Number Theory Seminar: Randomness of Sequences
The first part of this talk surveys different notions of randomness of one-dimensional sequences. A focal point will be on what is not known about local statistics which go beyond uniform distribution modulo one. The second part is reporting on joint work with Christopher Lutsko and Athanasios Sourmelidis. By employing Fourier analytic tools, this work provides a better understanding of how random slowly growing monomial sequences are - showing that their correlations are Poissonian. |
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Graph Theory and Combinatorics Seminar: Rational exponents for generalized extremal problems
Fix a target graph H and a family F of forbidden graphs. The generalized extremal number ex(n, H, F) is the maximum number of H-copies possible in an n-vertex graph which avoids F. Note that when H is an edge, ex(n, H, F) is the ordinary extremal number ex(n, F). After the systematic study of generalized extremal numbers was initiated by Alon and Shikhelman in 2016, the area has received substantial attention. In addition to explicit computation of ex(n, H, F) for specific choices of H, F, many questions in extremal graph theory (e.g., supersaturation, stability) naturally extend to the generalized setting.
In 2015, Bukh and Conlon applied the random algebraic method to show that, for any rational r in the interval [1,2], there is a family F such that ex(n, F) = Theta(n^r). Analogously, for a fixed target graph H and a rational number r within an appropriate interval, we may ask whether it is possible to find a forbidden family F for which ex(n, H, F) = Theta(n^r). In this talk, we present results on this question for some specific target graphs H, focusing on the case where H is a triangle, for which we show that all rational exponents in [1,3] are realizable. Joint work with Sean English and Bob Krueger.
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Canary Professorship: Candidate Presentation
Title: Chaos and turbulence in stochastic fluid mechanics: What is it and how could we study it? Abstract: In this survey-style talk I discuss the (old) idea of studying turbulence in stochastically-forced fluid equations. I will discuss efinitions of chaos, anomalous dissipation, and various other predictions by physicists that can be phrased as mathematically precise conjectures in this context. Then, I will discuss some recent work by my collaborators and I on various aspects, namely (1) a straightforward characterization of anomalous dissipation that implies the classical Kolmogorov 4/5 law for 3d NSE (joint with Michele Coti Zelati, Sam Punshon-Smith, and Franziska Weber); (2) the study of "Lagrangian chaos" and exponential mixing of scalars and how it leads to a proof of anomalous dissipation and of the power spectrum predicted by Batchelor in 1959 for the simpler problem of Batchelor-regime passive scalar turbulence (joint with Alex Blumenthal and Sam Punshon-Smith); (3) the more recent proof of "Eulerian chaos" for Galerkin truncations of the Navier-Stokes equations (joint with Alex Blumenthal and Sam Punshon-Smith). For other details, please see email sent on behalf of Jared Bronski dated April 27 (sent by P. Currid) |
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Graduate Commutative Algebra & Algebraic Geometry Seminar: Condensation and the Algebraic Geometry of Topological Algebras
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Department Colloquium: On sphere packings and the hard sphere model
I will present results on high-dimensional sphere packings and spherical codes and new bounds for the absence of phase transition at low densities in the hard sphere model. The techniques used take the perspective of algorithms and optimization and can be applied to problems in extremal and enumerative combinatorics as well. Lunch is 11:50am at Mandarin Woks, if you wish to attend, let Jozsef Balogh know before 11am, Dinner plans to be discussed after the lecture.
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IRisk Lab Spring 2022 Final Presentations
You are invited to attend the IRisk Lab's final presentations event during Reading Day, May 5. The event will be held in 1090 Lincoln Hall. The IRisk Lab is an industry-academic collaboration hub that facilitates integration of discovery-based learning experiences for students, and showcases state-of-the-art research in all areas of risk analysis and advanced analytics. Whether your are presenting or just want to learn more about the IRisk Lab, come celebrate this semester's research projects. Event Agenda
The organizers highly recommend that you take a COVID test and know the result before joining this event. If you have any questions about the event, please get in touch with professor Zhiyu (Frank) Quan.
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IGL Open House
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Conversation Series: Felix Leditzky
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Algebra, Geometry & Combinatorics: A Murnaghan-Nakayama rule for Grothendieck polynomials of Grassmannian type
The Grothendieck polynomials appearing in the K-theory of Grassmannians are analogs of Schur polynomials. We establish a version of the Murnaghan-Nakayama rule for Grothendieck polynomials of the Grassmannian type. This rule allows us to express the product of a Grothendieck polynomial with a power sum symmetric polynomial into a linear combination of other Grothendieck polynomials. |
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Spring 2022 Faculty meeting
The Spring regular meeting of the Department. Please pre-register with Becky Bishop to receive zoom link. rjbishop@illinois.edu |
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Graduate Student Homotopy Theory Seminar: Calculus for Algebraic Topologists
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AWM Ice Cream Social
Take this time to hang out with others in our department, celebrate the end of the school year, and congratulate our graduating class! You may also bring your family or a guest! |
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Symplectic and Poisson geometry seminar: Toric Hamiltonian actions in a Poisson context
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Graduate Commutative Algebra and Algebraic Geometry Seminar: Introduction to Geometric Invariant Theory II
This talk is the second of a series of talks on this topic. In the first talk, I introduced the quotient scheme and linearization of invertible sheaves. In the second talk, I will use them to introduce the stability of points under a group action. Then I will show how the 1-parameter subgroups determine the properties of our group action and stableness. Cookies will be provided as usual. |
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Algebraic Geometry Seminar: Virasoro constraints for moduli of sheaves
Descendent classes on moduli spaces of sheaves are defined via the Chern characters of the universal sheaf. I will present several conjectures and results concerning Virasoro constraints for integrals of the descendent classes. |
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Combinatorics Colloquium: The upper tail for triangles in random graphs
Abstract: Let $X$ denote the number of triangles in the random graph $G_{n,p}$. The problem of determining the asymptotics of the logarithimic upper tail probability of $X$, that is, $\log \Pr(X > (1+\delta)\mathbb{E}[X])$, for every fixed positive $\delta$ has attracted considerable attention of both the combinatorics and the probability communities. We shall present an elementary solution to this problem, obtained recently in a joint work with Matan Harel and Frank Mousset. The crux of our approach is a simple probabilistic argument, inspired by the work of Janson, Oleszkiewicz and Ruci\’nski, that reduces the estimation of this upper tail probability to a counting problem. |
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Elliptic Cohomology and Conformal Field Theory
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