Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration

We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type $\operatorname{SL}_n$ and $\operatorname{PGL}_n$. More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises p-adic integration relative to the fibres, and interprets canonical gerbes present on these moduli spaces as characters on the Hitchin fibres using Tate duality. Furthermore we prove for $d$ coprime to $n$, that the number of rank $n$ Higgs bundles of degree $d$ over a fixed curve defined over a finite field, is independent of $d$. This proves a conjecture by Mozgovoy--Schiffman in the coprime case.Comment: Various part of the article have been revise

Motivic and pp-adic Localization Phenomena

In this thesis we compute motivic classes of hypertoric varieties, Nakajima quiver varieties and open de Rham spaces in a certain localization of the Grothendieck ring of varieties. Furthermore we study the $p$-adic pushforward of the Haar measure under a hypertoric moment map $\mu$. This leads to an explicit formula for the Igusa zeta function $I_\mu(s)$ of $\mu$, and in particular to a small set of candidate poles for $I_\mu(s)$. We also study various properties of the residue at the largest pole of $I_\mu(s)$. Finally, if $\mu$ is constructed out of a quiver $\Gamma$ we give a conjectural description of this residue in terms of indecomposable representations of $\Gamma$ over finite depth rings. The connections between these different results is the method of proof. At the heart of each theorem lies a motivic or $p$-adic volume computation, which is only possible due to some surprising cancellations. These cancellations are reminiscent of a result in classical symplectic geometry by Duistermaat and Heckman on the localization of the Liouville measure, hence the title of the thesis.Comment: Thesi

Arithmetic and metric aspects of open de Rham spaces

In this paper we determine the motivic class---in particular, the weight polynomial and conjecturally the Poincar\'e polynomial---of the open de Rham space, defined and studied by Boalch, of certain moduli of irregular meromorphic connections on the trivial bundle on $\mathbb{P}^1$. The computation is by motivic Fourier transform. We show that the result satisfies the purity conjecture, that is, it agrees with the pure part of the conjectured mixed Hodge polynomial of the corresponding wild character variety. We also identify the open de Rham spaces with quiver varieties with multiplicities of Yamakawa and Geiss--Leclerc--Schr\"oer. We finish with constructing natural complete hyperk\"ahler metrics on them, which in the $4$-dimensional cases are expected to be of type ALF.Comment: 69 page

Geometric stabilisation via p-adic integration

In this article we give a new proof of Ng\^o's Geometric Stabilisation Theorem, which implies the Fundamental Lemma. This is a statement which relates the cohomology of Hitchin fibres for a quasi-split reductive group scheme $G$ to the cohomology of Hitchin fibres for the endoscopy groups $H_{\kappa}$. Our proof avoids the Decomposition and Support Theorem, instead the argument is based on results for $p$-adic integration on coarse moduli spaces of Deligne-Mumford stacks. Along the way we establish a description of the inertia stack of the (anisotropic) moduli stack of $G$-Higgs bundles in terms of endoscopic data, and extend duality for generic Hitchin fibres of Langlands dual group schemes to the quasi-split case.Comment: 49 page

Cosmesis and body image after single-port laparoscopic or conventional laparoscopic cholecystectomy: a multicenter double blinded randomised controlled trial (SPOCC-trial)

BACKGROUND: Emerging attempts have been made to reduce operative trauma and improve cosmetic results of laparoscopic cholecystectomy. There is a trend towards minimizing the number of incisions such as natural transluminal endoscopic surgery (NOTES) and single-port laparoscopic cholecystectomy (SPLC). Many retrospective case series propose excellent cosmesis and reduced pain in SPLC. As the latter has been confirmed in a randomized controlled trial, patient's satisfaction on cosmesis is still controversially debated. METHODS/DESIGN: The SPOCC trial is a prospective, multi-center, double blinded, randomized controlled study comparing SPLC with 4-port conventional laparoscopic cholecystectomy (4PLC) in elective surgery. The hypothesis and primary objective is that patients undergoing SPLC will have a better outcome in cosmesis and body image 12 weeks after surgery. This primary endpoint is assessed using a validated 8-item multiple choice type questionnaire on cosmesis and body image. The secondary endpoint has three entities: the quality of life 12 weeks after surgery assessed by the validated Short-Form-36 Health Survey questionnaire, postoperative pain assessed by a visual analogue scale and the use of analgesics. Operative time, surgeon's experience with SPLC and 4PLC, use of additional ports, conversion to 4PLC or open cholecystectomy, length of stay, costs, time of work as well as intra- and postoperative complications are further aspects of the secondary endpoint. Patients are randomly assigned either to SPLC or to 4PLC. Patients as well as treating physicians, nurses and assessors are blinded until the 7th postoperative day. Sample size calculation performed by estimating a difference of cosmesis of 20% (alpha = 0.05 and beta = 0.90, drop out rate of 10%) resulted in a number of 55 randomized patients per arm. DISCUSSION: The SPOCC-trial is a prospective, multi-center, double-blind, randomized controlled study to assess cosmesis and body image after SPLC

Cosmesis and Body Image in Patients Undergoing Single-port Versus Conventional Laparoscopic Cholecystectomy: A Multicenter Double-blinded Randomized Controlled Trial (SPOCC-trial)

