in this talk, we will introduce a new bootstrap algorithm for estimating linear combinations of coefficients in high dimensional threshold ridge regression model. in addition, we will also introduce a new definition about prediction and demonstrate how bootstrap calibration algorithm achieves this goal.

cut and project sets are well-studied models of almost-periodic discrete subsets of $r^d$. in 2014 marklof and strombergsson introduced a natural class of random processes which generate cut and project sets in a way which is invariant under the group $asl(d,r)$. using ratner's theorem and the theory of algebraic groups we classify all these measures. using the classification we obtain results analogous to those of siegel, rogers, and schmidt in geometry of numbers: summation formulas and counting points in large sets for typical cut and project sets. joint work with rene ruehr and yotam smilansky.

researchers around the globe are gathering biomedical information at a massive scale. we develop algorithms to compress this data that enable computation on the reduced representation. in this talk, i will discuss how we can leverage the low-dimensional true structure of biological data manifolds in order to build useable compact geometric summaries of this data. i will highlight our latest work on single-cell transcriptomic datasets, that enables an unprecedented scale of data to be effectively pooled from individuals and institutions across nations to enable novel life-saving discoveries.

in this talk, i will show that, given two distinct strictly stable self-expanders (of mean curvature flow) that are asymptotic to the same cone, there is a new self-expander trapped between these two. this is achieved by developing a min-max theory for the relative expander entropy. this is joint with jacob bernstein.

how much time does brownian motion spends at 0? even though it spends lebesgue 0 amount of time there, this is not the end of the story. the local time for brownian motion let us go deeper into this question, creating an object that keeps track of how much time brownian motion spends at 0. once established, we will discuss applications of local time such as tanaka's formula, change of variables formula, engelbert-schimdt 0-1 law and reflected diffusions.

we revisit the tears of wine problem for thin films in water-ethanol mixtures and present a new model for the climbing dynamics. the new formulation includes a marangoni stress balanced by both the normal and tangential components of gravity as well as surface tension which lead to distinctly different behavior. the combined physics can be modeled mathematically by a scalar conservation law with a nonconvex flux and a fourth order regularization due to the bulk surface tension. without the fourth order term, shock solutions must sastify an entropy condition - in which characteristics impinge on the shock from both sides. however, in the case of a nonconvex flux, the fourth order term is a singular perturbation that allows for the possibility of undercompressive shocks in which characteristics travel through the shock. we present computational and experimental evidence that such shocks can happen in the tears of wine problem, with a protocol for how to observe this in a real life setting.

an independent set of size $k$ in a finite undirected graph is a set of $k$ vertices of the graph, no two of which are connected by an edge. the structure of independent sets of size $k$ as $k$ varies is of interest in probability, statistical physics, combinatorics, and computer science. in 1987, alavi, malde, schwenk and erdos conjectured that the number of independent sets of size $k$ in a tree is a unimodal sequence (this number goes up and then it goes down), and this problem is still open. a variation on this question is: do the number of independent sets of size $k$ form a unimodal sequence for erdos-renyi random graphs, or random trees? by adapting an argument of coja-oghlan and efthymiou, we show unimodality for erdos-renyi random graphs, random bipartite graphs and random regular graphs (with high probability as the number of vertices in the graph goes to infinity, when the expected degree of a single vertex is large). the case of random trees remains open, as we can only show weak partial results there.

classically, the transcendence (and even the irrationality) of
odd zeta values is widely conjectured, but yet unproven. however, for
zeta values defined over the rational function field, jing yu succeeded
in proving their transcendence in 1991, and many other transcendence
results (including algebraic independence) followed in the intervening
years. in this work (joint with t. ngo dac), we prove the algebraic
independence of zeta values defined over the function field of an
elliptic curve. the main technique we use is to construct a tannakian
category of t-motives whose associated periods contain these zeta values
- thus we may exploit the existence of a motivic galois group to study
the transcendence degree. we also discuss the difficulties and pathway
to proving algebraic independence for zeta values of function fields of
arbitrary curves.

option contracts are a type of financial derivative that allow investors to hedge risk and speculate on the variation of an asset's future market price. in short, an option has a particular payout that is based on the market price for an asset on a given date in the future. in 1973, black and scholes proposed a valuation model for options that essentially estimates the tail risk of the asset price under the assumption that the price will fluctuate according to geometric brownian motion. more recently, demarzo et al., among others, have proposed more robust valuation schemes, where we can even assume an adversary chooses the price fluctuations. this framework can be considered as a sequential two-player zero-sum game between the investor and nature. we analyze the value of this game in the limit, where the investor can trade at smaller and smaller time intervals. under weak assumptions on the actions of nature (an adversary), we show that the minimax option price asymptotically approaches exactly the black- scholes valuation. the key piece of our analysis is showing that nature's minimax optimal dual strategy converges to geometric brownian motion in the limit.

i will introduce a program, begun by dan-cohen and wewers, to compute
minhyong kim's selmer varieties using mixed tate motives. the idea is as
follows. we care about the galois action on the unipotent fundamental
group of $\mathbf{p}^1 \setminus \{0,1,\infty\}$. this galois action
lives in a certain category of $p$-adic galois representations known as
mixed tate representations. we will see that this category is tannakian
and has a fairly simple description, in terms of its ext groups, which
are just bloch-kato selmer groups. the bloch-kato selmer groups are
$p$-adic vector spaces, but they also have a rational structure coming
from algebraic k-theory. the category of mixed tate motives gives us a
$\mathbb{q}$-linear tannakian category that underlies the
$\mathbb{q}_p$-linear tannakian category of galois representations. this
in turn allows us to define a selmer variety over $\mathbb{q}$ (rather
than $\mathbb{q}_p$), and a more explicit understanding of the category
of mixed tate motives allows us to compute this variety explicitly.

in this talk i'll teach the basics of making a website and how to connect with the ucsd server. if you want to have a working website by the end of the week, it will help to (1) talk to saul about getting your login information for the ucsd math servers, (2) have a text editor to make the website (i like notepad++), and (3) have a way to ssh to the math server (i like winscp). to stall for time i'll also talk about other resources that might be of use to a math phd student. other people are more than welcome to contribute ideas in this direction!

when $x$ is a grassmannian, marian-oprea-pandharipande and toda constructed alternate compactifications of spaces of maps from curves to $x$. the construction has been generalized to a large class of git quotients $x=w//g$ by ciocan-fontanine-kim-maulik and many others. it is called the theory of $\epsilon$-stable quasimaps. in this talk, we will introduce permutation-equivariant k-theoretic epsilon-stable quasimap invariants and prove their wall-crossing formulae for all targets in all genera. the wall-crossing formulae generalize givental's k-theoretic toric mirror theorem in genus zero. in physics literature, these k-theoretic invariants are related to the $3d n = 2$ supersymmetric gauge theories studied by jockers-mayr, and the wall-crossing formulae can be interpreted as relations between invariants in the uv and the ir phases of the $3d$ gauge theory. it is based on joint work with yang zhou.