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2:00pm to 3:00pm - 510R Rowland Hall - Combinatorics and Probability Jie Han - (Beijing Institute of Technology) Robustness of graph theory theorems Dirac's theorem says that any n-vertex graph with minimum degree at least n/2 contains a Hamiltonian cycle. A robust version of Dirac's theorem was obtained by Krivelevich, Lee and Sudakov: for such graphs, if we take a random subgraph where every edge is included with probability Clog(n)/n, for some large fixed constant C, then whp the resulting graph is still Hamiltonian. We will discuss some recent results along this line. |
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9:00am to 10:00am - Zoom - Inverse Problems Gabriel Paternain - (University of Cambridge) Geometric inverse problems in 2D: a transport twistor perspective |
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10:00am to 11:00am - RH 306 - Number Theory Anurag Sahay - (University of Rochester) Moments of the Hurwitz zeta function on the critical line The Hurwitz zeta function is a shifted integer analogue of the Riemann zeta function, for shift parameters $0<\alpha\leqslant 1$. We consider the integral moments of the Hurwitz zeta function on the critical line $\Re(s)=\tfrac12$. We will focus on rational shift parameters. In this case, the Hurwitz zeta function decomposes as a linear combination of Dirichlet $L$-functions, which leads us into investigating moments of products of $L$-functions. Using heuristics from random matrix theory, we conjecture an asymptotic of the same form as the moments of the Riemann zeta function. If time permits, we will discuss the case of irrational shift parameters, which will include some joint work with Winston Heap and Trevor Wooley and some ongoing work with Heap. |
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4:00pm - Natural Sciences II, Room 1201 - Colloquium Bryna Kra - (Northwestern University) Patterns in large sets of integers: Finite to infinite ***Distinguished Visitor Colloquium*** Resolving a conjecture of Erdos and Turan from the 1930's, in the 1970's Szemerédi showed that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Soon thereafter, Furstenberg used ergodic theory to give a new proof of this result, leading to the development of combinatorial ergodic theory. These tools have led to uncovering new patterns that occur in any sufficiently large set of integers, but until recently all such patterns have been finite. Based on joint work with Joel Moreira, Florian Richter, and Donald Robertson, we discuss recent developments for infinite patterns, including the resolution of a conjecture of Erdos. |
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3:00pm - RH 306 - Dynamical Systems Bryna Kra - (Northwestern University) Infinite patterns in large sets of integers: Dynamical approaches ***Special Dynamical Systems and Ergodic Theory Seminar*** Furstenberg's proof of Szemeredi's theorem introduced the Correspondence Principle, a general technique for translating a combinatorial problem into a dynamical one. While the original formulation suffices for certain patterns, including arithmetic progressions and some infinite configurations, higher order generalizations have required refinements of these tools. We discuss the new techniques introduced in joint work with Moreira, Richter, and Robertson that are used to show the existence of infinite patterns in large sets of integers. |
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3:00pm - RH 306 - Colloquium Bryna Kra - (Northwestern University) Infinite patterns in large sets of integers: Dynamical approaches ***Special Dynamical Systems and Ergodic Theory Seminar*** Furstenberg's proof of Szemeredi's theorem introduced the Correspondence Principle, a general technique for translating a combinatorial problem into a dynamical one. While the original formulation suffices for certain patterns, including arithmetic progressions and some infinite configurations, higher order generalizations have required refinements of these tools. We discuss the new techniques introduced in joint work with Moreira, Richter, and Robertson that are used to show the existence of infinite patterns in large sets of integers. |