Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, April 30, 2020.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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1  2  3  4  5  6  7             1  2  3  4                   1  2
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Thursday, April 30, 2020

11:00 am in 241 Altgeld Hall,Thursday, April 30, 2020

#### Locally Split Galois Representations and Hilbert Modular Forms of Partial Weight One

###### Eric Stubley (University of Chicago)

Abstract: The Galois representation attached to a p-ordinary eigenform is upper triangular when restricted to a decomposition group at p. A natural question to ask is under what conditions this upper triangular decomposition splits as a direct sum. Ghate and Vatsal have shown that for Galois representations coming from families of p-ordinary eigenforms, the restriction to a decomposition group at p is split if and only if the family has complex multiplication; in their proof, the weight one members of the family play a key role. I'll talk about work in progress which aims to answer similar questions in the case of Galois representations for a totally real field which are split at only some of the primes above p. In this work Hilbert modular forms of partial weight one play a central role; I'll discuss what is known about them and to what extent the techniques of Ghate and Vatsal can be adapted to this situation.

3:00 pm in 347 Altgeld Hall,Thursday, April 30, 2020

#### **Rescheduled due to COVID-19 campus-shutdown**

###### Wai Ling Yee   [email] (University of Windsor)

Abstract: To Be Announced

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.