We survey several well-known direct consequences of very large cardinal axioms.  In particular we intend to cover SCH (Solovay), the failure of the approachability property (Shelah), and the failure of Not So Very Weak Square (Foreman--Magidor).  If time permits, we will discuss a characterization of strong compactness due to Ketonen or the tree property (Magidor-Shelah).

Mathematical modeling and scientific computing are very important in improving the quality of radiotherapy. I my talk, I will go through some of the steps of the entire process of radiotherapy where mathematical modeling is important. In particular, I will talk about the following two topics in detail.
The first topic is on accurate radiation dose delivery in volumetric modulated arc therapy (VMAT) in cancer radiotherapy. It can be described as an optimization problem, where beam parameters, such as directions, shapes, and intensities, can be adjusted in simulations to yield desired dose distributions. Treatment plan optimization in this setting, however, can be quite complicated due to constraints arising from the equipment involved. We introduce a variational model in the VMAT setup for the optimization of beam shapes and intensities under these constraints. Our numerical tests on real data reveal that our algorithm shows great promise in the generation of desired dose distributions for treatment plans in cancer radiotherapy.
The second topic is on optimal marker selection for tumor motion estimation in lung cancer radiotherapy. We propose a novel mathematical model and an efficient algorithm to automatically determine the optimal number and locations of fiducial markers on patient’s surface (typically on the chest) for predicting lung tumor motion. Experiments on the 4DCT data of 4 lung cancer patients have shown that usually 6-7 markers are selected on patient’s external surface. Using these markers, the lung tumor positions can be predicted with an average 3D error of approximately 1mm. Both the number of markers and the prediction accuracy are clinically acceptable, indicating that our method can be used in clinical practice.

A subset $X$ in $\{0,1\}^n$ is a called an $\epsilon$-biased set if for any nonempty subset $T\subseteq [n]$ the following condition holds. Randomly choosing an element $x\in X$, the parity of its T-coordinates sum has bias at most $\epsilon$. This concept is essentially equivalent or close to expanders, pseudorandom generators and linear codes of certain parameters. For instance, viewing $X$ as a generator matrix, an $\epsilon$-biased set is equivalent to an $[|X|, n]_2$-linear code of relative distance at least $1/2-\epsilon$. For fixed $n$ and $\epsilon$, it's a challenging problem to construct smallest $\epsilon$-biased sets. In this talk we will first introduce several known constructions of $\epsilon$-bias sets with the methods from number theory and geometrical coding theory. Then we will present a construction with a conjecture, which is closely related to the subset sum problem over prime fields.

For a polynomial map f(x) from a field F to itself, we are interested in the size of the values that f misses, that is, the cardinality of F - f(F). For F = C (the complex numbers), if f misses one value, then f is a constant (this is the fundamental theorem of algebra). For F = C, if a holomorphic map f misses two values, then f is again a constant (this is Picard's little theorem). What about when f: F^n -> F^n is a polynomial vector map? When F is a finite field F_q of q elements, this problem becomes very interesting. There are extensive results and open problems available. For example, if a polynomial f of degree d>1 misses one value of F_q, then it must miss at least (q-1)/d values. In this lecture, we give a self-contained exposition of the main results and the open problems on the value set problem, and explain its link to different parts of mathematics.

A convolution of two singular continuous measures can be singular continuous or absolutely continuous (or of a mixed type). It is usually hard to determine which case is present for a specific pair of measures. It turnes out that for measures of maximal entropy of large Hausdorff dimension supported on dynamically defined Cantor sets generically the convolution is a.c. (this is a joint result with D.Damanik and B.Solomyak). This is in a sense a measure-theoretical counterpart of the claim (known as Newhouse Gap Lemma) that the sum of two sufficiently thick Cantor sets must contain an interval.