Seminars archive

The Sato-Tate conjecture is concerned with the statistical distribution of the number of points on the reduction modulo primes of a fixed elliptic curve defined over the rational numbers. It predicts that this distribution can be explained in terms of a random matrix model, using the Haar measure on the special unitary group SU(2). Thanks to recent work by Richard Taylor and others, this conjecture is now a theorem. The Sato-Tate conjecture generalizes naturally to abelian varieties of dimension g, where it associates to each such abelian variety a compact subgroup of the unitary symplectic group USp(2g), the Sato-Tate group, whose Haar measure governs the distribution of certain arithmetic data attached to the abelian variety. While the Sato-Tate conjecture remains open for all g>1, I will present recent work that has culminated in a complete classification of the Sato-Tate groups that can arise when g=2 (and proofs of the Sato-Tate conjecture in some special cases), and highlight some of the ongoing work in dimension 3. I will also present numerical computations that support the conjecture, along with animated visualizations of this data. This is joint work with Francesc Fit\'{e}, Victor Rotger, and Kiran S. Kedlaya, and also with David Harvey.

Let $\sigma(n)$ be the usual sum-of-divisors function. The ancient Greeks put the natural numbers into three categories: deficient numbers, for which $\sigma(n) < 2n$, abundant numbers, for which $\sigma(n) > 2n$, and perfect numbers, for which $\sigma(n)=2n$. While early discussion of these numbers has more in common with numerology than with number theory, the 20th century saw great progress in understanding how these numbers were distributed within the sequence of natural numbers. I will survey the problems, the known methods and results, and the (numerous!) still unresolved questions in this area. Some of this represents joint work with Mits Kobayashi and Carl Pomerance.

How do humans perform high-level reasoning and problem solving? Much of cognitive science research and almost all of AI research into problem solving has focused on the use of verbal or amodal propositional representations, despite the growth of evidence from neuroscience showing that many mental representations function as modal perceptual symbols. In this talk, I will discuss the role of iconic mental representations in high-level problem-solving tasks. I will first examine the notion of whether certain individuals with autism may have a bias towards "thinking visually." I will then focus on one problem-solving domain in particular: the Raven's Progressive Matrices test, which represents one of the single best psychometric measures of general intelligence that has yet been developed. I will describe previous computational theories of problem solving on the Raven's test, which have all been propositional in nature, and then present a new computational model, the ASTI model, which uses purely visual operations akin to those used in mental imagery. I will end by discussing implications of the model for our evolving understanding of cognition in autism, general human cognition, and computational views of intelligence.

The lecture is going to focus on extremal set theory. The general problem is concerned with the maximum possible size of a subset of the power set of a finite set $X$ of $n$ elements subject to some conditions. The simplest result is probably the following.\\ \\ Proposition 0. If $F$ is a subset of $2^X$, such that any two sets in $F$ have non-empty intersection then $|F| \leq 2^(n-1)$.\\ \\ One way to achieve equality is by taking all subsets containing a fixed element.\\ \\ Erdös-Ko-Rado Theorem. If $F$ is a collection of $k$-element subsets of $X$ such that any two sets in $F$ have non-empty intersection and $2k < n$ , then $|F| \leq {n-1 \choose k-1}$ with equality holding only if all subsets in $F$ contain a fixed element. We are going to discuss various generalizations and extensions of this result, some of which are still unsolved.

A field $F$ is called $C_r$ if every homogenous form of degree $n$ in more then $n^r$ variables has a non-trivial solution. Consider a central simple algebra $A$ of exponent $n$ over a field $F$. By the Merkurjev-Suslin theorem assuming $F$ contains a primitive $n$-th root of unity, $A$ is similar to the product of symbol algebras. The smallest number of symbols required is called the \emph{length} of $A$ and is denoted $l(A)$. If $F$ is $C_r$ we prove $l(A) \leq n^{r-1}-1$. In particular the length is independent of the index of $A$.

It is often implied or assumed that the ultimate goal of virtual reality (VR) systems is the perfect reproduction of reality. It seems that the objective is making graphics, audio, haptics, physics, and interaction as indistinguishable from the real world as possible. In other words, the hypothetical Star Trek Holodeck is seen as the gold standard of VR systems. The word "reality" is even part of the name of the field! However, increasing realism can have a steep cost, and evidence for the benefits of higher levels of realism is often lacking. In this talk, I want to take a step back to evaluate how, when, and why high levels of realism are desirable. I will review a decade of research in my group focusing on understanding the effects of various types of realism in VR systems, and will draw some potentially surprising conclusions about where research efforts should be focused and how much realism is enough. Along the way, I will share some lessons learned about life in academia and a career in research.

One of the simplests statements of Sidorenko's conjecture is an inequality bounding the homomorphism density from a bipartite graph to a non-empty graph. Due to extensive research the conjecture has been established for various classes of bipartite graphs. The proofs, however, are often complicated and seem to have little relation to one another. Recently (in a yet unpublished paper), Balázs Szegedy gave a new, unified proof of all known cases for which Sidorenko's conjecture has been established. Our aim in this talk is to give some idea of how this proof goes and to discuss which cases it covers.

Datacenter scale cloud platforms continue to exhibit increases in the scale and diversity of their infrastructure -- cores, servers, enclosures, I/O -- and the workloads they host -- number and type of applications, virtual machines, etc. This results in significant challenges in the ability to effectively manage these platforms, to provide high resource utilization while also meeting client performance expectations, and to further evaluate the effectiveness of such management operation. In this talk, I will present our ongoing efforts toward management of compute, I/O and energy resources in cloud platforms, and will provide data regarding the insights we gained in the behavior of the management processes themselves, including their failures. For the results gathered in our experimental approach we use publicly available cloud traces as well as a range of popular cloud and enterprise workloads. Our evaluation with real cloud applications illustrate additional challenges due to the interactions of the cloud-level management processes and those present in modern cloud runtimes, like Hadoop.

I will describe a method to compute models for Hilbert modular surfaces $Y_{-}(D)$, parametrizing principally polarized abelian surfaces with real multiplication by the full ring of integers in the quadratic field of discriminant D, through moduli spaces of elliptic K3 surfaces. I will also show a couple of applications of these explicit equations, in number theory and in geometry/dynamics.

Erdos, Faudree, Gould, Gyarfas, Rousseau and Schelp proved that for every complete graph of order $n$ with edges colored with three colors, there exist a set $X$ of 22 vertices and a color $c$ such that the number of vertices in $X$ or joined to a vertex of $X$ by an edge of color $c$ is at least $2n/3$. They also conjectured that the bound of 22 can be lowered to 3. We improve the bound to 4. The talk is based on joint work with Chun-Hung Liu, Jean-Sebastien Sereni, Peter Whalen and Zelealem Yilma.