In statistical methods for language and document modeling, there are

two major perspectives: representation at the document level, and

representation at the word level. At the document level, topic models

such as latent Dirichlet allocation (LDA) and hierarchical Dirichlet

process (HDP), based on the  word-document matrix, aim to discover

topics whose dimensionality is much lower than the size of the

vocabulary. At the word-level, language models such as n-grams and

neural word embedding, based on the word co-occurrence matrix, aim to

represent each word in a high-dimensional vector space. In this work,

we develop Dual Representation Topic Model (DRTM), a novel topic model

which combines the advantages of the two approaches. DRTM models

documents and words based on the locations of the individual words

within documents, as well as the local contexts of the words. DRTM

transforms each document into a network of words by generating edges

when words of near proximity have high semantic similarity. Then it

infers the topic for each edge - a pair of words - rather than

assigning topics for individual words as in traditional topic models.

This enables the model to learn a better document representation by

inferring the global topics while considering the local contexts of

individual words.

Black holes are perhaps the most celebrated predictions of general relativity. Miraculously, these complicated spacetimes arise as explicit (i.e., exact expression can be written down!) solutions to the vacuum Einstein equation. Looking these explicit black hole solutions leads to an intriguing observation: While the black hole exterior look qualitatively similar for every realistic black hole, the structure of the interior, in particular the nature of the `singularity' inside the black hole, changes drastically depending on whether the black hole is spinning (Kerr) or not (Schwarzschild).

A proposed picture for what happens in general is the so-called strong cosmic censorship conjecture of R. Penrose, which is a central conjecture in general relativity. In this colloquium, I will give a short introduction to general relativity and explain what this conjecture is. Time permitting, I will present some recent progress (joint work with J. Luk at Stanford) on related topics, using tools from nonlinear hyperbolic PDEs.

 Intelligent systems with deep learning have emerged as a key technique for a wide range of different applications including vision processing, autonomous driving and robot navigation. SoC implementations in deep learning-based intelligent systems give us higher performance and low-power operations in many applications.

Hilbert syzygy theorem says that any finitely generated graded module $M$ over the standard graded polynomial ring $S=K[x_1,ldots,x_n]$ has a finite free resolution

$$

0 leftarrow M leftarrow F_0 leftarrow F_1 leftarrow ldots leftarrow F_c leftarrow 0

$$

with $F_i = oplus_j S(-i)^{beta_{ij}}$ a free module with $beta_{ij}$ generators

in degree $j$. Hilbert proved his syzygy theorem to exhibit the polynomial nature of the Hilbert series:

$$

H_M(t) = sum_k dim M_k t^k = frac{sum_i (-1)^i sum_j beta_{ij}t^j}{(1-t)^n}

$$

In the talk I will report on the  question, what kind of more information about $M$

is encoded in the graded Betti numbers $beta_{ij}(M)$, what are the possible values

of these numbers, and, what can be said about extremal cases.

Theoretical Computer Science provides the sound foundation

and rigorous concepts underlying contemporary algorithm

design and software development -- for discrete problems:

Problems in the continuous realm commonly considered in Numerical Engineering are largely treated by 'recipes' and 'methods'

whose correctness and efficiency is usually shown empirically.

We extend and apply the theory of computation over discrete structures

to continuous domains: It turns out that famous complexity classes like

P, NP, #P, and PSPACE naturally re-emerge in the setting of real numbers,

sequences, continuous functions, operators, and Euclidean subsets

(including a reformulation of a Millennium Prize Problem as a numerical one).

We currently work towards a rigorous computability and complexity

classification for partial differential equations, namely over

Sobolev spaces that their solutions naturally 'live' in.