Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, March 30, 2021.

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events for the
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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, March 30, 2021

2:00 pm in Zoom Meeting (email daesungk@illinois.edu for info),Tuesday, March 30, 2021

#### Exponential concentration of overlap, free energy, and replica symmetry breaking for inhomogeneous Sherrington-Kirkpatrick Spin glass

###### Qiang Wu (UIUC)

Abstract: Multi-species Sherrington-Kirkpatrick Spin glass model, as an inhomogeneous generalization of classical SK model, was first introduced by Barra etal in 2015, later Panchenko set up the Parisi formula to compute limiting free energy at all temperatures. However, all those results can only hold when the disorder variance matrix is positive semi-definite. For the indefinite MSK model, nearly nothing rigorous is known, physicists conjectured that this model has some intrinsic difference with the positive definite case. In this talk, we will discuss some fluctuation results for the general MSK model in replica symmetric regime. First, A unified argument for the exponential concentration of overlap will be presented, and this concentration result further enables one to prove a CLT of free energy. Besides that, we also introduce a new species-wise cavity approach to study the fluctuation of overlap vectors, and this approach does not require positive definite assumption. The fluctuation results also suggest the phase boundary (known as AT line) of replica symmetry and replica symmetry breaking for general MSK model, we will conclude with an explicit form of conjectured AT equation.

2:00 pm in Zoom,Tuesday, March 30, 2021

#### Lower Bounds for Generalized Ramsey Numbers Through Color Energy Graphs

###### Sean English (UIUC)

Abstract: Let the generalized Ramsey number $f(n,p,q)$ denote the least integer $c$ such that $K_n$ can be edge-colored with $c$ colors such that every set of $p$ vertices spans a clique with at least $q$ colors. in this talk, we will discuss the method of color energy graphs, which borrows some ideas from the concept of additive energy used in additive combinatorics to find new lower bounds on generalized Ramsey numbers, and the connections between generalized Ramsey numbers and other interesting problems in extremal combinatorics.

Joint work with Jozsef Balogh, Emily Heath and Robert A. Krueger.