Department of

Mathematics


Seminar Calendar
for Special Colloquium events the year of Tuesday, September 29, 2020.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, January 21, 2020

4:00 pm in 245 Altgeld Hall,Tuesday, January 21, 2020

The Helly geometry of some Garside and Artin groups

Jingyin Huang   [email] (Ohio State University)

Abstract: Artin groups emerged from the study of braid groups and complex hyperplane arrangements, and they are connected to Coxeter groups, 3-manifold groups, buildings and many others. Artin groups have very simple presentation, yet rather mysterious geometry with many basic questions widely open. I will present a way of understanding certain Artin groups and Garside groups by building geometric models on which they act. These geometric models are non-positively curved in an appropriate sense, and such curvature structure yields several new results on the algorithmic, topological and geometric aspects of these groups. No previous knowledge on Artin groups or Garside groups is required. This is joint work with D. Osajda.

Wednesday, January 22, 2020

4:00 pm in 245 Altgeld Hall,Wednesday, January 22, 2020

Statistical reduced models and rigorous analysis for uncertainty quantification of turbulent dynamical systems

Di Qi   [email] (Courant Institute of Mathematical Sciences)

Abstract: The capability of using imperfect statistical reduced-order models to capture crucial statistics in turbulent flows is investigated. Much simpler and more tractable block-diagonal models are proposed to approximate the complex and high-dimensional turbulent flow equations. A rigorous statistical bound for the total statistical uncertainty is derived based on a statistical energy conservation principle. The systematic framework of correcting model errors is introduced using statistical response and empirical information theory, and optimal model parameters under this unbiased information measure are achieved in a training phase before the prediction. It is demonstrated that crucial principal statistical quantities in the most important large scales can be captured efficiently with accuracy using the reduced-order model in various dynamical regimes with distinct statistical structures.

Thursday, January 23, 2020

4:00 pm in 245 Altgeld Hall,Thursday, January 23, 2020

Semistable reduction in characteristic 0

Gaku Liu (Max Planck Institute for Mathematics in the Sciences)

Abstract: Semistable reduction is a relative generalization of the classical problem of resolution of singularities of varieties; the goal is, given a surjective morphism $f : X \to B$ of varieties in characteristic 0, to change $f$ so that it is "as nice as possible". The problem goes back to at least Kempf, Knudsen, Mumford, and Saint-Donat (1973), who proved a strongest possible version when $B$ is a curve. The key ingredient in the proof is the following combinatorial result: Given any $d$-dimensional polytope $P$ with vertices in $\mathbb{Z}^d$, there is a dilation of $P$ which can be triangulated into simplices each with vertices in $\mathbb{Z}^d$ and volume $1/d!$. In 2000, Abramovich and Karu proved, for any base $B$, that $f$ can be made into a weakly semistable morphism $f' : X' \to B'$. They conjectured further that $f'$ can be made semistable, which amounts to making $X'$ smooth. They explained why this is the best resolution of $f$ one might hope for. In this talk I will outline a proof of this conjecture. They key ingredient is a relative generalization of the above combinatorial result of KKMS. I will also discuss some other consequences in combinatorics of our constructions. This is joint work with Karim Adiprasito and Michael Temkin.

Friday, January 24, 2020

4:00 pm in 245 Altgeld Hall,Friday, January 24, 2020

Logical and geometric tameness over the real line.

Erik Walsberg (University of California, Irvine)

Abstract: There are now a number of important and well-understood examples of logically tame first order structures over the real numbers such as the ordered field of real numbers and the ordered field of real numbers equipped with the exponential function. In these examples subsets of Euclidean space which are (first order) definable are geometrically very well behaved. Recent research had yielded general theorems in this direction. I will discuss one result in this subject: A first order structure on the real line which expands the ordered vector space of real numbers and defines a closed set X such that the topological dimension of X is strictly less then the Hausdorff dimension of X defines every bounded Borel set. Informally: An expansion of the ordered real vector space which defines a fractal is maximally wild from the viewpoint of logic. Joint with Fornasiero and Hieronymi.

Monday, January 27, 2020

4:00 pm in 245 Altgeld Hall,Monday, January 27, 2020

A new approach to bounding L-functions

Jesse Thorner   [email] (University of Florida)

Abstract: Analytic number theory began with studying the distribution of prime numbers, but it has evolved and grown into a rich subject lying at the intersection of analysis, algebra, combinatorics, and representation theory. Part of its allure lies in its abundance of problems which are tantalizingly easy to state which quickly lead to deep mathematics, much of which revolves around the study of L-functions. These extensions of the elusive Riemann zeta function $\zeta(s)$ are generating functions with multiplicative structure arising from either arithmetic-geometric objects (like number fields or elliptic curves) or representation-theoretic objects (automorphic forms). Many equidistribution problems in number theory rely on one's ability to accurately bound the size of L-functions; optimal bounds arise from the (unproven!) Riemann Hypothesis for $\zeta(s)$ and its extensions to other L-functions. I will discuss some motivating problems along with recent work (joint with Kannan Soundararajan) which produces new bounds for L-functions by proving a suitable "statistical approximation" to the (extended) Riemann Hypothesis.

