we study regularity properties of the free boundary for solutions of the porous medium equation with the presence of drift. we show that if the initial data has super-quadratic growth at the free boundary, then the support strictly expands relative to the streamline, and that the movement is holder continuous in time. under additional information of directional monotonicity in space, we derive nondegeneracy of solutions and $c^{1,\alpha}$ regularity of free boundaries. finally several examples of singularities are given that illustrate differences from the zero drift case.

``who doesn't like those?''

combinatorial neural codes are 0/1 vectors that are used to model the co-firing patterns of a set of place cells in the brain. one wide-open problem in this area is to determine when a given code can be algorithmically drawn in the plane as a venn diagram-like figure. a sufficient condition to do so is for the code to have a property called k-inductively pierced. gross, obatake, and youngs recently used toric algebra to show that a code on three neurons is 1-inductively pierced if and only if the toric ideal is trivial or generated by quadratics. no result is known for additional neurons in the same generality.

in this talk, we study two infinite classes of combinatorial neural codes in detail. for each code, we explicitly compute its universal gröbner basis. this is done for the first class by recognizing that the codewords form a lawrence-type matrix. with the second class, this is done by showing that the matrix is totally unimodular. these computations allow one to compute the state polytopes of the corresponding toric ideals, from which all distinct initial ideals may be computed efficiently. moreover, we show that the state polytopes are combinatorially equivalent to well-known polytopes: the permutohedron and the stellohedron.

\textbf{small kam perturbations of integrable systems which are entropy expansive.} one of the greatest achievements in dynamics in the xx century is the kam theory. it says that after a small perturbation of a non-degenerate completely integrable system it still has an overwhelming measure of invariant tori with quasi-periodic dynamics. what happens outside kam tori remains a great mystery. it is easy, by modern standards, to show that topological entropy can be positive. it lives, however, on a zero measure set. we are now able to show that metric entropy can become infinite too, under arbitrarily small $c^{\infty}$ perturbations, answering an old-standing problem of kolmogorov. furthermore, a slightly modified construction resolves another long standing problem of the existence of entropy non-expansive systems. in these modified examples positive metric entropy is generated in arbitrarily small tubular neighborhoods of one trajectory. joint with s. ivanov and dong chen.

\noindent \textbf{metric approximations of length spaces by graphs with uniformly bounded local structure.} how well can we approximate an (unbounded) space by a metric graph whose parameters (degrees of vertices, lengths of edges, density of vertices etc) are uniformly bounded? we want to control the additive error. some answers are given (the most difficult case is $\mathbb{r}^2$) using dynamics and fourier series. joint with s. ivanov.

\noindent \textbf{on busemann's problem on minimality of flats in normed spaces for the buseman-hausdorff surface area.} busemann asked if regions in affine subspaces of normed spaces are area minimizers with respect to the busemann-hausdorff measure. this has been known for long for hyperplanes (codim=1), this is a classic result in convex geometry. sergei ivanov and me were able to prove this for 2-dimensional subspaces.

grothendieck's inequality guarantees that a certain discrete optimization problem-optimizing a bilinear form over +1, -1 inputs—is equivalent up to a constant, called grothendieck's constant, to a continuous optimization problem -- optimizing the same bilinear form over unit vectors in a hilbert space. this is important for convex relaxation, because the former contains np-hard problems such as max-cut, whereas the latter is a semidefinite program and can be solved in polynomial time. a world apart from convex
relaxation is algebraic computational complexity, where it is well-known that the exponent of matrix multiplication is exactly the sharp lower bound for the (log of) tensor rank of the strassen matrix multiplication tensor. we show that grothendieck's constant is the sharp upper bound on a tensor norm of strassen matrix multiplication tensor. hence these two important quantities from disparate areas of theoretical computer science—strassen's exponent of matrix multiplication and grothendieck's constant—are just different measures of the same underlying tensor. this allows us to rewrite grothendieck's inequality as a norm inequality for a 3-tensor, which in turn gives a family of natural generalized inequalities. we show that these are locally sharp and prove that grothendieck's inequality is unique.

