maximizing computer system performance relies on careful resource management: how to best allocate resources among jobs. effective resource allocation is most difficult in regimes with uncertainty. this talk examines three common types of uncertainty. we consider uncertainty in job sizes and ask how to optimally schedule jobs to minimize response time in such regimes. we next turn to uncertainty in the arrival rate and ask how we should adapt capacity provisioning and power management in data centers to handle unexpected load fluctuations. finally, we consider uncertainty in the system state and look at how job replication can help curtail unpredictability. a common thread in this talk is stochastic performance modeling and the insights it illuminates.

motivated by questions in pointwise ergodic theory, modern discrete harmonic analysis, as developed by bourgain, has focused on understanding the oscillation of averaging operators -- or related singular integral operators -- along polynomial curves. in this talk we present the first example of a discrete analogue of polynomially modulated oscillatory singular integrals; this begins to unify the work of bourgain, stein, and stein-wainger. the argument combines a wide range of techniques from euclidean harmonic analysis and analytic number theory.

we give a systematic and self-contained account of the construction of
geometrically decomposed bases and degrees of freedom in finite element
exterior calculus. in particular, we elaborate upon a previously overlooked
basis for one of the families of finite element spaces, which is of interest
for implementations. moreover, we give details for the construction of
isomorphisms and duality pairings between finite element spaces. these
structural results show, for example, how to transfer linear dependencies
between canonical spanning sets, or give a new derivation of the degrees of
freedom.

loosely defined, a sieve is a mathematical technique for finding the rate of growth of a set of objects with a quantifiable (and hopefully small) error term. sieve techniques have wide applications in number theory and combinatorics. we will first present the idea behind sieving and present a toy example of calculating the probability that an integer is squarefree. then we will discuss poonen’s recently developed algebro-geometric sieve, which allows one to compute the probability that a hypersurface intersects smoothly with a given projective variety x over a finite field.

an $(\alpha,\beta)$-spanner of a graph g is a subgraph h that distorts distances in g up to a multiplicative factor of $\alpha$ and an additive factor of $\beta$, where the goal is to construct an h with as few edges as possible. when $\beta=0$ we call h a multiplicative spanner, and when $\alpha=1$ we call h an additive spanner. it is known how to construct multiplicative spanners of essentially optimal size, but much less is known about additive spanners. in this talk we discuss a recent result which shows how to construct a (0,6)-additive spanner for any graph g.

high-dimensional high-order data arise in many modern scientific applications including genomics, brain imaging, and social science. in this talk, we consider the methods, theories, and computations for tensor singular value decomposition (tensor svd), which aims to extract the hidden low-rank structure from high-dimensional high-order data. first, comprehensive results are developed on both the statistical and computational limits for tensor svd under the general scenario. this problem exhibits three different phases according to signal-noise-ratio (snr), and the minimax-optimal statistical and/or computational results are developed in each of the regimes. in addition, we further consider the sparse tensor singular value decomposition which allows more robust estimation under sparsity structural assumptions. a novel sparse tensor alternating thresholding algorithm is proposed. both the optimal theoretical results and numerical analyses are provided to guarantee the performance of the proposed procedure.

at the beginning of section 11 in perelman's celebrated paper ``the entropy formula for the ricci flow and its geometric applications'', he made the assertion that for an ancient solution to the ricci flow with bounded nonnegative curvature operator, bounded entropy is equivalent to noncollapsing on all scales. we give a proof for this assertion.

today the main emphasis in local number theory (i.e the local
langlands) is on the finite places. in charactacteristic 0 the infinite
place is the ''elephant in the room''. this is especially true in the
whittaker theory in which serious difficulties separate the infinite from
the finite places. whittaker models were developed to help the study of
fourier coefficients at cusps of non-holomophic cusp forms (i.e maass cusp
forms) through representation theory. the first of these lectures will start
with an explanation of the role of whittaker models in the theory of
automorphic forms. it will continue with a description of the main results.
the second lecture will explain the proof of the whittaker plancherel
theorem.

in this talk, we will prove a classical theorem which states that the simple random walk on the integer lattice $\mathcal{z}^d$ is
recurrent in the case where $d=1,2$ and transient in the case where $d \geq 3$. in particular, we plan to focus on the combinatorial nature of the proof.

a numerical semigroup is a subset of the natural numbers which is closed under addition. consider a numerical semigroup s selected via the following random process: fix a probability p and a positive integer m , and select a generating set for s from the integers 1, 2, . . . , m where each generator has probability p of being selected. what properties can we expect the numerical semigroup s to have? for instance, how many minimal generators do we expect s to have? in this talk, we answer several such questions, and describe some surprisingly deep geometric and combinatorial structures that arise naturally in this process.
no familiarity with numerical semigroups or probability will be assumed for this talk.