OBJECTIVE: To evaluate cosmesis, body image, pain, and quality of life (QoL) after single-port laparoscopic cholecystectomy (SPLC) versus conventional 4-port laparoscopic cholecystectomy (4PLC). BACKGROUND: The impact of SPLC on improving cosmesis, body image, pain, and QoL has not been evaluated in double-blinded randomized controlled trials (RCT). This approach therefore remains controversial. METHODS: Between October 2011 and February 2014, 110 patients from 2 centers were randomly assigned to SPLC (n = 55) or 4PLC (n = 55). Primary endpoints were a validated cosmesis (3-24 points) and body image (5-20 points) score after 3 and 12 months. Secondary endpoints included operative duration, postoperative pain, complications, QoL, and length of hospital stay. Patients, physicians, and nurses were blinded until the seventh postoperative day. RESULTS: Demographics were equally distributed between both groups (mean age: 46 years, SD: 14, 62 females, 34 males). The SPLC-group showed superior mean cosmesis and body image compared with the 4PLC-group at 12-weeks (21 vs 16, P < 0.001 and 5 vs 6, P = 0.013, respectively) and at 1-year (24 vs 16, P < 0.001 and 5 vs 6, P < 0.017, respectively). Operation duration was longer in the SPLC-group (mean 101 vs 90 minutes, p = 0.031). Although postoperative pain was less in the SPLC-group (mean VAS 1 vs 2, p = 0.005), there were no differences in complications, and length of hospital-stay. CONCLUSIONS: This is the first multicenter double-blinded RCT reporting superior short- and long-term cosmetic and body image, postoperative pain, and QoL in SPLC compared with 4PLC. Although cost-effectiveness is still a subject of ongoing debate, SPLC should be offered to patients undergoing surgery for benign gallbladder disease

Prehabilitation in patients undergoing colorectal surgery fails to confer reduction in overall morbidity : results of a single-center, blinded, randomized controlled trial

Background: Currently, there are solely weak recommendations in the enhanced recovery after surgery (ERAS) protocol regarding the role of preoperative physical activity and prehabilitation in patients undergoing colorectal surgery. Studies in heterogenous groups showed contradictory results regarding the impact of prehabilitation on the reduction of postoperative complications. The aim of this study was to assess the impact of prehabilitation on postoperative complications in patients undergoing colorectal surgery within an ERAS protocol. Methods. Between July 2016 and June 2019, a single-center, blinded, randomized controlled trial designed to test whether physiotherapeutic prehabilitation vs. normal physical activities prior to colorectal surgery may decrease morbidity within a stringent ERAS protocol was carried out. The primary endpoint was postoperative complications assessed by Comprehensive Complications Index (CCI®). Primary and secondary endpoints for both groups were analyzed and compared. Results: A total of 107 patients (54 in the prehabilitation enhanced recovery after colorectal surgery [pERACS] group and 53 in the control group) were included in the study and randomized. Dropout rate was 4.5% (n = 5). Baseline characteristics were comparable between the pERACS and control groups. The percentage of colorectal adenocarcinoma was low in both groups (pERACS 32% vs. control 23%, p = 0.384). Almost all patients underwent minimally invasive surgery in both groups (96% vs. 98%, p = 1.000). There was no between-group difference in the primary outcome, as the mean CCI at 30-day postoperative in the pERACS group was 18 (SD 0–43) compared to 15 (SD 0–49) in the control group (p = 0.059). Secondary outcome as complications assessed according to Clavien-Dindo, length of hospital stay, reoperation rate, and mortality showed no difference between both groups. Conclusions: Routine physiotherapeutic prehabilitation has no additional benefit for patients undergoing colorectal surgery within an ERAS protocol. Trial registration: ClinicalTrial.gov: ID: NCT02746731; Institution Ethical Board Approval: KEK-ZH Nr. 2016–00,229

Memory performance-related dynamic brain connectivity indicates pathological burden and genetic risk for Alzheimer's disease

Background The incidence of Alzheimer’s disease (AD) strongly relates to advanced age and progressive deposition of cerebral amyloid-beta (Aβ), hyperphosphorylated tau, and iron. The purpose of this study was to investigate the relationship between cerebral dynamic functional connectivity and variability of long-term cognitive performance in healthy, elderly subjects, allowing for local pathology and genetic risk. Methods Thirty seven participants (mean (SD) age 74 (6.0) years, Mini-Mental State Examination 29.0 (1.2)) were dichotomized based on repeated neuropsychological test performance within 2 years. Cerebral Aβ was measured by 11C Pittsburgh Compound-B positron emission tomography, and iron by quantitative susceptibility mapping magnetic resonance imaging (MRI) at an ultra-high field strength of 7 Tesla (7T). Dynamic functional connectivity patterns were investigated by resting-state functional MRI at 7T and tested for interactive effects with genetic AD risk (apolipoprotein E (ApoE)-ε4 carrier status). Results A relationship between low episodic memory and a lower expression of anterior-posterior connectivity was seen (F(9,27) = 3.23, p < 0.008), moderated by ApoE-ε4 (F(9,27) = 2.22, p < 0.005). Inherent node-strength was related to local iron (F(5,30) = 13.2; p < 0.022). Conclusion Our data indicate that altered dynamic anterior-posterior brain connectivity is a characteristic of low memory performance in the subclinical range and genetic risk for AD in the elderly. As the observed altered brain network properties are associated with increased local iron, our findings may reflect secondary neuronal changes due to pathologic processes including oxidative stress