\textbf{counting collisions.}20 years ago the topic of my talk at the icm was a solution of a problem which goes back to boltzmann and ya. sinai. the conjecture of boltzmann-sinai states that the number of collisions in a system of $n$ identical balls colliding elastically in empty space is uniformly bounded for all initial positions and velocities of the balls. the answer is affirmative and the proven upper bound is exponential in $n$. the question is how many collisions can actually occur. on the line, there can be $n(n-1)/2$ collisions, and this is he maximum. since the line embeds in any euclidean space, the same example works in all dimensions. the only non-trivial (and counter-intuitive) example i am aware of is an observation by thurston and sandri who gave an example of 4 collisions between 3 balls in $r^2$. recently, sergei ivanov and me proved that there are examples with exponentially many collisions between $n$ identical balls in $r^3$, even though the exponents in the lower and upper bounds do not match.

\noindent \textbf{a survival guide for a feeble fish and homogenization of the g-equation.} how fish can get from a to b in turbulent waters which maybe much fasted than the locomotive speed of the fish provided that there is no large-scale drift of the water? this is related to g-equation and has applications to its homogenization. g-equation which is believed to govern many combustion processes. based on a joint work with s. ivanov and a. novikov.

regarding time as a scaling parameter, we prove the one-point, lower tail large deviation principle (ldp) of the kpz equation, with an explicit rate function. this result confirms existing physics predictions. we utilize a formula from [borodin gorin 16] to convert ldp of the kpz equation to calculating an exponential moment of the airy point process, and analyze the latter via stochastic airy operator and riccati transform

smoothing spline mixed-effects density models are proposed for the nonparametric
estimation of density and conditional density functions with clustered data.
the random effects in a density model introduce within-cluster correlation and
allow us to borrow strength across clusters by shrinking cluster specific density
function to the population average, where the amount of shrinkage is decided
automatically by data. estimation is carried out using the penalized likelihood
and computed using a markov chain monte carlo stochastic approximation algorithm.
we apply our methods to investigate evolution of hemoglobin density functions
over time in response to guideline changes on anemia management for dialysis
patients.

the ligand-receptor binding/unbinding is a complex biophysical process in which water plays a critical role. to understand the fundamental mechanisms of such a process, we have developed a new and efficient approach that combines our level-set variational implicit-solvent model with the string method for transition paths, and have studied the pathways of dry-wet transition in a model ligand-receptor system. we carry out brownian dynamics simulations as well as fokker-planck equation modeling with our efficiently calculated potentials of mean force to capture the effect of solvent fluctuations to the binding and unbinding processes. without the description of individual water molecules, we have been able to predict the binding and unbinding kinetics quantitatively in comparison with the explicit-water molecular dynamics simulations. our work indicates that the binding/unbinding can be controlled by a few key parameters, and provides a tool of efficiently predicting molecular recognition with application to drug design.

in 1970, manin observed that the brauer group br(x) of a variety
x over a number field k can obstruct the hasse principle on x. in other
words, the lack of a k-point on x despite the existence of points over
every completion of k is sometimes explained by non-trivial elements in
br(x). this so-called brauer-manin obstruction may not always suffice to
explain the failure of the hasse principle but it is known to be sufficient
for some classes of varieties (e.g. torsors under connected algebraic
groups) and conjectured to be sufficient for rationally connected varieties
and k3 surfaces.
a zero-cycle on x is a formal sum of closed points of x. a rational point
of x over k is a zero-cycle of degree 1. it is interesting to study the
zero-cycles of degree 1 on x, as a generalisation of the rational points.
yongqi liang has shown that for rationally connected varieties, sufficiency
of the brauer-manin obstruction to the hasse principle for rational points
over all finite extensions of k implies sufficiency of the brauer-manin
obstruction to the hasse principle for zero-cycles of degree 1 over k. in
this talk, i will discuss joint work with francesca balestrieri where we
extend liang's result to kummer varieties.

i will discuss the einstein-klein-gordon coupled system of general relativity, and the problem of global stability of the minkowski space-time. this is joint work with benoit pausader.

the minimal log discrepancy of an algebraic variety is an invariant which measures the singularites of the variety. for mild singularities the minimal log discrepancy is a non-negative real value; the closer to zero this value is, the more singular the variety. it is conjectured that in a fixed dimension, this invariant satisfies the ascending chain condition. in this talk we will show how boundedness of fano varieties imply some local statements about the minimal log discrepancies of klt singularities. in particular, we will prove that the minimal log discrepancies of klt singularities which admit an $\epsilon$-plt blow-up can take only finitely many possible values in a fixed dimension. this result gives a natural geometric stratification of the possible mld's on a fixed dimension by finite sets. as an application, we will prove the ascending chain condition for minimal log discrepancies of exceptional singularities in arbitrary dimension.