Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, April 11, 2017.

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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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Tuesday, April 11, 2017

11:00 am in 345 Altgeld Hall,Tuesday, April 11, 2017

#### Traces for periodic point invariants

###### Kate Ponto (U Kentucky)

Abstract: Up to homotopy, the Lefschetz number and its refinement to the Reidemeister trace capture the essential information about fixed points of an endomorphism. These invariants can be applied to iterates of an endomorphism to describe periodic points, but in this case they provide far less complete information. I will describe an approach to refining these invariants through refinements of the associated symmetric monoidal and bicategorical traces. This gives richer invariants that also apply to endomorphisms of spaces with more structure (such as bundles).

12:00 pm in 243 Altgeld Hall,Tuesday, April 11, 2017

#### The conjugacy problem for Mod(S)

###### Mark Bell (Illinois Math)

Abstract: We will discuss a new approach for tackling the conjugacy problem for the mapping class group of a surface. This relies on recently developed tools for finding tight geodesics in the curve complex. This is joint work with Richard Webb.

2:00 pm in 347 Altgeld Hall,Tuesday, April 11, 2017

#### Concentration of measure without independence: a unified approach via the martingale method

###### Maxim Raginsky (UIUC)

Abstract: The concentration of measure phenomenon may be summarized as follows: a function of many weakly dependent random variables that is not too sensitive to any of its individual arguments will tend to take values very close to its expectation. This phenomenon is most completely understood when the arguments are mutually independent random variables, and there exist several powerful complementary methods for proving concentration inequalities, such as the martingale method, the entropy method, and the method of transportation inequalities. The setting of dependent arguments is much less well understood. This talk, based on joint work with Aryeh Kontorovich, will focus on the martingale method for deriving concentration inequalities without independence assumptions. In particular, I will show how the machinery of so-called Wasserstein matrices together with the Azuma-Hoeffding inequality can be used to recover and sharpen several known concentration results for nonproduct measures.

2:00 pm in 241 Altgeld Hall,Tuesday, April 11, 2017

#### Andrew's recent papers on integer partitions and the existence of combinatorial proofs

###### Hsin-Po Wang (UIUC)

Abstract: We will start with introducing some combinatorial notions; and then attack George E. Andrew's recent papers[1][2][3] to see if we can come up with some (simpler) combinatorial proofs. Despite the papers, we will show that under certain conditions, we can always translate an algebraic proof into a combinatorial proof. [1] G. Andrews and G. Simay. The mth Largest and mth Smallest Parts of a Partition. http://www.personal.psu.edu/gea1/pdf/307.pdf [2] G. Andrews and M. Merca. The Truncated Pentagonal Number Theorem. http://www.personal.psu.edu/gea1/pdf/288.pdf [3] G. Andrews, M Bech and N. Robbins. Partitions with Fixed Differences Between Larger and Smaller Parts. http://www.personal.psu.edu/gea1/pdf/305.pdf

3:00 pm in 241 Altgeld Hall,Tuesday, April 11, 2017

#### Equivalence of Quasiconvexity and Rank-One Convexity

###### Terry Harris   [email] (UIUC Math)

Abstract: In 1952 Morrey conjectured that quasiconvexity and rank-one convexity are not equivalent, for functions defined on m by n matrices. For two by two matrices this conjecture is still open. I will outline a proof that equivalence holds on the subspace of two by two upper-triangular matrices, which extends the result on diagonal matrices due to Müller. This is joint work with Bernd Kirchheim and Chun-Chi Lin.

3:00 pm in 243 Altgeld Hall,Tuesday, April 11, 2017

#### Enriched Hodge Structures

###### Deepam Patel (Purdue University)

Abstract: It is well known the the category of mixed Hodge structures does not give the right answer when studying cycles on possibly open/singular varieties. In this talk, we will discuss how the category of mixed Hodge structures can be `enriched’ to a category appropriate for studying algebraic cycles on infinitesimal thickenings of complex analytic varieties. This is based on joint work with Madhav Nori and Vasudevan Srinivas.

3:00 pm in 241 Altgeld Hall,Tuesday, April 11, 2017

#### On the number of Hamiltonian subsets

###### Jaehoon Kim (University of Birmingham)

Abstract: In 1981, Komlós conjectured that among all graphs with minimum degree at least $d$, the complete graph $K_{d+1}$ minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when $d$ is sufficiently large. In fact we prove a stronger result: for large $d$, any graph $G$ with average degree at least $d$ contains almost twice as many Hamiltonian subsets as $K_{d+1}$, unless $G$ is isomorphic to $K_{d+1}$ or a certain other graph which we specify. This is joint work with Hong Liu, Maryam Sharifzadeh and Katherine Staden.

4:00 pm in 245 Altgeld Hall,Tuesday, April 11, 2017

#### Harmonic analysis techniques in several complex variables

###### Loredana Lanzani (Syracuse University)

Abstract: This talk concerns the application of relatively classical tools from real harmonic analysis (namely, the $T(1)$-theorem for spaces of homogenous type) to the novel context of several complex variables. Specifically, I will present recent joint work with E. M. Stein on the extension to higher dimension of Calderón's and Coifman-McIntosh-Meyer's seminal results about the Cauchy integral for a Lipschitz planar curve (interpreted as the boundary of a Lipschitz domain $D\subset{\mathbb C}$). From the point of view of complex analysis, a fundamental feature of the 1-dimensional Cauchy kernel $H(w, z) = \tfrac{1}{2\pi i}(w-z)^{-1}dw$ is that it is holomorphic as a function of $z\in D$. In great contrast with the one-dimensional theory, in higher dimension there is no obvious holomorphic analogue of $H(w, z)$. This is because geometric obstructions arise (the Levi problem), which in dimension one are irrelevant. A good candidate kernel for the higher dimensional setting was first identified by Jean Leray in the context of a $C^\infty$-smooth, convex domain $D$: while these conditions on $D$ can be relaxed a bit, if the domain is less than $C^2$-smooth (never mind Lipschitz!) Leray's construction becomes conceptually problematic. In this talk I will present (i) the construction of the Cauchy-Leray kernel and (ii) the $L^p(bD)$-boundedness of the induced singular integral operator under the weakest currently known assumptions on the domain's regularity -- in the case of a planar domain these are akin to Lipschitz boundary, but in our higher-dimensional context the assumptions we make are in fact optimal. The proofs rely in a fundamental way on a suitably adapted version of the so-called "$T(1)$-theorem technique" from real harmonic analysis. Time permitting, I will describe applications of this work to complex function theory -- specifically, to the Szego and Bergman projections (that is, the orthogonal projections of $L^2$ onto, respectively, the Hardy and Bergman spaces of holomorphic functions).