Tuesday, March 24, 2020

4:00 pm in Zoom Meeting https://illinois.zoom.us/j/314999665,Tuesday, March 24, 2020

Error Thresholds for Arbitrary Pauli Noise

Felix Leditzky (Institute for Quantum Computing, University of Waterloo; and Perimeter Institute)

Abstract: The error threshold of a one-parameter family of quantum channels is defined as the largest noise level such that the quantum capacity of the channel remains positive, which in turn guarantees existence of quantum error correction codes. Here, we study the error thresholds of channels arising from probabilistic Pauli errors. To this end, we determine lower bounds on the quantum capacity of these channels by evaluating the coherent information of so-called graph states affected by Pauli noise. Graph states are a subclass of stabilizer states and uniquely defined by a simple undirected graph. The main tools for our results are a) a simplified analysis of the channel action on graph states using the language of homomorphic group actions, and b) using strong generating systems for permutation groups to implement the algorithm in a computationally efficient manner. We provide an extensive analysis of known good codes such as repetition codes and cat codes in the whole Pauli channel simplex. Furthermore, we identify a novel family of quantum codes based on tree graphs with desirable error correction properties. This meeting will be a Zoom meeting! To participate, go to https://illinois.zoom.us/j/314999665 (please watch out for changes)

Tuesday, March 31, 2020

4:00 pm in Zoom Meeting,Tuesday, March 31, 2020

Quantum Resources What Are They and How Much Are They Worth?

Jamie Sikora (Perimeter Institute, Waterloo, ON)

Abstract: In this talk, I will discuss several natural quantum problems and, in particular, how the problems change as the quantum resources change. I will show how to take an economics perspective to assign a shadow price to each quantum resource. To do this, I will use optimization theory and show that shadow prices are often given for free if you know where to look for them. This will be a Zoom meeting (https://illinois.zoom.us/j/653578028). Faculty host: Marius Junge. A recording of the talk may be accessed at https://mediaspace.illinois.edu/media/t/0_fv18gz5b

Thursday, April 16, 2020

4:00 pm in Virtual Altgeld Hall,Thursday, April 16, 2020

Harnessing quantum entanglement

Laura Mancinska (University of Copenhagen)

Abstract: Entanglement is one of the key features of quantum mechanics. It lies at the heart of most cryptographic applications of quantum technologies and is necessary for computational speed-ups. However, given a specific information processing task, it is challenging to find the best way to harness entanglement and we are yet to uncover the full range of its potential applications. We will see that nonlocal games provide a rigorous framework for studying quantum entanglement and the advantage that it can offer. We will take a closer look at the question of how much entanglement can be needed to play a nonlocal game optimally. We will then use games requiring large amounts of entanglement to build protocols for certifying proper functioning of untrusted quantum devices. https://illinois.zoom.us/j/701331281 The meeting will be locked 4.15, contact me, if you need to get in.

Thursday, April 30, 2020

4:00 pm in Zoom Meeting (see abstract),Thursday, April 30, 2020

Quantum information, quantum groups, and counting homomorphisms from planar graphs

David Roberson   [email] (Technical University of Denmark)

Abstract: We introduce a game in which two cooperating parties attempt to convince a referee that two graphs G and H are isomorphic. Classical strategies that win this game correspond to actual isomorphisms of G and H. However, if the two parties are given access to certain quantum mechanical resources (local measurements on a shared entangled state), then they can sometimes win this game even when G and H are not isomorphic. This operationally defined notion of quantum isomorphism turns out to have an elegant algebraic description in terms of magic unitaries, a notion from the theory of quantum groups. Moreover, quantum isomorphism can be completely reformulated in terms of the quantum automorphism group of a graph. We will discuss how these connections allow us to prove that graphs G and H are quantum isomorphic if and only if they admit the same number of homomorphisms from any *planar* graph. This can be viewed as a quantum analog of a classical result of Lovasz from over 50 years ago: graphs G and H are isomorphic if and only if they admit the same number homomorphisms from any graph. Though the connection to quantum groups is crucial, the details of the proof of this result are mostly combinatorial. Please email Jane Bergman (jbergman@illinois.edu) or Jozsef Balogh (jobal@illinois.edu) for Zoom meeting link.