Seminars archive

In 1946 Erdos asked to determine or estimate the minimum number of distinct distances determined by an n-element planar point set V. He showed that a square integer lattice determines \Theta(n/\sqrt{log n}) distinct distances, and conjectured that any n-element point set determines at least n^{1−o(1)} distinct distances. In 2010-2015, Guth and Katz answered Erdos’s question by proving that any n-element planar point set determines at least \Omega(n/log n) distinct distances. In this talk, we consider a variant of this problem by Erdos and Gyarfas. For integers n, p, q with p \geq q \geq 2, determine the minimum number D(n,p,q) of distinct distances determined by a planar n-element point set V with the property that any p points from V determine at least q distinct distance. In a recent paper, Fox, Pach and Suk prove that when q = {p \choose 2} - p + 6, D(n,p,q) is always at least n^{8/7 - o(1)}. We will discuss a recent improvement of their result and some new bounds for a related (graph theoretic) Ramsey problem of Erdos and Shelah which arise. This is joint work with Adam Sheffer.

Medication dosing is a comprehensive problem with uncertainties. Every patient has unique condition, meanwhile some drugs have narrow therapeutic windows. Mis-dosing might result in preventable adverse event. Therefore, a robust decision support system would be helpful to clinicians by providing advisable dosing suggestions. Heparin is one of the sensitive drugs. In order to build up the decision support system for heparin patients, we present a clinician in the loop framework with deep reinforcement learning algorithm. There are two main objectives in this thesis, the first one is providing individualized dosing suggestion based on the multi-dimensional features of patients. The second one is evaluating the dosing predicted by our decision support system. We implemented several experiments to achieve these objectives. The data used in the experiments including simulated data, MIMIC-II intensive care unit data and Emory hospital intensive care unit data. There are two important processes with respect to our objectives. In the training process, the decision support system learned from the dosing executed by clinicians and the corresponding response of patients. In the evaluating process, we explored the results from several aspects and focused on the causality between variables and outcomes. The experimental results suggested that given the states of patients, our medication dosing support system is able to provide a reasonable recommendation

We discuss an extension of field patching to proper schemes. Then, we will introduce Tannaka duality for stacks as first developed by Lurie and then refined by Hall and Rydh. Their work allows us to patch morphisms from proper schemes to nice stacks, in particular certain moduli stacks. As an application of this result, we prove that patching holds for relative torsors which allows us to give a characterization for local-global principles for torsors over proper schemes. This is joint work with Daniel Krashen and Max Lieblich.

Given two $k$-graphs $H$ and $F$, a perfect $F$-packing in $H$ is a collection of vertex-disjoint copies of $F$ in $H$ which together cover all the vertices in $H$. In the case when $F$ is a single edge, a perfect $F$-packing is simply a perfect matching. For a given fixed $F$, it is generally the case that the decision problem whether an $n$-vertex $k$-graph $H$ contains a perfect $F$-packing is NP-complete.\\ \\In this talk we describe a general tool which can be used to determine classes of (hyper)graphs for which the corresponding decision problem for perfect $F$-packings is polynomial time solvable. We then give applications of this tool. For example, we give a minimum $l$-degree condition for which it is polynomial time solvable to determine whether a $k$-graph satisfying this condition has a perfect matching (partially resolving a conjecture of Keevash, Knox and Mycroft). We also answer a question of Yuster concerning perfect $F$-packings in graphs.

Mathematics is a powerful tool that addresses real-life applications arising from various disciplines. In this event, organized by the Emory SIAM Student Chapter, you will be able to taste the versatility of Math. Professors conducting cutting-edge research in Biology, Sociology, and Physics will share how Mathematics contributes to their work. The talks will be followed by an open discussion and refreshments.

We work over an algebraically closed field k of characteristic p greater than 0. In 1994, Serre showed that if semi-simple representations V_i of a group \Gamma are such that \sum ( dim(V_i) - 1 ) less than p, then their tensor product is semi-simple. In the late nineties, Serre generalized this theorem comprehensively to the case where \Gamma is a subgroup of G(k), for G a reductive group, and answered the question of “complete reducibility” of \Gamma in G (Seminaire Bourbaki, 2003). In 2014, Deligne generalized the results of Serre (of 1994) to the case when the V_i are semi-simple representations of a group scheme \mathfrak{G}. In my talk I will present the recent work of mine with Deligne and Parameswaran where we consider the case when \mathfrak{G} is a subgroup scheme of a reductive group G and generalize the results of Serre and Deligne. A key result is a structure theorem on “doubly saturated” subgroup schemes \mathfrak{G} of reductive groups G. As an application, we obtain an analogue of classical Luna's etale slice theorem in positive characteristics.

A celebrated 19th century result of Cayley and Salmon is that a smooth cubic surface over the complex numbers contains exactly 27 lines. Over the real numbers, it is a lovely observation of Finashin–Kharlamov and Okonek–Teleman that while the number of real lines depends on the surface, a certain signed count of lines is always 3. We extend this count to an arbitrary field k using an Euler number in A1-homotopy theory. The resulting count is valued in the Grothendieck-Witt group of non-degenerate symmetric bilinear forms. This is joint work with Jesse Kass.

Accelerated optimization methods have gained wide applicability within the machine learning and optimization communities. They are known for leading to optimal convergence rates among schemes that use only use gradient (first order) information in the convex case. In the non-convex case, they appear to provide robustness to shallow local minima. The intuitive idea is that by considering a particle with mass that moves in an energy landscape, the particle will gain momentum and surpass shallow local minimum and settle in in more wider, deeper local extrema in the energy landscape. Although these techniques have been widely used, it was only within the last few years that theoretical attempts have been made to understand them and put them in a mathematical framework. Recent work has shown that accelerated methods may be formulated with variational principles, although in finite dimensions. Motivated by the success of accelerated methods in finite dimensional problems, we formulate optimization problems on infinite dimensional manifolds of diffeomorphisms using a generalization of this approach. The talk will mainly be about the mathematical formulation and some simple examples to illustrate the advantages of this approach. We note very large speed-ups in optical flow computation compared with standard approaches, and robustness to local minimum. Finally, we outline considerations for generalizing this approach to video data and applications in motion-based object segmentation, which require one to optimize diffeomorphisms not just defined on the image domain, but evolving regions of interest that encompass the domain of each of the objects in the scene.\\ \\Bio: Ganesh Sundaramoorthi received the PhD in Electrical and Computer Engineering from Georgia Institute of Technology, Atlanta, USA, and BS in Computer Engineering and BS Mathematics from the same institution in 2003. He was then a postdoctoral researcher in the Computer Science Department at the University of California, Los Angeles between 2008 and 2010. In 2011, he was appointed Assistant Professor of Electrical Engineering and Assistant Professor of Applied Mathematics and Computational Science at King Abdullah University of Science and Technology (KAUST). His research interests include computer vision and its mathematical foundations with recent interest in shape and motion analysis, video analysis, invariant representations for visual tasks, and applications. He was an area chair for IEEE ICCV 2017 and IEEE CVPR 2018.