4:00pm to 5:00pm - RH 306 - Applied and Computational Mathematics Tony Wong - (UCLA) First passage time problems in the spatial searching and binding processes of T-cells with antigen-presentation cells in a lymph node Quick antigen detection is essential for our adaptive immune system to function properly. We model the T-cell migration and their search for antigens from binding with antigen-presentation cells (APC) in a lymph node. A successful search must encounter the cognate antigens from many APCs within a stochastic time span (before departure from the lymph node). As such, we study the first passage time statistic of binding and exiting events of T-cells in various setting, for example, multi-stage binding and spatially heterogeneous distribution of APC. This is a joint work with Ikchang Cho, Maria D’Orsogna and Tom Chou. |
3:00pm to 3:50pm - RH 306 - Analysis Xuan Duong - (Macquarie University, Australia) Calder\'on-Zygmund decomposition, Hardy spaces associated with operators and weak type estimates Let $(X, d, \mu )$ be a metric space with a metric $d$ and a doubling measure $\mu$. Assume that the operator $L$ generates a bounded holomorphic semigroup $e^{-tL}$ on $L^2(X)$ whose semigroup kernel satisfies the Gaussian upper bound. Also assume that $L$ has bounded holomorphic functional calculus on $L^2(X)$. Then the Hardy spaces $H^p_L(X)$ associated with the operator $L$ can be defined for $0 < p \le 1$. In this talk, we revisit the Calder\'on-Zygmund decomposition and show that a function $f \in L^1(X)\cap L^2(X)$ can be decomposed into a good part and a bad part which is in $H^p_L(X)$ for some $0 < p <1$. An important result of our variants of Calder\'on-Zygmund decompositions is that if a sub-linear operator $T$ is bounded from $L^r(X)$ to $L^r(X)$ for some $r > 1$ and also bounded from $H^p_L(X)$ to $L^p(X)$ for some $0 < p < 1$, then $T$ is of weak type $(1,1)$ and bounded from $L^q(X)$ to $L^q(X)$ for all $1< q <r$.
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2:00pm to 3:00pm - 510R Rowland Hall - Combinatorics and Probability Yiyun He - (UCI) Online differential privacy We present a polynomial-time algorithm for online differentially private synthetic data generation. For a data stream within the hypercube [0,1]^d and an infinite time horizon, we develop an online algorithm that generates a differentially private synthetic dataset at each time t. This algorithm achieves a near-optimal accuracy bound in the 1-Wasserstein distance. |
2:00pm to 2:50pm - RH 510R - Algebra Maureen Zhang - (UCI) Ext algebras of twisted tensor products in Koszul cases Given any associative k-algebra A, the Ext algebra is a graded k-vector space with algebra structure given by the Yoneda product. The Ext algebra served as an important homological invariant for various reasons. Moreover, it is known that taking the Ext algebra is a process that distributes nicely across tensor product, i.e. Ext(A ⊗B)≅Ext(A) ⊗Ext(B). Therefore it is natural to ask if the same statement holds for the non-commutative analogue of tensor product, twisted tensor product. |
4:00pm - RH 306 - Colloquium Andreas Seeger - (UW-Madison) Problems on spherical maximal functions Abstract: This will be a survey on some old and some new problems on spherical averages, regarding pointwise convergence, p-improving properties of local variants, and consequences for sparse domination. Emphasis will be put on how the results depend on suitable notions of fractal dimension of the dilation sets. |