Functions of many variables may be very complex. And optimizing them can be exceedingly difficult or slow. Think for instance of the following very simple question: How hard is it to find the minimum of a cubic polynomial of many variables? If you chose the cubic polynomial randomly, it is a very hard problem. I will survey recent work describing this phenomenon and its consequences, first from a topology point of view, and introduce an important topological phase transition. The mathematical tool underlying these complexity questions is given by Random Matrix Theory, through the classical tool of random geometry, i.e. the Kac-Rice formulae. I will illustrate this phenomenon in the case of random functions on the high-dimensional sphere. These random functions happen to be the energy landscapes of important models of statistical physics of disordered media, i.e spherical spin glasses. These random landscapes are also central in an important question of data-science, i.e the analysis of spiked tensors. The questions is here to detect and estimate a signal in a noisy tensor. I will discuss recent progress on the topological complexity, and on the dynamic exploration properties of such random landscapes.
The relevant work is joint with mathematical colleagues (Auffinger, Cerny, Jagannath, Montanari, Nica, Subag, Zeitouni) or physicists (Biroli, Cammarota, Fyodorov, Khorunzenko and Ros) and Computer Scientists (Yann Le Cun and his team at Facebook).
*Please note that tea and coffee reception will start at 4:00 PM. All are welcome!
NYU Shanghai STEM seminar series is a weekly seminar series on every Wednesday. Please see below schedule of STEM seminar series in 2017 Fall Semester.
- October 11: Gerard Ben Arous, Professor of Mathematics
- October 18: Gus Xia, Assistant Professor of Computer Science
- October 25: Brad Weslake, Associate Professor of Philosophy
- November 1: William Glover, Assistant Professor of Chemistry
- November 8: Chuck Newman, Professor of Mathematics & Vladas Sidorvicius, Professor of Mathematics
- November 15: Tim Byrnes, Assistant Professor of Physics
- November 29: Xing Tian, Assistant Professor of Neural and Cognitive Sciences & Zheng Zhang, Professor of Computer Science
- December 6: Yves Le Jan, Visiting Professor of Mathematics
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