevant to the study of conformally or CR invariant boundary value problems and other problems in geometric analysis involving submanifolds. The technical details of the theory are substantial (especially in the CR case). I will try to emphasize some of the concrete geometric ideas behind the approach, giving insight into the original work of Elie Cartan.

Critical lattice models are believed to converge to a free field in the scaling limit, at or above their critical dimension. This has been (partially) established for Ising and $Phi^4$ models for $d \geq 4$. We describe a simple spin model from uniform spanning forests in $\mathbb{Z}^d$ whose critical dimension is 4 and prove that the scaling limit is the bi-Laplacian Gaussian field for $d\ge 4$. At dimension 4, there is a logarithmic correction for the spin-spin correlation and the bi-Laplacian Gaussian field is a log correlated field. The proof also improves the known mean field picture of LERW in d=4, by showing that the renormalized escape probability (and arm events) of 4D LERW converge to some "continuum escaping probability". Based on joint works with Greg Lawler and Xin Sun.

Dynamic Conditional Beta (DCB) is an approach to estimating regressions with time varying parameters. The conditional covariance matrices of the exogenous and dependent variable for each time period are used to formulate the dynamic beta. Joint estimation of the covariance matrices and other regression parameters is developed. Tests of the hypothesis that betas are constant are non-nested tests and several approaches are developed including a novel nested model. The methodology is applied to industry multifactor asset pricing and to global systemic risk estimation with non-synchronous prices.

Free registration is required to attend. Registration information is available at

http://www.math.ucsd.edu/$\sim$rosenblattconf/rosenblattlecture.html

Gross showed that to every Hermitian symmetric tube domain we
may associate a canonical variation of Hodge structure (VHS) of Calabi-Yau
type. The construction is representation theoretic, not geometric, in
nature, and it is an open question to realize this abstract VHS as the
variation induced by a family of polarized, algebraic Calabi-Yau
manifolds. In order for a geometric VHS to realize Gross's VHS it is
necessary that the invariants associated to the two VHS coincide. For
example, the Hodge numbers must agree. The later are discrete/integer
invariants. Characteristic forms are differential-geometric invariants
associated to VHS (introduced by Sheng and Zuo). Remarkably, agreement of
the characteristic forms is both necessary and sufficient for a geometric
VHS to realize one of Gross's VHS. That is, the characteristic forms
characterize Gross's Calabi-Yau VHS. I will explain this result, and
discuss how characteristic forms have been used to study candidate
geometric realizations of Gross's VHS.

In the late 1980's, after Jones' define his polynomial, there is a
revolution in this area, followed by Witten's reinterpreting Jones
polynomial by using Chern-Simons theory and predicting new quantum
invariants. Finally Reshetikhin-Turaev was the first one to define a
mathematically rigorous theory of such complex-valued invariant for closed
3-manifolds. More importantly, Reshetikhin-Turaev define their invariant
not only at roots of unity $q(1)$ originally considered by Witten but also at
other roots of unity. Later Turaev-Viro defined a real valued invariants
for closed 3-manifolds by triangulation both at $q(1)$ and other roots of
unity.

In 1997, Kashaev discover his invariants of hyperbolic knots will become
exponentially large as $N->infinity$ and he further conjectured that the
growth rates corresponds to hyperbolic Volume of complement of that knot in
$S^3$. In 2001, H. Murakami-J. Murakami extend Kashaev's Volume Conjecture
from hyperbolic knots to all knots and hyperbolic volume to simplicial
volume by using colored Jones polynomials.

For many years, Witten-Reshetikhin-Turaev invariants evaluated at $q(1)$ was
considered to be only polynomial growth and its asymptotic expansion is
called WAE Conjecture (Witten's Asymptotic Expansion). Last year, in a
joint work with T. Yang, we first define a real valued Turaev-Viro type
invariant for 3-manifold with boundary by using ideal triangulation. Then
we discovered this Turaev-Viro type invariant and Reshetikhin-Turaev
invariant evaluated at other roots of unity (especially at $q(2)$) will have
exponentially large phenomenon and the the growth rates corresponds to
Volume of 3-manifold with boundary and Volume of closed 3-manifold
respectively.

Thankful to the new tool developed by Ohtsuki recently, asymptotic
expansion of Kashaev invariants (including Volume Conjecture) up to 7
crossing has been solved. This new tool can also be used to attack my
volume Conjecture with Tian Yang. I will give a brief introduction for all
these new developments.

Finally we expect Reshetikhin-Turaev at roots of unity other than $q(1)$
could have a different Geometry/Physics interpretation than original
Chern-Simons theory given by Witten in 1989.

Direct observations of the modern geomagnetic field enable us to understand its role in protecting us from the depredations of the solar wind and associated space weather, while paleomagnetic studies provide geological evidence that the field is intimately linked with the history and thermal evolution of our planet. In the past the magnetic field has reversed polarity many times: such reversals occur when its overall strength decays, and there are departures from the usual spatial structure which at Earth's surface predominantly resembles that of an axially aligned dipole. Reversals are one element of a continuum of geomagnetic field behavior which also includes geomagnetic excursions (often viewed as unsuccessful reversals), and paleosecular variation. The fragmentary and noisy nature of the geological record combined with distance from the field's source in Earth's liquid outer core provide a limited view, but one that has been partially characterized by time series analysis, and development of stochastic models describing the variability. Analyses of changes in the dipole moment have revealed distinct statistical characteristics associated with growth and decay of field strength in some frequency ranges. Paleomagnetic studies are complemented by computationally challenging numerical simulations of geomagnetic field variations. Access to details within the numerical model allow the evolution of large scale physical processes to be studied directly, and it is of great interest to determine whether these computational results have Earth-like properties. The parameter regime accessible to these simulations is far from ideal, but their adequacy can be assessed and future development guided by comparisons of their statistical properties with robust results from paleomagnetic observations. Progress in geomagnetic studies has been greatly facilitated by the application of statistical methods related to stochastic processes and time series analysis, and there remains significant scope for continued improvement in our understanding. This is likely to prove particularly important for understanding the scenarios that can lead to geomagnetic reversals.

Free registration is required to attend. Registration information is available at

http://www.math.ucsd.edu/$\sim$rosenblattconf/rosenblattlecture.html